Logical Repre s entations
for Automate d Re asoning
ab out Spatial Relationship s
Brandon Bennett
Submitte d in accordance with the requirements for the degree of
Do ctor of Philo sophy
The Univers ity of Lee ds
Scho ol of Computer Studie s
Octob er
The candidate con rms that the work submitte d i s hi s own and that appropr iate
cre dit has b een given where reference has b een made to the work of others
Ab stract
Thi s the s i s inve stigate s logical repre s entations for de scr ibing and reasoning ab out spatial s ituations
Previously prop os e d theor ie s of spatial regions are inve stigate d in some detail e sp ecially the
st order theory of Randell Cui and Cohn The di culty of achieving e ective automate d
reasoning with the s e systems i s obs erve d
A new approach i s pre s ente d bas e d on enco ding spatial relations in formulae of order pro
p os itional logics It i s prove d that entailment which i s valid according to the standard s emantics
for the s e logics i s also valid with re sp ect to the spatial interpretation Cons equently well known
mechani sms for prop os itional reasoning can b e applie d to spatial reasoning Sp eci c enco dings
of top ological relations into b oth the mo dal logic S and the intuitioni stic prop os itional calculus
are given The complexity of reasoning us ing the intuitioni stic repre s entation i s examine d and
a pro ce dure i s pre s ente d which i s shown to b e of O n complexity in the numb er of relations
involve d
In order to make thi s kind of repre s entation su ciently expre ss ive the concepts of model con
straint and entailment constraint are intro duce d By means of thi s di stinction a order formula
may b e us e d e ither to ass ert or to deny that a certain spatial constraint holds of some s ituation It
i s shown how the pro of theory of a order logical language can b e extende d by a s imple meta level
generali sation to accommo date a repre s entation involving the s e two typ e s of formula
A numb er of other topics are dealt with a deci s ion pro ce dure bas e d on quanti er elimination
i s given for a large class of formulae within a st order top ological language reasoning mechani sms
bas e d on the composition of spatial relations are studie d the non top ological prop erty of convexity
i s examine d b oth f rom the p oint of view of its st order character i sation and its incorp oration into
a order spatial logic It i s sugge ste d that order repre s entations could b e employe d in a s imilar
manner to enco de other spatial concepts
There is no branch of mathematics however abstract that wil l not eventual ly be applied
to the phenomena of the real world
Lobachevsky quote d in the Amer ican Mathematical Monthly Feb
Acknowle dgements
Work on thi s the s i s was partially supp orte d by the CEC under the Bas ic Re s earch Action
MEDLAR Pro ject the SERC under grants GR H and GR G and the EPSRC
under grant GR K
I would like to give sp ecial thanks to Profe ssor Tony Cohn for constant encouragement dur ing
my work on thi s the s i s and for providing me with many opp ortunitie s to meet other re s earchers in
the eld
Thanks also to my mother Gaymer ing Bennett for pro of reading the manuscr ipt
Finally I would like to thank my wife Deb orah for b ear ing with me dur ing my long career as a
student
Contents
Intro duction
The Domain of Spatial Reasoning
Spatial Concepts and Information
Geometry of Points and Line s and its Pr imitive Concepts
Top ology
Shap e
Convexity and Containment
Pos ition and Or ientation
A Br ief Hi story of Spatial Reasoning
Or igins
Development
New Foundations
Conceptual and Formal Frameworks
Bas ic Elements in a Spatial Theory
Mo de s of Formali sm
Logical Theor ie s of Space and Spatial Logics
Spatial Reasoning in Computer Science
Commons ens e Knowle dge
Reasoning ab out Phys ical Systems
Spatial Reasoning in Rob otics
Spatial Reasoning and Computer Vi s ion
Temp oral Reasoning
Automating Spatial Reasoning
Complexity of Mathematical Theor ie s
Tractability and Decidability
The Content of thi s The s i s
Assume d Background and Notations Employe d
Axiomatic Theor ie s of Spatial Regions
Point Set Top ology
The Or igins of Region Bas e d Theor ie s
CONTENTS
Le sniewski s Mereology
Other Mereological Systems
Tarski s Geometry of Solids
Clarke s Theory
Fus ions and Quas i Bo olean Op erators
Top ological Functions
Points
The Region Connection Calulus RCC
Functional Extens ion of the Bas ic Theory
The Sorte d Logic LLAMA
Sorts in the RCC Theory
Two Additional Axioms
Further Development of RCC
Other Relevant Work on Region Bas e d Theor ie s
de Laguna s Theory
Grzegorczyk s Undecidability Re sults
Some Recent Re s earch in the Field
Analys i s of the RCC Theory
RCC in Relation to thi s The s i s
Identity and Extens ionality
The Quas i Bo olean Functions
The Status of the Function De nitions
RCC without Functions or Sorts
The Complement Function
Relation to Ortho dox Bo olean Algebras
A Single Generator for Bo olean Functions
Intro duction of a Null Region
Atoms and the NTPP Axiom
Mo dels of the RCC Theory
Graph Mo dels of the C relation
Mo dels in Point Set Top ology
Interpreting RCC in Point Set Top ology
The Bo olean Algebra of Regular Op en Point Sets
A Dual Top ological Interpretation
Completene ss and Categor icity
A Revi s e d Vers ion of the RCC Theory
CONTENTS
A Order Repre s entation
Spatial Interpretation of Order Calculi
Set Semantics for the Class ical Calculus
An Entailment Corre sp ondence
Reasoning with Non Universal Equations
Repre s enting Top ological Relationships in C
Mo del and Entailment Constraints
Cons i stency of C Situation De scr iptions
Repre s enting RCC Relations
Non Null Constraints
Repre s entations of the RCC Relations
Reasoning with C
Determining Entailments
Complexity of the Reasoning Algor ithm
A Mo dal Repre s entation
The Spatial Interpretation of Mo dal Logics
Overview of the Approach Taken
Semantics for Order Mo dal Logics
Mo dal Logics
The Logic S
Kr ipke Semantics
Mo dal Algebras
Algebraic Mo dels
Power Set Algebras
Mapping Between Algebraic and Logical Expre ss ions
Entailment among Mo dal Algebraic Equations
Relating S Mo dal Algebraic Entailment to De ducibility
Top ological Closure Algebras
Closure and Inter ior Algebras
RCC Relations Repre s entable in Inter ior Algebra
Us ing Inequalitie s to Extend Expre ss ive Power
Enco ding Closure Algebraic Constraints in S
RCC Relations Repre s entable in S
Extende d Mo dal Logics L
Convexity of Mo dal Algebras
A Corre sp ondence Theorem for S
Repre s enting RCC Relations in S
Regular ity and Bo olean Combination of Regions
Eliminating Entailment Constraints
CONTENTS
An Example of an Entailment Enco de d in C
The Utility of L as Compare d with L
An Intuitionistic Repre s entation and its Complexity
The Top ological Interpretation of I
Relation b etween I and S
Corre sp ondence Theorem for I
Intuitioni stic Repre s entation of RCC Relations
The I Enco ding
The Regular ity Constraint and Bo olean Functions Co de d in I
E cient Top ological Reasoning Us ing I
Sequent Calculus for I
Hudelmaier s Rule s
Spatial Reasoning Us ing Hudelmaier s Rule s
Further Optimi sation
Complexity of the Improve d Algor ithm
Implementation and Performance Re sults
Neb el s Complexity Analys i s
Quanti er Elimination
Quanti er Elimination Pro ce dure s
Quanti er Elimination in RCC
Extending the Pro ce dure
Limitations and Us e s of the Pro ce dure
Convexity
Beyond Top ology
The Convex Hull Op erator conv
Containment Relations De nable with conv
st Order axioms for conv
Enco ding conv x in I
Mo dal Repre s entation of Convexity
Practicality of the Mo dal Repre s entation
Comp o s ition Bas e d Re asoning
Comp os ition Table s
Soundne ss and Completene ss of a Comp os ition Table
Formal Theor ie s and Comp os ition Table s
The Extens ional De nition of Comp os ition
Comp os ition Table s and CSPs
Comp os ition Table s for RCC Relations
CONTENTS
RCC
RCC
RCC
Compactne ss of RCC
Exi stential Imp ort in RCC Comp os itions
Relation Algebras
De ning Spatial Relations
Further Work and Conclus ions
What has b een Achieve d
Further Work
Complete Spatial Theor ie s
E ective Mo dal and Intuitioni stic Reasoning
Extending Expre ss ive Power
Reasoning with One Piece and other Simplicity Constraints
Points and Dimens ionality
The Relation Between Logic and Algebra
Comp os itional Reasoning and Relation Algebra
Spatial Reasoning in a More General Framework
A General Theory of the Phys ical World
Spatial Information and Change
Vague and Uncertain Information
Relating Qualitative and Metr ic Repre s entations
Applications
Top ological Inference in a GIS Prototyp e
Conclus ion
A Elementary Geometry
A Tarski s Axiom System
A Pr imitive Geometr ical Concepts
B An Alter native Pro of of MEconv
C Prolog Co de
C Generating all Conjunctions of RCC Relations
C Conjunctions of the RCC Relations and the ir Negations
C An I Theorem Prover for Spatial Sequents
C A Sp ecial Purp os e O n Algor ithm for Spatial Sequents
D Re dundancy in Comp o s ition Table s
Bibliography
Li st of Figure s
Some s igni cant relations among p oints
Bas ic relations in the RCC theory
Shap e s di stingui she d by Gotts us ing the RCC theory
Qualitative or ientation in a relative co ordinate system
Top ological relations repre s entable in C
A spatial reasoner implemente d in Prolog us ing I
Illustration of convex hulls in dimens ions
Nine re nements of EC
Comp os ition of EC and TPP i s not fully extens ional
Trans ition network for e ight top ological relations
A prototyp e geographical information system
D Poss ible con gurations of symmetr ic and asymmetr ic relations
Li st of Table s
De ne d relations in Clarke s theory
De ne d relations in the RCC theory
An equational theory of Bo olean algebras
De nitions of four top ological relations in C
The C enco ding of some RCC relations
Some constraints expre ss ible as closure algebra equations
Seven relations de ne d by inter ior algebra equations
Alter native de nitions for clos e d regions
The RCC relations repre s ente d as inter ior algebra constraints
Closure Axioms and Corre sp onding Mo dal Schemata
Inter ior Axioms and Corre sp onding Mo dal Schemata
Seven relations de ne d by inter ior algebra equations and corre sp onding S formulae
The S enco ding of some RCC relations
S enco ding bas e d on the clos e d s et interpretation of RCC
Repre s entation of the RCC relations in I
Some RCC relations de ne d in I including the RCC relations
Class ical de scr iption of intuitioni stic binary claus e entailment
Comp os ition table for the RCC Relations
Comp os ition table for the RCC relations
Comp os ition table for the RCC relations
Comp os itional inference s among I formulae
Equational axioms for a Relation Algebra
D Comp os ition table re dundancy gure s for four relation s ets
Chapter
Intro duction
Although spatial relationships p ervade our comprehens ion of the world we are almost completely
unaware of how we manipulate spatial information Our f amiliar ity with spatial prop ertie s and
arrangements of everyday ob jects make s the logical connections b etween di erent spatial relation
ships so transparent that it i s extremely di cult to apprehend and make explicit the structure of
thi s conceptual f ramework
Spatial reasoning has a key role to play in a wide var iety of computer applications For example
it i s of crucial imp ortance in the following areas
geographical information systems GIS
rob ot control
computer aide d de s ign and manuf actur ing CAD CAM
virtual world mo delling and animation
me dical analys i s and diagnos i s systems
In current computer systems repre s entation of spatial information i s bas e d almost entirely
on numer ical co ordinate s and parameters However to sp ecify the b ehaviour of the system a
programmer will often nee d to evaluate high level qualitative relationships holding b etween data
ob jects e g to te st whether one region overlaps another Such information can b e extracte d when
nee de d f rom numer ical data structure s by sp ecial purp os e algor ithms Wr iting such algor ithms may
often b e quite straightforward for a comp etent programmer but as large systems are develop e d
problems are likely to emerge There i s p otentially in nite var iety in the form that spatial data
can take so a large numb er of s imilar algor ithms op erating on slightly di erent typ e s of data must
1
b e wr itten More s er iously the heterogene ity of data ob jects means that apparently equivalent
prop ertie s of di erent data typ e s may diverge in extreme cas e s and thi s can lead to co ding errors
which are di cult to identify
The pr imary caus e of the s e problems i s that current programming systems provide no general
1
The object oriented paradigm of computer programming can help overcome this problem but only if great care
is taken in e stablishing a hierarchical organisation of data type s even then it may be di cult to integrate new
unfore s een data type s neatly into an existing structure
CHAPTER INTRODUCTION
f ramework for manipulating high level qualitative spatial information In order to te st whether a
particular qualitative relationship holds e g the s ensor i s in contact with the blo ck the pro
grammer must rst know ab out the details of how ob jects and the ir lo cations are repre s ente d and
then formulate some te st involving value s containe d in the relevant data structure s Thi s te st will
generally take the form of an equation or inequality or p erhaps some Bo olean combination of
equations and or inequalitie s Such te sts determine qualitative spatial relationships according
to the intende d interpretation of data structure s in the databas e If a programmer could directly
employ qualitative spatial vo cabulary in place of complex te st op erations many co ding tasks would
b e greatly s impli e d but providing such a f acility i s very f ar f rom straightforward It require s the
formulation of an adequate theory of spatial relations together with an e ective means of computing
inference s according to the theory
In addition to its application in the context of well e stabli she d kinds of computer system spa
tial reasoning i s of crucial imp ortance to the eld of Arti cial Intelligence AI In attempting
to construct computer programs that s imulate intelligent b ehaviour many re s earchers have con
clude d that as well as nee ding general purp os e reasoning mechani sms such systems must p oss e ss
a large amount of background knowle dge and furthermore in order to draw cons equence s f rom
thi s information detaile d theor ie s character i s ing the logical prop ertie s of the concepts and rela
tions involve d will b e require d Spatial relations clearly form an extremely imp ortant conceptual
domain they are involve d in a very high prop ortion of f acts ab out the real world Hence in the
development of thi s logici st approach to AI theor ie s of spatial relations will play a central role
AI re s earch into spatial reasoning i s at the pre s ent largely di sso ciate d f rom relate d branche s
of mathematics geometry top ology and logic Thi s i s partly b ecaus e mathematical formalisms
in the s e areas do not naturally lend thems elve s to e ective automate d computation of inference s
Another f actor i s the di culty of ass imilating the s e highly develop e d and complex di scipline s into
the relatively young and as yet rather f ragmente d eld of AI From the standp oint of AI spatial
reasoning i s often s een as clos ely asso ciate d with the cognitive pro ce ss ing capabilitie s of humans
and other animals Mathematical theor ie s on the other hand give a very abstract character i sation
of reasoning which i s indep endent f rom biological or psychological pro ce ss e s However cons id
erations of the cognitive plausibility of repre s entations and algor ithms employe d in a computer
program to provide reasoning capabilitie s are clos ely connecte d to cons iderations of the computa
tional complexity of formal de ductive systems
In thi s the s i s I shall adopt a mathematical view of the problem However we shall s ee that
certain conceptual f rameworks which were in f act motivate d by arguments of cognitive plaus ibility
do lead to formal systems which are computationally manageable Thus for example the idea
of taking certain s ets of relations as b e ing of sp ecial s igni cance in the class i cation of spatial
2
s ituations and of taking the composition of two relations as a pr imary mo de of de duction app ears
to b e b oth cognitively plaus ible and to lead to formal systems in which many us eful inference s can
2
Given two relations R and R the ir comp osition R R is the stronge st relation such that for any three
1 2 1 2
ob jects a b c if R a b and R b c hold then R a c must hold The nature and signi cance of relational
1 2 3
comp osition will be studie d in chapter
CHAPTER INTRODUCTION
b e compute d e ectively s ee for example Freksa b and Her n andez
The re st of thi s intro ductory chapter will b e structure d as follows First I motivate the enterpr i s e
of automating spatial reasoning by exhibiting some of the more s igni cant of the wide var iety of
spatial concepts and sugge sting reasoning tasks and applications for which the s e concepts are
s igni cant I then give a br ief hi story of spatial reasoning in which I outline the ma jor approache s
to the sub ject that have b een develop e d by mathematicians and philosophers Thi s i s followe d by
a cons ideration of the relationships b etween di erent conceptual f rameworks and formal systems
for repre s enting and reasoning ab out space I then survey work on spatial reasoning in computer
science particularly f rom the p ersp ective of AI We shall s ee that in addition to problems of
adequate formal repre s entation automate d reasoning ab out spatial information f ace s cons iderable
problems of computational complexity Finally I give a br ief overview of each of the subs equent
chapters of the the s i s
The Domain of Spatial Re asoning
Spatial Concepts and Information
Spatial information i s pre s ente d to us by means of two very di erent mo de s s ensory p erception
and lingui stic de scr iption We acquire knowle dge of spatial relationships e ither by some more
or le ss unconscious pro ce ss ing or transformation of state s pro duce d in our s ensory organs in
re sp ons e to b ombardment by particle s f rom the outs ide world or by b e ing told or reading ab out
the spatial arrangement of parts of the world The former sensory kind of information has b een
intens ively studie d by re s earchers in computer vision and robotics with some succe ss but it i s
the latter propositional form of spatial information that will b e the concer n of thi s the s i s I shall
pursue repre s entations which can expre ss information such as i s containe d in the following Engli sh
s entence s
Yorkshire i s part of England
The hip b one i s connecte d to the thigh b one
The y i s in the b ottle
I shall not however b e concer ne d with the particular ways in which a natural language ex
pre ss e s spatial information but with preci s ely sp eci e d formal repre s entations with de nite rule s
of logical inference Neverthele ss it will b e s een that the s e formal expre ss ions can b e interprete d
in terms of certain natural language expre ss ions and moreover that logically valid de ductions
corre sp ond to arguments which are intuitively sound under thi s interpretation
Geometry of Points and Line s and its Pr imitive Concepts
The geometry of p oints and line s i s the most ancient branch of spatial reasoning Here the abstract
dimens ionle ss p oint i s the bas ic element and all other spatial entitie s must b e constructe d out of
p oints One of the olde st theor ie s of thi s mo de of geometry i s that of Euclid whos e axiomatic
CHAPTER INTRODUCTION
system i s still us e d to day
a) b)
c) d)
Figure Some s igni cant relations among p oints
Figure pre s ents four diagrams showing s imple but very s igni cant relationships which can
hold b etween p oints The gure s are of cours e two dimens ional but analogous relations can hold
in or more dimens ions Diagram a shows the betweenness relation holding among three p oints
if order i s di sregarde d one has the col linearity relation b depicts the equidi stance of two pairs
of p oints Betweenne ss and equidi stance are the two pr imitive s in Tarski s formalisation of
elementary geometry s ee app endix A c shows the relation of equidi stance of two p oints f rom
a third a s ituation eas ily constructe d on pap er us ing a compass In f act in Euclidean geometry
b oth the relations a and b and hence all relations of elementary Euclidean geometry can b e
de ne d in terms of relation c The ter nary relation of equilaterality d can also s erve as the sole
pr imitive for Euclidean geometry of three or more dimens ions Tarski and Beth
If a co ordinate f rame and metr ic are sp eci e d for a Euclidean space algebraic metho ds can
b e applie d to geometr ical problems Points line s and surf ace s are then repre s ente d by means
of equations and inequalitie s relating the co ordinate s of p oints Thi s analytic geometry i s the
most widely us e d repre s entation for spatial information it forms the bas i s of almost all spatial
repre s entation and reasoning mechani sms employe d in current computer systems
Top ology
Topology may b e regarde d as a sub eld of geometry but it i s f ar more abstract than the geometry of
p oints and line s The top ological prop ertie s of a spatial ob ject are thos e that do not vary dep ending
on scale or or ientation A go o d illustration of such invar iance i s provide d by cons ider ing a drawing
on a rubb er sheet the top ological prop ertie s of the drawing are thos e which are pre s erve d while
3
the sheet i s arbitrar ily stretche d and deforme d
Figure illustrate s particularly s igni cant top ological relations which can hold b etween two
regions although the diagram shows D regions analogue s of the s e relations apply to or higher
3
By virtue of the very abstract way in which the theory of top ology has been develope d top ological concepts
have also been applie d to areas of mathematics which are very far remove d f rom this rubber sheet interpretation
CHAPTER INTRODUCTION
dimens ional regions All of the s e relations are de nable in the RCC for Randell Cohn and Cui
or alter natively Region Connection Calculus theory of spatial regions Randell Cui and Cohn
which will b e inve stigate d in detail in chapter Ess entially the same s et of relations has
b een indep endently identi e d as s igni cant in the context of Geographical Information Systems
Egenhofer and Franzosa Egenhofer Clementini Sharma and Egenhofer
The relations form a jointly exhaustive and pairwise disjoint JEPD s et which means that
any two regions stand to each other in exactly one of the s e relations JEPD s ets are imp ortant
in the composition bas e d approach to reasoning ab out binary relations which will b e explore d in
chapter Thi s class i cation can b e re ne d to intro duce additional di stinctions b etween relations
For instance amongst pairs of EC exter nally connecte d regions we could di stingui sh thos e con
necte d at a b oundary s egment f rom thos e connecte d at a s ingle p oint Many such relations are
also de nable in the RCC theory
Top ological relationships are of a very general character and can b e us e d to give a high level
de scr iption of all manner of spatial s ituations For example us eful geographical information con
cer ning countr ie s province s and countie s and the relationships b etween them can b e expre ss e d in
terms of the s e relations Non spatial information can also often b e repre s ente d metaphor ically in
terms of top ological prop ertie s e g the range of application of colour terms might b e de scr ib e d
in terms of regions in a colour space
a a
b b a b a b
DC(a,b) EC(a,b)TPP(a,b) TPPi(a,b)
a b a b a b a b
PO(a,b) EQ(a,b)NTPP(a,b) NTPPi(a,b)
Figure Bas ic relations in the RCC theory
Repre s entation and e ective automate d reasoning ab out top ological relations will b e the main
concer n of thi s the s i s However we shall s ee that repre s entations and algor ithms develop e d pr imar
ily for e cient top ological reasoning can b e extende d to handle other asp ects of spatial information
Formal character i sation of top ological relationships has traditionally b een carr ie d out by axiomat
i s ing certain prop ertie s of s ets of p oints However such an axiomati sation assume s a theory of s ets
The re sulting theory i s extremely complex and cons equently impractical as a bas i s for an automate d
reasoning system An alter native approach to formali s ing top ological notions i s that of algebraic
top ology in which the ob jects of the theory are n dimens ional p olygons and p olyhe dra Thi s may
CHAPTER INTRODUCTION
Doughnut (or Solid Torus) Torus Doughnut with gap (topologically, a solid block)
Three doughnuts with degenerate holes
Four doughnuts with degenerate hole-surrounds
Double doughnut Loop Cylinder-surface Block minus block
Figure Shap e s di stingui she d by Gotts us ing the RCC theory
prove to b e more suitable for computational reasoning than the p oint s et repre s entation but in its
pre s ent form it i s also f ar to o complex In view of the imp ortance of top ological concepts and
the di culty of carrying out automate d reasoning us ing standard mathematical notations much
of thi s the s i s will b e concer ne d with the development of alter native repre s entations for top ological
relationships
Shap e
Character i s ing the shap e of ob jects or regions s eems to involve a wide sp ectrum of spatial concepts
Although the shap e of a region may b e regarde d as indep endent of its s ize and or ientation the
relative prop ortions and p os itions of the parts of a region are e ss ential to its shap e so s ize and
or ientation are in thi s way asp ects of shap e In f act if in de scr ibing any spatial s ituation we are
only intere ste d in di stingui shing o ccupie d regions f rom f ree space and are not concer ne d with the
overall scale then thi s can b e accompli she d by character i s ing the shap e of the o ccupie d or f ree
space Thus repre s enting and reasoning ab out arbitrary shap e s encompass e s a very large part
if not the whole of the domain of spatial reasoning
Neverthele ss a numb er of formali sms for de scr ibing shap e have b een develop e d The s e can b e
divide d into two broad class e s rstly there are the constructive repre s entations in which complex
shap e s are de scr ib e d by structure d combinations of pr imitive comp onents and s econdly there
are approache s which might b e calle d constraining s ince shap e s are character i s e d in terms of
CHAPTER INTRODUCTION
prop ertie s holding of a region and the s e prop ertie s are constraine d to conform to some theory
A well known form of the constructive approach which i s bas e d on numer ical vector repre s
entations of ob jects i s Constructive Solid Geometry Requicha and Tilove Requicha
More abstract example s of the approach include the many kinds of shape grammar that have b een
develop e d A rather di erent metho d of shap e construction i s de scr ib e d by Leyton He sp e
ci e s a process grammar which generate s shap e s by means of a s er ie s of deformations starting f rom
an initial di sc shap e Constraining approache s to shap e include thos e bas e d on axiomatic theor ie s
such as the st order RCC theory Randell Cui and Cohn Gotts has shown how
many top ologyically di stinct shap e s can b e di stingui she d in terms of thi s theory s ee gure
Another approach to shap e de nition us ing RCC i s de scr ib e d in Cohn
Convexity and Containment
A limite d but s igni cant sub domain of prop ertie s concer ning shap e compr i s e s thos e concepts
relate d to the notion of convexity An ob ject may b e convex or may have a certain numb er of
concavitie s Even such a s eemingly meagre range of di stinctions can s erve to di scr iminate b etween
many di erent kinds of spatial region Cohn Davi s Gotts and Cohn
In de scr ibing convexity relate d prop ertie s it i s us eful not only to b e able to say that a region i s
convex but also to b e able to identify the smalle st convex region which contains any given region
Thi s i s the convex hull of the region The extende d RCC theory employs a convex hull op erator
whos e interpretation i s the function f rom regions to the ir convex hulls In the pre s ent work I shall
only b e concer ne d with thos e notions of convexity and containment which are de nable in terms
of the convex hull op erator Thus not only will many asp ects of shap e b e overlo oke d but also the
4
treatment of convexity will b e limite d
Several us eful relationships concer ning the containment of one region within another may b e
de ne d in terms of convex hulls For instance if a region a do e s not overlap b but i s a part of
the convex hull of b we may say that b contains a Thi s give a preci s e although arguably
unnatural sp eci cation of a containment relation in terms of convex hull together with some
s imple top ological relations Convexity and containment will b e cons idere d in detail in chapter
Po s ition and Or ientation
Pos ition and or ientation are very imp ortant kinds of spatial information which can b e preci s ely
repre s ente d by means of numer ical co ordinate s However there are also a wide range of qualitative
relationships involving the s e concepts Figure illustrate s an analys i s of qualitative or ientation
due to Freksa b a depicts a s ituation in which an obs erver o i s heading towards a
landmark l and s ee s a hous e h which i s further away than and to the r ight of the landmark
Figure b i s a qualitative repre s entation of the relative p os ition of the hous e with re sp ect to the
obs erver at the lower inters ection in the gr id and landmark the upp er inters ection qualitat
4
A detaile d examination of many subtle di cultie s that ar is e when one tr ie s to precis ely character is e di erent
kinds of cavity can be found in Casati and Varzi
CHAPTER INTRODUCTION
ively di erent relative lo cations can b e di stingui she d by means of thi s repre s entation as indicate d
in c Qualitative repre s entations of or ientation have also b een inve stigate d by Her n andez
Whilst p os ition and or ientation are clearly very imp ortant for many mo de s of spatial reasoning
further cons ideration of the s e asp ects of spatial information i s b eyond the scop e of thi s the s i s
h l
o
a) b) c)
Figure Qualitative or ientation in a relative co ordinate system
A Br ief Hi story of Spatial Re asoning
I shall now de scr ib e some of the more succe ssful approache s to the character i sation of correct
reasoning ab out spatial relationships The ideas pre s ente d here pre date or are indep endent f rom
the us e of electronic computers More recent approache s to spatial reasoning taken by re s earchers
in computer science e sp ecially in the eld of AI will b e reviewe d later in s ection
Or igins
I shall give only br ief account of the early hi story of spatial reasoning further details can b e found
in any go o d hi story of mathematics such as that of Boyer
Geometry literally earth land measurement date s back to the Egyptians Egyptian mathem
atics was of a largely practical kind concer ne d with s imple calculations very often of a spatial
character e g determining the area of a piece of land The relations b etween lengths areas and
volume s were studie d b ecaus e of the ir value in commercial and architectural applications The idea
that all geometr ical reasoning might b e bas e d on the application of a small numb er of fundamental
pr inciple s app ears to have or iginate d in the ancient Greek civili sation The almost mythical charac
ter Thale s who live d around B C i s often cre dite d with b e ing the rst p erson to demonstrate
general pr inciple s of mathematical and particularly geometr ical reasoning
The idea of character i s ing valid reasoning in terms of logical mo de s of inference was taken up by
many Greek thinkers and develop e d surpr i s ingly rapidly so that within a century Pythagoras
B C and hi s followers had constructe d very r igorous pro ofs of many theorems in numb er theory
and geometry Laws of valid argument were also studie d indep endently of any particular sub ject
matter Early philosophers such as Plato B C reali s e d that s equence s of s entence s that
followe d certain patter ns always s eeme d to constitute a convincing argument Thi s i s the bas i s
of formal logic Many pr inciple s of reason such as modus ponens and the law of the excluded
CHAPTER INTRODUCTION
midd le were identi e d Ar i stotle B C analys e d syl logisms which make up a s igni cant
f ragment of quanti cational logic
At thi s time there was intens e inve stigation of how pr inciple s of r igorous logical inference
should b e applie d to reasoning ab out spatial relationships Geometers strove to elicit the elements
of geometry that i s a s et of fundamental de nitions and p ostulate s f rom which all geometr ical
truths could b e logically der ive d Many attempts were made to sp ecify the s e elements until nally a
system was di scovere d which s eeme d to yield all that was require d Euclid s Elements was wr itten
round ab out B C while Euclid was a teacher in the Museum at Alexandr ia an institution
e stabli she d by Ptolemy I De spite a certain amount of quibbling ab out a p ostulate concer ning
parallel line s Euclid s axiomatic geometry has b een in us e r ight up to the pre s ent day
Development
For almost two millennia geometry was extens ively develop e d but it did not really go b eyond the
p otentialitie s of its Euclidean foundation until it was inve stigate d by De scarte s In
La Geometrie an app endix to hi s Discours de la M ethode De scarte s intro duce d the idea
of a coordinate system in which p oints are identi e d with pairs in D or tr iple s in D of real
numb ers Thi s interpretation provide s the foundation for what i s now known as analytic geometry in
which line s surf ace s and volume s are repre s ente d by means of algebraic equations and inequalitie s
involving the Carte s ian co ordinate s of p oints The uniformity of algebraic repre s entation f acilitate s
general and very e ective metho ds for solving large class e s of geometr ical problems
The th century saw a dramatic revolution in geometry Euclid s fth p ostulate which state s
that for any p oint and any line there exi sts a unique line pass ing through the p oint and parallel to
the rst line had long b een the sub ject of inve stigation b ecaus e it had long b een hop e d that it could
b e der ive d f rom the other much s impler p ostulate s of the theory The formal apparatus involve d
in repre s enting and reasoning ab out Euclidean geometry had by thi s time b ecome very preci s e
and the general prop ertie s of formal systems had also b ecome clearer In particular the notions
of logical equivalence indep endence and cons i stency of axiom s ets were now well understo o d It
was nally e stabli she d that Euclid s fth p ostulate concer ning the exi stence of unique parallels
was indep endent of the other s impler p ostulate s so that cons i stent systems could b e constructe d in
which it did not hold Lobachevsky to ok the b old step of prop os ing a system of hyp erb olic
5
geometry which explicitly contradicts the fth p ostulate
The end of the th century also saw the birth of a radically new approach to the mathematical
de scr iption of spatial relationships The eld of p oint s et top ology was or iginate d by Cantor
as an application of s et theory to the study of Euclidean space Inve stigations in top ology by
Hausdor Kuratowski and many others lead to the clar i cation of many concepts
6
in analys i s e g limits of in nite s equence s The applications of mo der n top ology are for the
5
The signi cance of non Euclidean geometry is clearly explaine d in Trudeau which also give s an illumin
ating view of the status of geometr ical theor ie s
6
A thorough intro duction to basic top ology can be found in Kuratowski
CHAPTER INTRODUCTION
most part f ar remove d f rom spatial relationships in the phys ical world but are concer ne d with
abstract mathematical structure s The complexity of p oint s et top ology means that although it i s
a p owerful to ol for the mathematician it has not as yet yielde d e ective general purp os e metho ds
for reasoning ab out spatial relationships
An alter native approach to character i s ing top ological prop ertie s known as algebraic topology
was create d by Henr i Poincar e in the last years of the th century It was initially develop e d
more or le ss indep endently of p oint s et top ology although in the s some uni cation of the
approache s was attaine d Alexandro and Hopf The bas i s of the approach to top ology i s
to us e an algebraic ob ject often some kind of group to de scr ib e the structure of a top ological
space Thi s formali sm i s in many re sp ects more amenable to computational manipulation than
the p oint s et approach and it i s very likely that algebraic metho ds can provide a p owerful to ol
for the development of spatial reasoning algor ithms s ee e g Pigot Bertolotto Flor iani and
Marzano However further cons ideration of the theory of algebraic top ology i s b eyond the
scop e of thi s the s i s
New Foundations
Dur ing the early part of the th century the metho ds of logical analys i s reache d a state of extreme
preci s ion and were applie d to many branche s of mathematical and to a le ss er extent phys ical
science Russ ell and Whitehead were b oth keen that the metho ds of logic should b e applie d not
only to well e stabli she d ob jective phys ical theor ie s but also to the development of phenomenological
theor ie s de scr ibing the world as it i s p erce ive d through s ens e data How such theor ie s should b e
constructe d was and i s still f ar f rom clear One idea exp ounde d by Whitehead in hi s b o ok The
Concept of Nature Whitehead i s that in a theory of the world of s ensory exp er ience the
bas ic entitie s of the logical repre s entation should corre sp ond directly to phenomena the s e b e ing
ob jects of consciousne ss which are p erce ive d via divers e s ens e data but are conce ive d as integral
ob jects or events e g a cloud or the ight of a bird across the sky
Treating such things as bas ic entitie s i s at o dds with the theoretical systems which have b een
develop e d to formalis e class ical science In the s e systems the bas ic entitie s are typically p oints
of space instants of time and numer ical quantitie s such as mass and velo city in f act p oints
and instants are generally also identi e d with numer ical co ordinate s The spatial relationships
b etween p oints are character i s e d by well known geometr ical theor ie s and mathematical structure s
Moreover thi s analys i s allows sp eci cation of phys ical laws in terms of di erential equations which
form the axioms of nearly all phys ical theor ie s But the analys i s also means that formal ob jects
corre sp onding to phys ical b o die s or events must b e built up s et theoretically in terms of the s e bas ic
entitie s A complex and irregular region e g that o ccupie d by a cloud then b ecome s an in nite
s et of p oints which may b e extremely di cult or even imp oss ible to character i s e
Under the alter native phenomenological approach ob jects and events b ecome the bas ic entitie s
of a theory Geometry i s now concer ne d with relationships b etween the regions o ccupie d by b o die s
and dynamical laws must b e formulate d in terms of causal relationships b etween events di erential
CHAPTER INTRODUCTION
equations are replace d by qualitative relationships Attempts to construct theor ie s of thi s kind have
b een made by many philosophers and logicians as well as more recently by computer scienti sts
but thi s pro ject has met with s evere di cultie s In f act I think it i s f air to say that there i s no
widely accepte d phys ical theory bas e d on thi s typ e of ontology Thi s i s p erhaps not surpr i s ing
given rstly the relatively recent conception of the idea and s econdly the di culty in nding us e s
for such theor ie s that would make the ir construction more than just a philosophical exerci s e
An application which promi s e s to motivate development of phenomenon bas e d theor ie s i s AI
Not only do e s thi s ontology app ear to b e clos er to that employe d in human reasoning as evidence d
by the structure of our ordinary language but it also s eems that it may b e more appropr iate as
a vehicle for automate d reasoning ab out real world s ituations which if de scr ib e d in the terms of
class ical phys ics would b e unmanageably complex Neverthele ss de spite cons iderable e ort f rom
AI re s earchers the qualitative theor ie s emb e dde d in AI systems do not app ear to have a p ower
and generality comparable to class ical theor ie s of for example dynamics or electromagnetics
One explanation for the lack of progre ss may b e that re s earchers have assume d that given
the r ight formal f ramework sp ecifying theor ie s of real world phenomena will b e straightforward
much work has b een directe d towards providing general purp os e formal systems that are amenable
to computation but comparatively little has b een concer ne d with providing theor ie s of sp eci c
conceptual domains However in recent years intere st in such domain sp eci c theor ie s has grown
rapidly By analogy with the role of p oint bas e d geometry in class ical phys ical theor ie s it i s to
b e exp ecte d that character i sation of the geometr ical relationships that may hold b etween extende d
ob jects will b e of fundamental imp ortance to many of the s e conceptual theor ie s Construction of
general theor ie s of the s e relationships i s one of the pr imary goals of the sub eld of AI known as
Qualitative Spatial Reasoning henceforth QSR
A detaile d account of formal theor ie s of spatial regions will b e given in the next chapter
Conceptual and Formal Frameworks
Let us now examine the plurality of p oss ible f rameworks for repre s enting and reasoning ab out
spatial information and the relationships b etween the s e f rameworks
The hi story of spatial reasoning shows that formalisation of its mo de s of inference can b e carr ie d
out f rom a var iety of di erent p ersp ective s Given our mo der n understanding of logical systems
it i s obvious that for any axiomatic theory there are in nitely many syntactically di stinct but
logically equivalent axiomati sations of the theory In the context of geometry thi s i s well illustrate d
by Euclid s fth p ostulate which when taken together with hi s other four p ostulate s has b een
prove d logically equivalent to a host of other p oss ible axioms e g that the angle s of a tr iangle add
up to
At a more fundamental level there are also many di erent concepts or s ets of concepts that
could b e taken as pr imitive s in a formal system Given two s ets of pr imitive s A and B it may
happ en that each concept of B can b e de ne d by means of purely logical equivalence s in terms
of the concepts of A in thi s cas e the s et A i s at least as expre ss ive as B Moreover two s ets of
CHAPTER INTRODUCTION
concepts may b e equal in expre ss ive p ower and so could s erve as alter native s ets of pr imitive s for
7
e ss entially the same theory
Analys i s of Euclid s geometry le d to s everal equivalent systems employing di erent pr imitive
relations b etween p oints equidi stance of two pairs of p oints equidi stance of two p oints f rom a
third mutual equidi stance of three p oints the relation b etween ve p oints which lie on the surf ace
of the same sphere However in all the s e formulations one pr imitive notion remains constant
that of point Commitment to the notion of p oint i s eas ily overlo oke d in most axiomatic systems
of geometry b ecaus e it i s often assume d that the domain of st order quanti cation coincide s with
the totality of p oints so there i s no nee d to actually employ a pre dicate p oint x Neverthele ss as
we shall s ee in the next chapter a numb er of axiom systems have b een prop os e d in which regions
rather than p oints make up the domain of quanti cation So in formulating a theory of spatial
relationships or any other theory we have a large degree of f ree dom not only in how we state
its axioms and which pr imitive pre dicate s we employ but also in cho os ing the typ e of ob jects that
make up the domain of elementary individuals of the theory
Nico d s do ctoral the s i s Geometry in the Sensible World op ens with a p enetrating analys i s
of the relationship b etween alter native systems of geometry bas e d on di erent pr imitive notions
Here he intro duce s the ideas of intrinsic and extrinsic complexity of formal systems The former
re s ide s in the structure of the system its elf whereas the latter dep ends on how s imply the elements
and concepts of the formal theory can b e matche d to ob jects and prop ertie s in the domain of
application of the theory Thus for sp ecifying a theory of phys ical pro ce ss e s a formal system in
which p oints are the bas ic elements may b e inter nally s imple but b ecaus e abstract p oints cannot
b e p erce ive d directly or preci s ely lo cate d in the phys ical world it would b e deeme d exter nally
complex
Bas ic Elements in a Spatial Theory
Five of the most promi s ing candidate s to s erve as bas ic elements in a theory of spatial relationships
are given in the following table
Objects Existential Character Proponents
Points abstract Euclid De scarte s
Regions spatial Clarke Randell Cui and Cohn
Bo die s phys ical Snee d
Things lingui sti c metaphy s ical Whitehead Simons
Sens e data s ensory Whitehead Nico d
The most e stabli she d ontological foundation for spatial reasoning i s to construe points as the
bas ic elements out of which more complex spatial ob jects are in some s ens e comp os e d Points are
usually regarde d as abstract theoretical entitie s b ecaus e they have no phys ical extens ion nor mass
The idea of developing a geometry bas e d up on sense data was pursue d by Whitehead and Nico d
7
A number of imp ortant theorems concerning the de nability of concepts and the completeness of conceptual
f rameworks are given in Tarski b
CHAPTER INTRODUCTION
under the in uence of Russ ell s epi stemological theor ie s according to which the bas ic elements of
reason must b e correlate d with s imple s ens e data such as colour patche s in the vi sual eld Russ ell
Within such an ontology p oints if they are to exi st at all must b e somehow constructe d
in terms of s ens e data
Taking regions as bas ic may b e s een as a compromi s e b etween p oint bas e d and s ens e data
bas e d ontologie s in that although regions are str ictly abstract partitions of a space they s eem to
b e much clos er to s ens e data than are p oints A given region may p oss e ss a certain prop erty
greenne ss say and thi s will b e p erce ive d as a green patch Although Whitehead and Nico d
saw s ens e data as pr imary they also gave axiomatisations whos e ob jects are abstract regions
the corre sp ondence b etween the s e regions and actual s ens e data would then have to b e given by
an auxiliary de nitional theory c f the chapter The geometry of p ersp ective s in Nico d s the s i s
Laguna and Tarski also develop e d theor ie s of regions which they slightly mi sleadingly calle d
solids but did not app ear to b e so concer ne d with the epi stemological status of regions The ir
theor ie s are pre s ente d more as alter native abstract systems of geometry in which the status of
p oint and region i s inverte d with re sp ect to s et theoretic construction
Region bas e d formalisms have often b een pre s ente d as relating to arbitrary solids de Laguna
Tarski Thi s might sugge st that the ob jects of the s e theor ie s are phys ical b o die s
for solidity i s surely a phys ical prop erty which could not apply to an abstract region However
a theory of phys ical b o die s would have to take into account the mater ial structure and prop ertie s
of such ob jects rather than treating them as abstract volume s Such formali sations of phys ical
ob jects are at a relatively undevelop e d stage although a numb er of formal theor ie s of Newtonian
mechanics have b een prop os e d e g Montague A di scuss ion of the problems involve d in
sp ecifying phys ical theor ie s in a fully formal f ramework can b e found in Snee d
A nal exi stential p ersp ective on the ob jects of spatial reason i s given to us by our lingui stic
de scr iptions of ob jects in space Such ob jects are generally individuate d by means of count nouns
e g table cup saucer each of which carr ie s its own cr iter ia for identi cation The s e lingui stic
class i cations and the ir asso ciate d cr iter ia of recognition der ive f rom the practical s igni cance of
certain typ e s of phys ical entity conditione d to some extent by more or le ss arbitrary lingui stic
conventions The utility of thi s class i cation i s to a large extent determine d by the phys ical nature
of the world the mater ial prop ertie s of the world give r i s e to natural ways of class ifying it and
breaking it into chunks However it may b e argue d the s e phys ical circumstance s give r i s e to
a f ramework of metaphys ical categor ie s which must underly any lingui stic de scr iption of spatial
entitie s
Mo de s of Formali sm
At a still more fundamental level the very b oundary b etween a logical repre s entation language and
a theory expre ss e d in that language may b e shifte d Three kinds of repre s entation together with
the ir apparatus for information manipulation are summar is e d in the following table
CHAPTER INTRODUCTION
axiomatic theorem proving
algebraic analytic co ordinate s and equations
purely logical spatial logic and pro of pro ce dure s
Applying the axiomatic metho d to spatial reasoning involve s formulating a spatial theory in
some general purp os e logical language such as st order logic and then proving theorems in
that system It has b een found that theorem proving in all but the s imple st logical language s i s
intractable The algebraic approach i s the one that i s most commonly adopte d Information i s
co de d in p olynomial equations and or inequalitie s Di sjunctive and quanti cational constructs are
avoide d so that the expre ss ive p ower of the system i s limite d E ective metho ds for manipulating
and extracting information f rom equations and inequalitie s are well known The p oss ibility of a
purely logical approach i s not widely appreciate d It will b e di scuss e d in the next s ection
Logical Theor ie s of Space and Spatial Logics
The vo cabulary of a formal language can b e divide d into two categor ie s of atomic expre ss ion
which may b e calle d logical and non logical In st order logic the logical symb ols are the truth
functional connective s and quanti ers and the non logical vo cabulary cons i sts of constants and
pre dicate s Var iable s may b e regarde d as notational device s asso ciate d with quanti ers as a
means of indicating the ir scop e We have s een how in repre s enting a theory in a formal language
there may b e many p oss ible s ets of non logical pr imitive s in terms of which the theory could b e
sp eci e d However there i s also a more radical kind of alter native formulation concepts of the
theory may b e enco de d directly into logical symb ols or complex logical structure s of the formal
language In doing thi s we arr ive at a true spatial logic rather than merely a theory of spatial
relations sp eci e d in a general purp os e logical language
In chapter we shall inve stigate thi s p oss ibility at some length The most novel and substantial
re sults of thi s the s i s concer n the repre s entation of spatial relationships in terms of non class ical
order logics One advantage of such enco dings i s that one often imme diately obtains a deci s ion
pro ce dure for the spatial theory
Spatial Re asoning in Computer Science
In most exi sting computer programs repre s entation and manipulation of spatial data i s very largely
numerical Ob jects and regions are repre s ente d by s ets of co ordinate s and information i s extracte d
f rom thi s data by means of ar ithmetic and tr igonometr ic computations
Numer ical repre s entation may b e well suite d for some purp os e s in particular where the spatial
information preci s ely de scr ib e s some de nite s ituation and where the output require d f rom the
system i s its elf pr imar ily numer ical However in many cas e s us eful spatial information do e s not
de scr ib e a unique phys ical s ituation but qualitatively character i s e s a s ituation as b e ing of a par
ticular typ e Extracting information f rom such data require s logical reasoning ab out the concepts
CHAPTER INTRODUCTION
involve d in de scr ibing a s ituation and hence require s a r igorous formal theory of qualitative
spatial relationships
From a computational p oint of view qualitative theor ie s of spatial relations are relatively un
develop e d Neverthele ss some s igni cant work has b een done Randell and Cohn and
Randell Cui and Cohn sp ecify a st order theory of spatial regions bas e d on a pr imitive
relation of connecte dne ss C x y together with a numb er of quas i Bo olean functions De spite
containing very few non logical pr imitive s thi s theory has b een found to b e quite expre ss ive indee d
a large numb er of s igni cant spatial relations can b e de ne d exclus ively in terms of the relation
C Gotts Egenhofer pre s ents a much more limite d f ramework in which a numb er of
top ological relations can b e repre s ente d He also shows how some s imple inference rule s can b e
us e d to generate the composition of any pair of the s e relations s ee chapter for a full di scuss ion
of comp os ition bas e d reasoning
Commons ens e Knowle dge
Many in uential AI re s earcher s have argue d that repre s entation of so calle d commons ens e know
le dge i s of key imp ortance in developing intelligent computer systems and a f air prop ortion
of the s e re s earcher s have employe d formal repre s entations and axiomatic theor ie s as a means of
enco ding thi s knowle dge s ee e g Haye s Haye s b Guha and Lenat Qualitat
ive spatial concepts are p ervas ive in everyday de scr iptions of the world so axiomatic theor ie s of
commons ens e knowle dge will have to incorp orate many axioms gover ning the logical b ehaviour of
spatial prop ertie s and relations
A very large numb er of theor ie s have b een constructe d so detaile d de scr iptions cannot b e given
here Many of the pap ers which shap e d thi s eld of AI are containe d in the collections Hobbs
Blenko Croft Hager Kautz Kub e and Shoham and Hobbs and Mo ore A more
recent reference on formal repre s entations of commons ens e knowle dge i s Davi s
Re asoning ab out Phys ical Systems
Another domain of knowle dge repre s entation that has rece ive d cons iderable attention i s that of
physical systems Reasoning ab out phys ical systems may b e treate d as a typ e of commons ens e
reasoning or alter natively one may attempt to formalis e the kind of reasoning employe d by phys
ici sts which involve s manipulation of mathematical equations as well as the us e of commons ens e
pr inciple s Although spatial prop ertie s are of fundamental imp ortance to the character i sation of
phys ical systems work in thi s area has tende d to fo cus on the ir dynamical b ehaviour rather than
the ir static prop ertie s Key pap ers in thi s area can b e found in Weld and De Kleer
Spatial Re asoning in Rob otics
Spatial reasoning i s clearly of key imp ortance in the eld of rob otics But b ecaus e of the complexity
of the domain the us e of formal repre s entations has b een limite d Most rob ot control systems rely
on algor ithms which are f rom a logical p oint of view rather ad hoc
CHAPTER INTRODUCTION
However certain metho ds for class ical rob ot path planning do make us e of logical repre s enta
tions A general repre s entation for phys ical ob jects can b e given in terms of semi algebraic sets
The s e are s ets of p oints de ne d by st order formulae whos e atoms are p olynomial equalitie s or
inequalitie s The cons i stency of s ets of such expre ss ions can b e determine d by the deci s ion pro
ce dure for algebra and geometry given by Tarski who showe d how quanti ers could b e
eliminate d f rom the s e formulae The us e of thi s deci s ion pro ce dure for computing colli s ion f ree
paths for a rob ot in an arbitrary workspace character i s e d in terms of s emi algebraic s ets i s de
scr ib e d by Latomb e Other us e s of quanti er elimination metho ds in geometr ical reasoning
are di scuss e d by Ar non
It i s likely that spatial reasoning formalisms akin to thos e develop e d in thi s the s i s will ultimately
play an imp ortant role in rob otic control systems But b efore thi s can b e done it will b e nece ssary
to develop repre s entations with which one can expre ss and reason ab out b oth spatial and dynamical
asp ects of phys ical systems Thi s i s b eyond the scop e of the pre s ent re s earch
Spatial Re asoning and Computer Vi s ion
The eld of computer vi s ion i s an extremely active area of AI re s earch and has pro duce d systems
which are actually us e d in applications Vi s ion i s clearly very clos ely relate d to spatial reasoning
Neverthele ss very little of the re s earch done in thi s area i s of direct relevance to the concer ns of
thi s the s i s
Computer vi s ion i s concer ne d pr imar ily with extracting information f rom s ensor data The
s ensor data would typically take the form of two dimens ional pixel image s Var ious typ e s of
information may b e extracte d but the most common tasks would b e to construct some kind of
three dimens ional mo del of the scene or to lo cate typ e s of ob ject or region in the scene Spatial
reasoning on the other hand i s concer ne d with manipulating spatial information and in particular
in nding cons equence s holding among spatial prop os itions
Although the concer ns of vi s ion and spatial reasoning are rather di erent there i s some inter
action b etween the problems of the two elds For instance in extracting D information f rom a
D scene the ability to draw inference s f rom and to te st the cons i stency of D spatial informa
tion may b e very us eful in narrowing down the range of p oss ible interpretations of a scene Thi s
technique would b e akin to that us e d by Waltz for nding D de scr iptions of shade d D
drawings
Temp oral Re asoning
Temp oral reasoning i s a di stinct and very active area of re s earch Neverthele ss space and time are
often cons idere d to b e very clos ely relate d asp ects of reality so it i s us eful to cons ider s imilar itie s
b etween spatial and temp oral formalisms
8
Temp oral reasoning has b een develop e d in a numb er of di erent ways Or iginating with the
work of Pr ior tense logics have b een develop e d in which temp oral relationships
8
A survey of temp oral logics and the ir applications can be found in Galton
CHAPTER INTRODUCTION
b etween state s of a airs are mo delle d in terms of prop os itional op erators Thi s analys i s of tens e
i s much the same as that given by modal logics in re sp ect of concepts such as nece ss ity and b elief
which are likewi s e repre s ente d in terms of prop os itional op erators Galton further analys e s
the structure of temp oral op erators by means of a language in which prop os itions and events are
di stinct typ e s of expre ss ion
Until recently language s such as tens e logic where temp orality i s mo delle d by sp ecial categor
ie s of logical op erator have not b een widely employe d by AI re s earchers Theor ie s of actions and
change have rather b een repre s ente d in more standard notation st order logic or some var iant
with s emantic prop ertie s b e ing sp eci e d by axioms or capture d by sp ecial purp os e inference rule s
The b e st known work in thi s area i s that of Allen Allen identi e d a s et of thirteen JEPD relations
which can hold b etween two temp oral intervals and studie d reasoning pro ce dure s bas e d on the
comp os ition of the s e relations Allen Allen A st order theory de scr ibing the s e tem
p oral intervals and the ir relationship to actions and events was also develop e d Allen Allen
and Haye s
Whilst st order theor ie s may b e very us eful in e stabli shing a sound theoretical f ramework for
repre s enting information in some domains requirements of computational tractability mean that
for most practical purp os e s it has b een found that le ss expre ss ive more domain sp eci c language s
must b e us e d The s e come in two bas ic var ietie s on the one hand we have constraint language s
capable of repre s enting and reasoning with relational f acts involving a xe d s et of temp oral relations
e g the Allen relations or p erhaps some tractable subs et of di sjunctions of the s e relations
on the other hand we have language s containing temp oral op erators but le ss expre ss ive than st
order logic e g prop os itional or Hor n claus e language s Formali sms of b oth the s e kinds are now
extremely wide spread and well known in AI
The content of thi s the s i s re ects many parallels b etween the p oss ible approache s which can
b e taken to repre s enting spatial information and approache s which have b een applie d to temp oral
information Construction of the RCC theory of spatial regions was greatly in uence d by the
works of Allen and Haye s Allen and Haye s Haye s Haye s a Haye s b and
cons equently its development followe d a s imilar patter n a st order theory was pre s ente d and
inve stigate d then to provide a reasoning mechani sm us eful constraint language s were identi e d
within which comp os ition bas e d reasoning could b e conducte d The most or iginal part of thi s
the s i s develops an alter native route to spatial reasoning via order logical language s with spatial
op erators Hence spatial as well as temp oral reasoning can b e carr ie d out within the broad
9
f ramework of mo dal logic
I envi sage that as the eld of spatial reasoning i s develop e d it will b ecome increas ingly linke d
to temp oral reasoning In order to repre s ent and reason ab out changing s ituations a combine d
spatio temp oral formali sm i s clearly nee de d Reasoning ab out action and change has very often
9
In fact it is perhaps more revealing to realis e that what is common between all the s e mo de s of reasoning is that
they are all repre s entable in the very general f ramework of Boolean algebras with additional monadic operators
Logical language s whos e s emantics can be speci e d in terms of such algebras form a very natural class of formal
systems whos e expre ssive p ower is greater than that of prop ositional logic but which are still in many cas e s decidable
The us e of such language s in spatial reasoning will be inve stigate d in chapters and
CHAPTER INTRODUCTION
b een pre s ente d in formali sms in which there i s a bas ic category of expre ss ion referr ing to events
In providing s emantics for such formalisms events have very often b een identi e d with temp oral
intervals However the temp oral extent of an event i s only one dimens ion of its exi stence I b elieve
that events or at least most kinds of event are spatial just as much as temp oral entitie s and that
an adequate s emantics for events must take into account thi s spatial character
The mo dal repre s entation of spatial relations develop e d in chapter i s in many re sp ects s imilar
to prop os itional tens e logics If prop os itions in a tens e logic are regarde d as dimens ional regions
on a time line it i s clear that temp oral op erators are clos ely relate d to spatial relationships The
main di erence b etween tens e logics and the spatial logics that I shall pre s ent i s that a tens e logical
formula i s evaluate d to b e true or f als e at a particular time
Automating Spatial Re asoning
Automate d reasoning has attracte d a great deal of attention f rom computer scienti sts f rom the
s ixtie s onwards Signi cant advance s have b een made in developing pro of metho ds which are
10
well suite d to computation
De spite thi s progre ss fundamental problems remain Most re s earchers in thi s area have fo cus e d
on general purp os e st order theorem proving However it i s known that reasoning with thi s
formali sm i s undecidable Thi s means that although pro of algor ithms for st order logic can b e
sp eci e d which are guarantee d to generate a pro of of any theorem in nite time there can b e no
algor ithm that can determine whether any arbitrary st order formula i s a theorem in nite time
Thi s i s b ecaus e whatever pro of pro ce dure i s us e d there will always b e a class of non theorems for
which the algor ithm do e s not terminate Unle ss thi s di culty can somehow b e circumvente d it i s
unlikely that general purp os e st order theorem provers will ever b e us e d in practical applications
There are e ss entially two ways of avoiding the undecidability problem one i s to us e a general
purp os e logical language which i s le ss expre ss ive than st order logic the other i s to us e some
sp ecial purp os e repre s entation de s igne d for reasoning in a particular conceptual domain Thi s
the s i s combine s b oth the s e approache s I fo cus on repre s enting information in the re str icte d domain
of spatial relationships but in order to reason ab out the s e relations I show in chapter that they
can b e enco de d in a formali sm which i s normally regarde d as a general purp os e order language
Complexity of Mathematical Theor ie s
As we have s een spatial reasoning has long b een a concer n of mathematicians Indee d the elds
of geometry and top ology are extremely well develop e d and are of direct relevance to automate d
reasoning ab out spatial s ituations But the problem with nearly all mathematical theor ie s i s that
they are to o complex to reason with e ectively Top ology i s built up on a large amount of s et
theory so any naive reasoning algor ithm bas e d on standard formulations of top ology will have as its
s earch space virtually all of mathematics Whilst rather more succinct st order axiomati sations
10
General texts on Automate d Reasoning which de scr ibe the s e metho ds include Bibel and Du y
CHAPTER INTRODUCTION
of elementary geometry exi st e g Tarski the s e are still f ar to o complex to b e tackle d by
exi sting theorem proving technique s
The nee d to employ such axiom systems can b e avoide d by employing the metho ds of analytic
geometry Line s and regions can then b e repre s ente d in terms of formulae compr i s ing p olynomial
equations and inequalitie s relating the Carte s ian co ordinate s of p oints If such an approach i s to
b e e ective the logical form of the s e formulae must b e s everely re str icte d normally one s imply has
a s et of equations inequalitie s which i s implicitly taken as a conjunction in which all var iable s are
universally quanti e d Under the s e re str ictions di sjunctive information cannot b e repre s ente d nor
i s it p oss ible to sp ecify relationships involving more subtle quanti cational structure Surpr i s ingly
if one intro duce s Bo olean op erators and arbitrary quanti cation the re sulting language known as
the Tarski language do e s actually remain decidable by means of a quanti er elimination metho d
Tarski Algor ithms for quanti er elimination in the Tarski language have b een the sub ject
of cons iderable inve stigation Collins Ar non Cavine ss and Johnson Mi shra
and although the general problem i s intractable pro ce dure s have b een found which are e ective
for large class e s of formulae
Tractability and Decidability
The ma jor problem in developing a us eful formalism for reasoning ab out spatial information indee d
for any domain i s the trade o b etween expre ss ive p ower and computational tractability Whilst
Egenhofer s repre s entation do e s allow for certain inference s to b e compute d e ectively the scop e
of the theory i s limite d On the other hand although the formalism pre s ente d in Randell Cui and
Cohn i s very expre ss ive s ince it i s pre s ente d in st order logic reasoning in the calculus i s
extremely di cult however the us e of pre calculate d comp os ition table s for relations de nable in
the theory do e s enable certain kinds of inference to b e compute d e ciently
It i s common in computer science to equate tractability with p olynomial time computability
But to a logician thi s will probably s eem an overly harsh re str iction s ince pro of pro ce dure s in
nearly all intere sting logics are at least exp onentially hard In thi s the s i s I shall b e pr imar ily
concer ne d with nding decidable repre s entations for spatial information but we shall s ee in chapter
that by re str icting the range of spatial relations which may b e repre s ente d to a class including all
the RCC relations illustrate d in g a p olynomial time reasoning algor ithm can b e obtaine d
The Content of thi s The s i s
The pr incipal aim of thi s the s i s i s to inve stigate f rameworks for repre s enting spatial information
that are b oth expre ss ive enough to b e us eful for solving real problems and are in some s ens e
tractable I fo cus on top ological relationships which I cons ider to b e the most fundamental of
spatial concepts but I also examine the non top ological prop erty of convexity The re st of the
the s i s i s organi s e d into the following chapters
CHAPTER INTRODUCTION
Axiomatic Theor ie s of Spatial Regions
In the next chapter I survey previously prop os e d theor ie s of spatial regions I rst give a br ief
de scr iption of class ical p oint s et top ology in which regions are treate d as s ets of p oints I then
cons ider theor ie s in which regions are taken as bas ic entitie s The earlie st of the s e are the systems
of Le sniewski and of Whitehead put forward at the b eginning of thi s the th century Also
covere d are the theor ie s of Tarski and Clarke I go on to de scr ib e in some detail the
more recent theory of Randell Cui and Cohn the RCC theory which i s a mo di cation of
Clarke s calculus and was formulate d with computational applications sp eci cally in mind
Analys i s of the RCC Theory
The RCC theory i s now inve stigate d in some detail I examine the axiom s et and sugge st certain
mo di cations which s eem to b e require d Mo dels of the theory in terms of class ical p oint s et
top ology are given and the p oss ibility of constructing a complete theory i s cons idere d I obs erve
that no adequate st order theory can b e e ither complete or decidable I sugge st a new theory
constructe d so as to avoid certain technical problems ar i s ing in the or iginal RCC theory
A Order Repre s entation
Since st order theor ie s such as RCC are undecidable they cannot b e us e d as a bas i s for e ective
reasoning Thus the repre s entation language or language s us e d in a spatial reasoning system
must b e more re str icte d in the ir expre ss ive p ower order logical calculi are normally regarde d as
propositional logics but as we shall s ee a spatial interpretation of expre ss ions of the s e formalisms
can b e given in which the non logical constants refer to spatial regions rather than prop os itions
Thi s idea i s intro duce d us ing the class ical prop os itional logic which can b e interprete d as a Bo olean
calculus of spatial regions The formali sm of class ical logic i s then augmente d to provide a language
C which i s capable of expre ss ing a cons iderably larger class of spatial f acts I give a deci s ion
pro ce dure for thi s language obtaine d by adding s imple meta level reasoning to the bas ic pro of
theory of class ical order logic
A Mo dal Repre s entation
Further extending the f ramework prop os e d in chapter I show how mo dal op erators can b e
interprete d so as to corre sp ond with further op erations on spatial regions which are nee de d to
capture more subtle di erence s b etween di erent spatial relationships Sp eci cally we shall s ee
how the op erator of the mo dal logic S can b e interprete d as a top ological inter ior op erator I
then give an enco ding for a large class of top ological relations also expre ss ible in RCC into an
augmente d form of the S language which I call S Thi s provide s a deci s ion pro ce dure for a
quite expre ss ive spatial language
CHAPTER INTRODUCTION
An Intuitionistic Repre s entation and its Complexity
Whilst the mo dal logic repre s entation of spatial reasoning exempli e s a general metho dology for
us ing order language s in knowle dge repre s entation its us e for any practical application would
require an e cient theorem prover for S In thi s chapter I de scr ib e the implementation of a spatial
reasoning system us ing a repre s entation in terms of order intuitioni stic formulae The core of
the system i s a Gentzen style s equent calculus which i s a re str iction of a well known rule s system
for the full order intuitioni stic calculus The intr ins ic complexity of reasoning algor ithms us ing
thi s intuitioni stic repre s entation has b een studie d by Neb el a Neb el lo oke d at reasoning
us ing a tableau metho d and has shown that the inference s nee de d for reasoning with the f ragment
of the logic nee de d to repre s ent a large class of spatial relations including in particular the bas ic
relations cons idere d in chapter can b e compute d with a p olynomial time algor ithm
Quanti er Elimination
In thi s short chapter I pre s ent a partial deci s ion pro ce dure for st order theor ie s of the connection
relation Thi s i s bas e d on the metho d of quanti er elimination Thi s technique can b e us e d as a
prepro ce ss ing step applie d to a re str icte d class of st order spatial formulae pr ior to translation
into the S or intuitioni stic enco dings
Convexity and Containment
The main re sults of the the s i s apply pr imar ily to the s igni cant but by no means comprehens ive
range of spatial relations de nable f rom the pr imitive relation of connecte dne ss However s imilar
metho ds can b e applie d to other asp ects of spatial reasoning and probably to other areas of know
le dge repre s entation In thi s chapter I explain how the technique s of order repre s entation can
b e extende d to handle non top ological information concer ning the convexity of regions Thi s illus
trate s metho ds by which the technique s given for e ective reasoning with top ological relationships
can b e extende d to handle non top ological information
Comp o s ition Bas e d Re asoning
In thi s chapter I lo ok at spatial reasoning bas e d on the notion of relational comp os ition I examine
the us e of composition tables to compute inference s and the ir relation to st order theor ie s I also
pre s ent a relation algebra formalism for top ological relations in which the role of the comp os ition
op eration i s much more prominent than in st order repre s entations
Further Work and Conclus ions
In the concluding chapter I evaluate the us efulne ss of the logical repre s entations and reasoning
systems pre s ente d in thi s the s i s I ass e ss the prosp ects for development of more expre ss ive repre s
entations for spatial reasoning which are computationally viable and lo ok at how spatial reasoning
might b e incorp orate d into more general reasoning systems Potential applications areas includ
CHAPTER INTRODUCTION
ing Geographical Information Systems GIS Rob ot Motion Planning and Computer Vi s ion are
cons idere d and I de scr ib e a prototyp e GIS with a limite d qualitative spatial reasoning capability
As sume d Background and Notations Employe d
In thi s the s i s I assume a knowle dge of class ical logic and s et theory I also make us e of e stabli she d
work in the areas of algebra mo del theory mo dal logic and intuitioni stic logic so acquaintance
with the s e elds will b e us eful Standard formal notations of logic and s et theory are employe d
Other notations will b e intro duce d and explaine d when require d
Chapter
Axiomatic Theor ie s
of Spatial Regions
Thi s chapter surveys in some detail a numb er of formal theor ie s of spatial regions First I
br ie y explain class ical p oint s et top ology in which regions are character i s e d as s ets of p oints
The re st of the chapter i s concer ne d with theor ie s in which extende d regions are treate d as
bas ic order entitie s Although some very eminent logicians have prop os e d and inve stigate d
region bas e d formali sms they are still f ar le ss well understo o d than p oint bas e d theor ie s The
following systems will b e de scr ib e d in some detail Le sniewski s Mereology Tarski s Geometry
of Solids Clarke s theory of the Connection relation and the Region Connection Calculus
RCC Several other formali sms will also b e cons idere d
Point Set Top ology
Class ical p oint s et top ology i s bas e d on s et theory The bas ic order elements of the theory are
p oints Regions are identi e d with s ets of p oints In developing the theory the pr inciple mathem
atical ob jects cons idere d are topological spaces The s e are s ets of elements p oints asso ciate d with
an auxiliary structure determining the top ological prop ertie s of the space A top ological space can
b e formally de ne d in a numb er of ways Perhaps the s imple st i s as a s et of s ets which include s
the empty s et and i s clos e d under arbitrary unions and nite inters ections Thi s i s the s et of open
s ets of the space The large st op en s et which i s the same as the union of all op en s ets i s calle d
the univers e of the top ology A top ology can thus b e repre s ente d by a structure T hU O i where
U i s the univers e and O i s the s et of op en s ets
In a top ological space T hU O i given an arbitrary subs et S of U the interior of S i s the
large st memb er of O that i s a subs et of S The inter ior function i on a top ology hU O i maps
every subs et of U to its inter ior a memb er of O Becaus e of the conditions on the s et of op en
s ets i must sati sfy the axioms PSTi given b elow In PSTi U i s a meta symb ol referr ing
to whatever i s the universal s et of the top ological space under cons ideration i e for top ology
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
1
T hU O i we have U U X and Y are any subs ets of the univers e
PSTi i X X X
PSTi i i X i X
PSTi i U U
PSTi i X Y i X i Y
Given a s et U any function i that maps subs ets of U to subs ets of U and ob eys the ab ove
axioms determine s a unique top ology hU O i the elements of O are s imply thos e subs ets S of U
such that i S S Hence any top ology hU O i can b e alter natively character i s e d by a structure
hU ii where i i s an inter ior function
A s et i s calle d closed i it i s the complement of some op en s et The closure of a s et i s the
smalle st clos e d s et of which it i s a subs et The closure function c mapping arbitrary subs ets of
a space to the ir closure s must sati sfy the equations PSTc given b elow The s et of clos e d s ets
of a space or the closure function c can each b e us e d as further alter native ways of sp ecifying
i X and the top ology of a space Inter ior and closure functions are inter de nable c X
i X c X Here and throughout the s equel X i s the complement of X w r t the univers e
PSTc X c X c X
PSTc c c X c X
PSTc c
PSTc c X Y c X c Y
As was mentione d in s ection the language of s et theory in which p oint s et top ology and
many other mathematical theor ie s are formulate d i s highly intractable Hence thi s formali sm i s
not well suite d to computational applications Neverthele ss it may b e p oss ible to nd us eful sub
language s of s et theory for which e ective reasoning pro ce dure s can b e constructe d In s ection
I shall de scr ib e a purely algebraic sub language of the formali sm of p oint s et top ology which i s
b oth decidable and quite expre ss ive
Be ing built directly on s et theory p oint s et top ology has an unambiguous s et theoretic s e
mantics Thi s make s it a us eful to ol for studying the mo del theory of other spatial language s In
the re st of thi s chapter and the following chapter I shall cons ider s everal theor ie s whos e s emantics
are not so well de ne d If it i s p oss ible to interpret such a language in p oint s et theoretic terms thi s
imme diately give s it a preci s e though indirect s emantics Hence such an interpretation can form
the bas i s for soundne ss and completene ss pro ofs The metho ds of top ological reasoning de scr ib e d
in chapters and are b oth justi e d in thi s way
1
In consider ing a single top ological space the symb ol U is not really nece ssary since we can always refer directly
to the universal s et However in chapters and I shall often us e U to make statements ab out class e s of algebras
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
The Or igins of Region Bas e d Theor ie s
The early years of the th century saw intens e activity in attempting to apply the metho ds of formal
logic and s et theory to mathematics and phys ics Russ ell s epi stemology and ideas ab out logical
pr imitive s were very in uential at that time and Whitehead s b o ok Concept of Nature Whitehead
prop os e d a view of phys ics and geometry which i s a radical revi s ion of traditional conceptions
he sought to found the s e di scipline s on sense data which according to Russ ell can b e the
only referents of truly pr imitive terms To de scr ib e the spatial asp ects of s ens e data Whitehead
prop os e d the construction of a geometry in which spatial regions rather than p oints would b e the
bas ic entitie s Sens e data could then b e said to o ccupy spatial regions whereas p oints would b e
abstract entitie s der ive d theoretically f rom regions
In hi s b o ok Process and Reality Whitehead sugge ste d that a general theory of ob jects
events and pro ce ss e s could b e develop e d bas e d on the pr imitive relation of connectedness and
he sp eci e d a large numb er of logical prop ertie s of thi s pr imitive Since the only well develop e d
phys ical theor ie s are formulate d in terms of var iable s ranging over p oints in space and time
Whitehead prop os e s the metho d of extensive abstraction intro duce d in the earlier work White
head as a means of constructing p oints f rom regions of space or space time The idea i s
to de ne a p oint in terms of certain in nitely ne ste d s ets of regions a s imilar approach to char
acter i s ing p oints in terms of regions has b een followe d by Clarke and i s de scr ib e d b elow in
s ection
Nico d s do ctoral the s i s Geometry in the Sensible World develop e d Whitehead s approach
2
in a numb er of directions Nico d adopte d and mo di e d Whitehead s metho d of extens ive abstrac
tion for the construction of p oints f rom regions He also prop os e d some highly or iginal approache s
to constructing geometr ical systems f rom a phenomenological standp oint One of the s e i s a char
acter i sation of geometry f rom the p oint of view of a b e ing equipp e d only with a kinae sthetic s ens e
of its own movement in space Another take s into account the viewp oint and p ersp ective of an
obs erver in de scr ibing geometr ical entitie s It i s also intere sting to note that the chapter of the
the s i s on Temp oral Relations and the Hyp othe s i s of Durations contains a di scuss ion of temp oral
relationships b etween intervals and prop os e s a class i cation which i s e ss entially the same as that
adopte d much later by Allen Another logician in uence d by Whitehead was Theo dore de
Laguna who gave a theory of the geometry of solids Thi s will b e br ie y de scr ib e d in s ection
Contemp orary with the inve stigations of Whitehead and hi s followers the Poli sh logician and
philosopher Stani slav Le sniewski was conducting an extens ive enquiry into ontology and logical
repre s entations He was particularly concer ne d with character i s ing the part whole relation b etween
ob jects and was cr itical of the s et theoretic treatment of thi s relationship Hi s theory was intende d
to de scr ib e entitie s of any kind but in thi s chapter I shall only b e concer ne d with its application
to spatial regions
2
Russ ell regarde d Nico d as p otentially one of the greate st logicians of the th century and lo oke d to him in
particular to carry forward the pro ject of founding logical theor ie s of the physical world on the basis of s ens e data
Tragically Nico d die d prematurely so on after the publication of his do ctoral the sis
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Le sniewski s Mereology
Mereology a formal theory of the part whole relation was or iginally pre s ente d by Le sniewski
in hi s own logical calculus which he calle d Ontology Thi s calculus i s bas e d on pr inciple s
which are rather di erent f rom thos e of the standard pre dicate calculus The pr incipal di stinctive
feature s of Ontology are rstly that terms do not nece ssar ily denote a s ingle ob ject they may refer
to nothing a unique individual or any numb er of di stinct individuals and s econdly quanti cation
i s not asso ciate d with exi stential commitment it has a more substitutional avour For certain
purp os e s Le sniewski s Ontology has di stinct advantage s over standard logic For example in the
spatial domain one may wi sh to employ a function the region of inters ection of x and y but thi s i s
a partial function s ince if x and y are di sjoint no such region exi sts In standard logic terms always
denote a unique individual so partial functions are not legitimate but in Ontology such functions
3
pre s ent no problem A full de scr iption of Le sniewski s Ontology i s b eyond the scop e of thi s the s i s
s ee Simons for a detaile d account However the content of the theory of Mereology i s not
b ound to the form in which it was initially state d Hence I now pre s ent a formulation of Mereology
due to Tarski state d in standard class ical logic
Mereology i s built on the s ingle pr imitive relation P x y whos e interpretation i s that x i s a
part of y In terms of thi s the relations of prop er part PP and di sjointne ss DJ are de ne d
as well as SUM which i s a relation b etween a s et of individuals and an individual I shall us e small
Roman letters for var iable s ranging over individuals and small Greek letters for var iable s ranging
over s ets of individuals The de nitions can then b e given formally as
Mdef PP x y P x y x y
def
Mdef DJ x y z P z x P z y
def
Mdef SUM x y y P y x
def
z P z x y y DJ y z
In addition to the usual pr inciple s of class ical logic and the theory of s ets the system i s require d
to sati sfy the following sp eci cally mereological p ostulate s
Mp o st x y z P x y P y z P x z
Mp o st x x x SUM x
The s e ensure rstly that the part relation i s trans itive and s econdly and rather controvers ially
that for any non empty s et of individuals there i s a unique individual which i s the sum of that s et
Pro ofs of a numb er of theorems der ivable f rom the s e axioms e g that P i s re exive can b e found
in Wo o dger App endix E A short coming of the theory of mereology bas e d as it i s on
the part relation i s that no di stinction can b e made b etween the relations of connecte dne ss and
overlapping if two regions do not overlap they are s imply di screte
3
Alternative formalisms in which partial functions can be handle d are sorted classical logic s ee e g Cohn
and free logic Bencivenga
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Other Mereological Systems
A numb er of theor ie s have b een develop e d which contain mereological pr imitive s equivalent or
s imilar to Le sne iwski s Wo o dger s The Axiomatic Method in Biology us e s the theory
exactly as given ab ove Leonard and Go o dman Leonard and Go o dman devi s e d a formali sm
which they calle d the calculus of individuals bas e d up on a pre dicate which holds when two
individuals are di screte Thi s system i s e ss entially the same as Le sne iwski s but us e s di erent
notation and contains many additional de nitions The theory i s applie d to a numb er of problems
involving relations b etween individuals groups and ens emble s that cannot b e handle d by ordinary
quanti cation Thi s formali sm also app ears in Go o dman s b o ok The Structure of Appearance
which prop os e s an approach to formal de scr iption of the world bas e d on pr inciple s of
logical nominalism a reluctance to admit the exi stence of abstract entitie s such as s ets
Tarski s Geometry of Solids
Building on Le sne iwski s mereology by intro ducing a new sphere pr imitive Tarski gave a
4
theory of the geometry of solids which i s emb e dde d by means of de nitions into an axiomat
i sation of elementary Euclidean geometry such as that given in Tarski
Tarski starts by p ostulating a domain of spheres I us e the pre dicate SPH x to mean x i s
a sphere over which he de ne s the relations of external tangency ET internal tangency IT
external diametricity ED internal diametricity ID and concentricity CONC ED a b c holds
when a and b are exter nally tangent to c and touch diametr ically opp os ite p oints on c s b oundary
The s e relations are de ne d as follows
SGdef ET a b SPH a SPH b DJ a b
def
x y P a x P a y DJ b x DJ b y P x y P y x
SGdef IT a b SPH a SPH b PP a b
def
x y P a x P a y P x b P y b P x y P y x
SGdef ED a b c SPH a SPH b ET a c ET b c
def
x y DJ x c DJ y c P a x P b y DJ x y
SGdef ID a b c SPH a SPH b SPH c IT a c IT b c
def
x y DJ x c DJ y c ET a x ET b y DJ x y
SGdef CONC a b SPH a SPH b a b
def
PP a b x y ED x y a IT x b IT y b
ID x y b
PP b a x y ED x y b IT x a IT y a
ID x y a
The next step in Tarski s formulation i s to constrain the theory to b e compatible with Euclidean
geometry To do thi s he de ne s the notions of point and equidistance which can s erve as the only
4
As mentione d in s ection this is perhaps better thought of as a theory of volume s since the entitie s of the
theory are allowe d to inter penetrate each other and the property of solidity is not considere d The same applie s to
de Laguna s theory which will be de scr ibe d in s ection
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
5
pr imitive s in such a theory of geometry Both the s e concepts can b e de ne d in terms of the
relations de ne d ab ove
SGdef A p oint i s de ne d as the s et of all sphere s concentr ic with a given sphere
POINT x x y y CONC x y
def
SGdef Equidi stance of two p oints f rom a third ab bc i s de ne d as follows
ab bc x x b y y a y c P y x DJ y x
def
Identi cation with the corre sp onding notions in Euclidean geometry i s then achieve d by the
following p ostulate
SGp o st The notions of p oint and of equidi stance of two p oints f rom a third sati sfy all the
p ostulate s of ordinary Euclidean geometry of three dimens ions
Having xe d the structure of the s et of p oints we still nee d to sp ecify how solids are relate d
to thi s structure
6
SGdef A solid i s an arbitrary sum of sphere s
SOLID x X SUM X x y y X SPH y
def
SGdef The p oint i s inter ior to the solid a
INTER a x x P x a
def
We now correlate the s et of inter ior p oints of a solid with the geometr ically de nable concept
of a regular open set of p oints To do thi s I de ne inter ior int and closure cl functions on s ets
of p oints capital Greek letters The de nitions us e the relation xy y z which i s de nable f rom
xy y z s ee app endix A In the usual top ology of Euclidean space the inter ior p oints of a s et
are thos e that can b e surrounde d by an op en ball all of whos e p oints are within the s et Thi s i s
the bas i s of the following de nitions
SGdef int x x y y x z z x xy z
def
SGdef cl x x y y x z z x xy z
def
SGdef ROPEN int cl
def
The next two p ostulate s stipulate that the inter ior p oints of solids are to b e identi e d with
regular op en s ets of p oints
5
Tarski s own formulation of elementary geometry which is given in appendix A employs equidistance and
betweenness as pr imitive relations but betweenne ss can in fact be de ne d in terms of equidistance so with the
addition of such a de nition that axiom s et could be us e d
6
In fact in Tarski s theory all order entitie s are solids so this pre dicate de nition could be replace d with a
universal axiom
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
SGp o st If x i s a solid then the class of all inter ior p oints of x i s a non empty regular op en
s et
x SOLID x f j INTER x g ROPEN
SGp o st If a class of p oints i s a non empty regular op en s et there exi sts a solid x such
that i s the class of all its inter ior p oints
ROPEN x SOLID x f j INTER x g
The s e two p ostulate s ensure a one to one corre sp ondence b etween solids and non empty regular
op en s ets of p oints Thus the categor ical axioms of elementary geometry which x the structure of
the domain of p oints are us e d to determine the structure of the domain of solids
Finally the mereological part relation P must b e xe d in terms of p oint geometry by identifying
it with s et inclus ion among the s ets of inter ior p oints asso ciate d with solids
SGp o st If a and b are solids and all the inter ior p oints of a are at the same time inter ior to
b then a i s part of b
INTER a INTER b P a b
As a logical foundation for a conceptual scheme Tarski s theory has the great mer it of b e ing
categorical which means that all its mo dels are i somorphic Hence the theory can b e regarde d as
completely xing the meanings of all the concepts covere d by its vo cabulary However the theory i s
only made categor ical by indirect means rstly the notions of point equidistance and betweenness
are intro duce d by a s er ie s of de nitions then it i s stipulate d that the s e de ne d concepts ob ey the
axioms of Euclidean geometry Tarski He admits that the re sulting system i s not ideal
The p ostulate system given ab ove i s f ar f rom s imple and elegant it s eems very likely that thi s
p ostulate system can b e e ss entially s impli e d by us ing intr ins ic prop ertie s of the geometry of
solids Tarski
What make s Tarski s system so unwieldy as a to ol for actually reasoning ab out spatial regions
i s the hidden complexity involve d in SGdef and SGp o st The s e br ing in the whole of Euclidean
geometry as a means of xing the structure of the space of regions Reasoning with the axioms
of elementary geometry i s in its elf very hard although it i s known to b e decidable no e ective
7
general reasoning metho d i s known for thi s system but in thi s context the complexity i s f ar wors e
b ecaus e the p oints constraine d by the Euclidean geometr ical axioms corre sp ond to sets of sphere s
in the solid geometry Thus if p oints were eliminate d f rom the system by unpacking the Euclidean
axioms in terms of the de nition of p oint the re sulting formali sm would b e an enormously complex
nd order theory
7
Reasoning in elementary geometry can be carr ie d out by translating geometr ic relations into algebraic p olynomial
equations and inequalities constraining the Carte sian co ordinate s of p oints Consistency of such equations can be
te ste d using a decision pro ce dure also due to Tarski
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Relation Interpretation De nition of R x y
DC x y x i s di sconnecte d f rom y C x y
P x y x i s a part of y z C z x C z y
PP x y x i s a prop er part of y P x y P y x
O x y x overlaps y z P z x P z y
DR x y x i s di screte f rom y O x y
EC x y x i s exter nally connecte d to y C x y O x y
TP x y x i s a tangential part of y P x y z EC z x EC z y
NTP x y x i s a nontangential part of y P x y z EC z x EC z y
Table De ne d relations in Clarke s theory
Clarke s Theory
The formali sm develop e d by Clarke i s an attempt to construct a system more ex
pre ss ive than that of Leonard and Go o dman and s impler than that of Tarski bas e d
on the pr imitive relation of connecte dne ss us e d by Whitehead The domain of the theory
i s spatial or spatio temp oral regions and the C pr imitive i s constraine d to ob ey the following two
axioms
Cax x C x x y C x y C y x
Cax x y z C z x C z y x y
The rst of the s e ensure s the relation i s re exive and symmetr ic whilst the s econd i s an axiom of
extensionality which state s that if two regions are connecte d to exactly the same other regions then
they must b e the same From the C relation Clarke de ne s s everal other us eful spatial relations
The s e are given in table
Fus ions and Quas i Bo ole an Op erators
A fus ion op erator f i s then de ne d as follows
Cdef x f X y C y x z z X C y z
def
Thi s means that the fus ion of a s et of regions i s that region which i s connecte d to all and only
thos e regions that are connecte d to at least one region in the s et The intende d interpretation of
f x may b e regarde d as the same as Le sniewski s SUM x although the latter i s de ne d
in terms of P rather than C
The theory also contains an axiom ensur ing that for every non empty s et of regions a fus ion
region exi sts
Cax X X x x f X
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Thi s axiom would b e very o dd in a completely standard st order theory s ince in such a theory it
i s normally assume d that all well forme d terms denote an exi sting individual all functions b e ing
unique and total Clarke however intro duce s a slight mo di cation into the logical interpretation
of quanti cation in hi s theory Sp eci cally the rule of universal instantiation which normally
allows one to replace a universally quanti e d var iable by any ground term i s re str icte d so that one
can only replace the var iable by e ither an individual constant or a complex term for which it i s
8
provable that x x
Clarke then de ne s functions s imilar to Bo olean op erators as follows
Cdef sum x y f fz j P z x P z y g
def
Cdef pro d x y f fz j P z x P z y g
def
Cdef compl x f fy j C y x g
def
The de nition of compl entails that every region i s di sconnecte d f rom its own complement
x C x compl x Ccompl
The pr inciple Ccompl i s cons i stent with an interpretation of regions as arbitrary p oint s ets compl
as s et complement and C x y as true when x and y share a p oint However if one i s intere ste d in
e stabli shing a naturali stic theory of regions one might prefer the complement function to b e such
that regions always connect with but do not overlap the ir complements
Top ological Functions
Clarke i s now able to de ne the top ological op erators of inter ior closure and exter ior as functions
f rom regions to regions
Cdef i x f fy j NTP y x g
def
Cdef c x f fy j C y i compl x g
def
Cdef ex x f fy j NTP y compl x g
def
An additional axiom i s concer ning the s e top ological functions i s given by Clarke as follows
Cax x z NTP z x
y z C z x O z x C z y O z y C z pro d x y O z pro d x y
It i s provable that the condition z C x z C y z i s equivalent to NTP x y and also that
NTP x x x i x Thus thi s axiom ass erts rstly that every region has a non tangential
part and s econdly that the pro duct of two op en regions i s its elf op en
8
This re str iction may be regarde d as enforcing a rudimentary sort theory quanti ers range over a sort region
and all individual constants refer to entitie s of this sort However functions such as f may have as the ir value
e ither a region or an entity whos e sort which we may call nul l is disjoint f rom region
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Points
Clarke subs equently extende d hi s or iginal theory of spatial regions by the intro duction of
points The s e are not bas ic entitie s of the system but are identi e d with certain s ets of regions
Thi s i s e ss entially the metho d of extensive abstraction rst prop os e d by Whitehead and taken up
by Nico d and de Laguna Clarke stipulate s that a s et of regions i s a p oint which we note as
PT i it sati s e s the following conditions
Cp oint x y x y C x y
Cp oint x y x y O x y pro d x y
Cp oint x y x P x y y
Cp oint x y sum x y x y
He further require s that any pair of connecte d regions must share at least one p oint
Cp oint x y C x y PT x y
The notion of a p oint s b e ing incident in a region i s de ne d s imply as
IN x PT x
def
so that p oint i s identi e d with the s et of regions in which it i s incident
A numb er of problems ar i s e f rom Clarke s treatment of p oints One i s that Cp oint i s intuit
ively f als e if a p oint i s incident in two overlapping regions thi s do e s not nece ssar ily imply that it
i s incident in the ir pro duct the regions might b e exter nally connecte d at one or more p oints that
are not incident in the region of overlap A further problem note d by Biacino and Gerla
i s that thi s treatment of p oints leads to a collaps e of C to O b ecaus e every pair of connecte d regions
must also overlap The pro of which do e s not dep end on the di scre dite d Cp oint i s as follows
pro of Supp os e C a b then f rom Cp oint we have a b Now cons ider
the region r sum compl a compl b Supp os e r i s equal to the univers e From
Cp oint we can der ive that every p oint incident in some region i s incident in the
9
univers e so the p oint must b e incident in sum compl a compl a By Cp oint
thi s means that e ither compl a or compl b so s ince a b we
have e ither a compl a or b compl b Cp oint then require s
that e ither C a compl a or C b compl b and b oth the s e alter native s contradict the
Ccompl pr inciple Thus z cannot equal the univers e Thi s means that there exi sts a
region w such that w compl r compl sum compl a compl b w must b e part
of b oth a and b So we can conclude that O a b
Thus Clarke s intro duction of p oints has the unintende d cons equence that connection i s s imply
equivalent to overlap The domain of the theory i s then e ss entially a Bo olean algebra with the null
9
The p ossibility of an empty p oint not incident in any region doe s not appear to be rule d out by Cp oint
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
element remove d and the top ology of regions i s di screte It i s apparent that the Ccompl pr inciple
i s instrumental in the collaps e and thi s must cast further doubt on the de nition of compl One
way to avoid the s e problems would b e to us e the following alter native de nition of complement
compl x f fy j O y x g
def
However thi s would render incorrect the de nitions of the top ological functions and it i s doubtful
whether such functions could b e re intro duce d even by mo di e d de nitions The theory would then
b ecome more like the RCC theory de scr ib e d in the next s ection In the RCC theory di stinctions
b etween op en and clos e d regions are not expre ss ible
The Region Connection Calulus RCC
With the intention of providing a logical f ramework for the incorp oration of spatial reasoning into
AI systems Clarke s formali sm was inve stigate d and mo di e d in the works Randell and Cohn
and Randell A more radical re working of the theory was pre s ente d in Randell
Cui and Cohn and it i s thi s vers ion which i s de scr ib e d here The new theory i s known as
the Region Connection Calculus RCC The re s earch rep orte d in thi s the s i s has b een very much
in uence d by thi s theory
Like Clarke s theory RCC i s bas e d on a pr imitive connecte dne ss relation C x y and the
univers e of quanti cation i s intende d to b e a domain of spatial regions The relation C x y i s
re exive and symmetr ic which i s ensure d by the following two axioms
RCC xC x x Cref
RCC xy C x y C y x Csym
Relation Interpretation De nition of R x y
DC x y x i s di sconnecte d f rom y C x y
P x y x i s a part of y z C z x C z y
PP x y x i s a prop er part of y P x y P y x
EQ x y x i s identical with y P x y P y x
O x y x overlaps y z P z x P z y
DR x y x i s di screte f rom y O x y
PO x y x partially overlaps y O x y P x y P y x
EC x y x i s exter nally connecte d to y C x y O x y
TPP x y x i s a tangential prop er part of y PP x y z EC z x EC z y
NTPP x y x i s a nontangential prop er part of y PP x y z EC z x EC z y
Table De ne d relations in the RCC theory
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Us ing C x y further dyadic relations are de ne d as shown in table The relations P
PP TPPand NTPP b e ing non symmetr ical supp ort invers e s For the invers e s the notation i
i s us e d where fP PP TPP NTPPg The s e relations are de ne d by de nitions of the form
i x y y x Of the de ne d relations DC EC PO EQ TPP NTPP TPPi and NTPPi
def
10
have b een proven to form a JEPD s et Randell Cohn and Cui a Thi s s et i s known as
RCC As the s et i s JEPD any two regions stand in exactly one of the s e e ight relations
It can b e s een that the RCC de nitions are almost the same as thos e of Clarke The new relations
PO TPP and NTPP have b een intro duce d in order to partition all p oss ible binary relations into a
JEPD s et The relation TP include s EQ as a sp ecial cas e and the universal region i s b oth equal
to and an NTP of its elf Also the de ne d relation EQ take s the place of the logical equality
us e d by Clarke Cons equence s of thi s change will b e examine d in s ection in the next chapter
Functional Extens ion of the Bas ic Theory
RCC also incorp orate s a numb er of functions on regions as well as a constant denoting the universal
region The functions are calle d quas i Bo olean s ince they are intende d to generate an algebra
very s imilar to a standard Bo olean algebra but with no least element i e no null region The
functions are sp eci e d as follows
u y z C z y
def
sum x y z w C z w C w x C w y
def
compl x y z C z y NTPP z x O z y P z x
def
pro d x y z u C u z v P v x P v y C u v
def
di x y w z C z w C z pro d x compl y
def
x y y x means x x x More will b e said ab out the s e functions and where
def
thi s form of de nition in s ection
The Sorte d Logic LLAMA
It i s imp ortant to note that all the quas i Bo olean functions except for sum are partial with re sp ect
to the domain of regions Thi s give s r i s e to a technical problem in that the standard pro of theory
and s emantics of st order logic i s bas e d on an assumption that all function symb ols corre sp ond to
total functions To avoid thi s di culty Randell Cui and Cohn employ the sorted st order
11
logic LLAMA as de scr ib e d by Cohn
The sorte d logic allows the domain of di scours e to b e partitione d into a numb er of bas e sorts
each cons i sting of a non empty s et of entitie s of a particular kind For each relation symb ol in
the vo cabulary of a theory certain combinations of argument sorts are sp eci e d When the relation
i s combine d with arguments whos e sorts accord with one of the s e combinations the re sulting
10
See s ection
11
Logic Lacking A Meaningful Acronym
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
prop os ition i s said to b e wel l sorted if the argument sorts do not agree with the sp eci cation the
prop os ition i s il l sorted Likewi s e it i s sp eci e d that application of function to a tuple of arguments
will give a well sorte d term only for certain sort combinations of the s e arguments Every function
application will also have a result sort which i s the sort of the entity denote d by the term forme d
by that application In general the re sult sort will b e an arbitrary extens ional function of the
sorts of the arguments given to a function A Bo olean combination of prop os itions i s well sorte d
i all its constituents are well sorte d Thi s give s us a general notion of a well sorte d quanti er f ree
12
formula
In the LLAMA formali sm quanti ers and var iable s are not thems elve s asso ciate d with any
sort re str iction rather the range of any particular quanti cation i s determine d by the context of
var iable s as arguments of sorte d functions and relations Supp os e a pre dicate i s forme d by replacing
one or more terms in a formula with a new var iable symb ol If a quanti er i s then applie d to the
pre dicate the quanti er range s over all entitie s in the domain which are such that if a constant
denoting that entity were substitute d in the pre dicate in place of each o ccurrence of the quanti e d
var iable the re sulting formula would b e well sorte d If the domain of p oss ible well sorte d value s i s
empty then the entire formula i s ill sorte d and cons idere d not to b e a well forme d formula of the
language
In the cas e where multiple quanti ers o ccur in a formula the s ituation i s more complex Here
the interpretation cannot b e analys e d in terms of succe ss ive applications of a s ingle quanti cation
op eration rather multiple quanti ers s erve to quantify over all s equence s of individuals such that
the formula when instantiate d with thi s s equence i s a well sorte d ground formula Thus in a
formula x y x y quanti cation can b e regarde d as b e ing over all pairs of entitie s ha bi such
that the formula a b i s well sorte d Thi s treatment of quanti cation applie s directly only to
prenex formulae with all quanti ers at the f ront but any formula can b e transforme d into an
equivalent prenex formula and the range s of quanti cation determine d f rom thi s
A further feature of LLAMA which make s it particularly expre ss ive i s that for each sort there
i s a sortal predicate The s e pre dicate s can b e us e d to sp ecify explicit sortal re str ictions on var iable s
in a formula in addition to thos e determine d f rom the sorts of the ordinary relations and functions
Sorts in the RCC Theory
In cons ider ing the purely spatial asp ects of the RCC theory we may assume that there are just
two di sjoint and non empty bas e sorts REGION and NULL plus the top sort thi s i s the join
of REGION and NULL all entitie s are of thi s sort and the b ottom sort no entity i s of thi s
13
sort We now declare that the arguments of all relations in the RCC theory are of sort REGION
12
Note that this character isation give s us a sorte d logic which is polymorphic This means that the permitte d
sorts of argument place s are not individually re str icte d but may depend on other arguments e g a pre dicate
SPOUSE x y might be allowe d to have arguments of sorts hmale f emalei or hf emale malei but not hmale malei
or hf emale f emalei Likewis e the re sult sort of a function can vary depending on its arguments
13
It is intende d that the theory be embe dde d in a more comprehensive formalism incorp orating temp oral intervals
and physical ob jects as well as spatial regions This theory would make us e of a much r icher sort structure
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
and the arguments and retur n value s of all the quas i Bo olean functions are of sort
Two Additional Axioms
Making us e of the sorte d f ramework a further axiom i s given which links the quas i Bo olean functions
to the relational part of the theory The axiom state s that the pro duct of two regions i s null if and
only if the two regions are di screte i e non overlapping
x y NULL pro d x y DR x y
Finally an exi stential axiom ensure s that every region has a non tangential prop er part
x y NTPP y x NTPP
The NTPP axiom rule s out the p oss ibility of atomic mo dels of the theory in which there i s
a class of regions atoms which have no prop er parts Several p oss ibilitie s are cons idere d for
mo difying the theory so as to allow the exi stence of atoms The s e will b e cons idere d in s ection
Further Development of RCC
As well as mo difying certain axioms of Clarke s theory Randell Cui and Cohn develop the ir
new theory so as to cover further non top ological information They intro duce a new convex hul l
function which enable s prop ertie s involving convexity and containment to b e repre s ente d I shall
examine thi s op erator in chapter The theory i s also extende d so as to de scr ib e p oss ible mo de s of
continuous change which can o ccur in spatial con gurations Thi s i s done by identifying p oss ible
trans itions which can o ccur amongst the top ological relations holding b etween the regions o ccupie d
by b o die s dur ing some continuous pro ce ss I shall comment on thi s in s ection
Other Relevant Work on Region Bas e d Theor ie s
I conclude thi s chapter by br ie y mentioning a numb er of other works which are relevant to the
study of region bas e d theor ie s of space
de Laguna s Theory
In s ection I referre d to de Laguna s geometry of solids Thi s theory i s bas e d on the
pr imitive relation x can connect y and z I shall wr ite thi s as CC x y z Thi s relation i s true
if it would b e p oss ible by di splacement and or rotation to br ing x in to such a p os ition that
it connects i e touche s or overlaps b oth y and z The CC pr imitive i s extremely expre ss ive
s ince it allows de nitions of b oth connecte dne ss C x y z CC z x y and relative length
def
Longer x y z w CC y z w CC x z w Unfortunately thi s theory do e s not s eem to
def
have b een explore d or develop e d by any subs equent re s earcher in the eld
CHAPTER AXIOMATIC THEORIES OF SPATIAL REGIONS
Grzegorczyk s Undecidability Re sults
Grzegorczyk s pap er Undecidability of some Topological Theories Grzegorczyk contains
s everal imp ortant and very general re sults ab out the undecidability of certain kinds of spatial
theory Thi s quite technical pap er s eems to b e rarely cite d by later re s earcher s and came to
14
my attention at a very late stage of my work on thi s the s i s Although f rame d in terms of
somewhat di erent formal apparatus f rom that found in the other spatial theor ie s surveye d in thi s
chapter Grzegorczyk s undecidability re sults app ear to apply assuming appropr iate notational
mo di cations to a very wide range of p oss ible spatial theor ie s The nature and rami cations of
the s e re sults will b e cons idere d in s ection
Some Recent Re s e arch in the Field
The formalism of Bo chman i s a s igni cant departure f rom all the others mentione d A
pr incipal feature i s that two typ e s of bas ic mereological element are p ostulate d ob jects and
connections The part relation i s pr imitive Ob jects can have other ob jects and or connections
as parts whereas connections are atomic having only thems elve s as parts A connection relation
i s then de ne d by saying that ob jects a and b are connecte d just in cas e there exi sts a connection
such that every ob ject of which i s a part also share s a part with a and a part with b
A survey by Gerla covers most of the formali sms de scr ib e d in thi s chapter but cons iders
them f rom a rather di erent p ersp ective fo cus ing on the ir corre sp ondence to certain kinds of
class ical top ological space
A mo di cation and development of Clarke s theory i s prop os e d by Asher and Vieu who
give an axiom s et bas e d on the C pr imitive which i s prove d to b e sound and complete with re sp ect
to a class of mo dels bas e d on p oint s et top ology A novel prop erty of thi s theory i s the de nability
of a relation of weak contact which i s supp os e d to hold when two b o die s touch each other but
are not phys ically joine d
Borgo Guar ino and Masolo give a theory of spatial regions bas e d on three pr imitive
concepts the part relation the prop erty of b e ing a top ologically s imple region and the binary
relation of congruence Thi s theory combine s asp ects of the connection bas e d theor ie s der ive d f rom
Clarke with the approach taken in Tarski s Geometry of Solids whereby the logic of regions can
b e tie d by means of de nitions to the class ical Euclidean geometry of p oints
Re sults of Pratt and Scho op concer ning a complete axiomatic theory of the D Euclidean
plane are of direct relevance to thi s the s i s particularly the next chapter but the ir pap er was
publi she d to o recently to b e fully cons idere d in the pre s ent work However I shall make some
comments in s ection
Antother recent pap er by Stell and Worb oys cons iders the structure of s ets of regions
in terms of Heyting algebras Thi s work i s clos ely relate d to the approach I shall de scr ib e in
chapters and e sp ecially where I us e the intuitioni stic logic I to repre s ent top ological
relations I can also b e interprete d in terms of Heyting algebras
14
Thanks to Nick Gotts
Chapter
Analys i s of the RCC Theory
In thi s chapter I examine in more detail the st order RCC theory de scr ib e d in s ection I
start with a cr itique of its axioms The cons equence s of st order axioms are often f ar f rom
obvious so some new meta level notations are develop e d to f acilitate analys i s of the theory
Us ing the s e to ols I inve stigate the structure s of p oss ible mo dels of the axioms I go on to
sugge st an alter native axiom system which i s in s everal re sp ects eas ier to manipulate than the
or iginal theory At the end of the chapter I sp ecify a partial deci s ion pro ce dure for the revi s e d
theory bas e d on the metho d of quanti er eliminatio n
RCC in Relation to thi s The s i s
Although the RCC theory was intende d as a language for b oth repre s enting and reasoning ab out
spatial information in its initial development repre s entation was the pr imary fo cus It was so on
reali s e d that whilst the theory i s very expre ss ive reasoning with RCC i s extremely di cult My
re s earch has b een directe d towards addre ss ing thi s problem Quite early in my inve stigation of
the RCC reasoning problem I di scovere d a computationally feas ible metho d for reasoning ab out
certain spatial relationships Thi s do e s not make any us e of the actual RCC axiom system but
us e s a radically di erent formali sm to repre s ent and reason ab out a large class of spatial relations
all of which are also de nable in the RCC system The repre s entation bas e d on a top ological
interpretation of intuitioni stic logic i s de scr ib e d in detail in chapter
My intuitioni stic enco ding partly solve d the RCC reasoning problem but was only capable of
handling a small alb e it s igni cant subs et of the spatial relationships expre ss ible in RCC Thus the
p oss ibility of nding a much more comprehens ive reasoning algor ithm p oss ibly one that would
cover everything expre ss ible in RCC still remaine d Furthermore many puzzle s concer ning
the RCC formali sm b ecame apparent It was clear that the axioms did not character i s e a s ingle
unique mo del The intende d mo del was to accord with our intuitive na ve ideas ab out regions
of xe d dimens ion exi sting in a top ologically s imple space However the dimens ionality and
global top ology of the space was not xe d by the axioms Moreover the exi stential imp ort of the
theory app eare d to b e to o weak to determine exactly which con gurations of regions are p oss ible
CHAPTER ANALYSIS OF THE RCC THEORY
The que stion aros e as to whether RCC could b e extende d to yield an syntactically complete
theory s ee s ection with a unique denumerable mo del i e an categorical theory As well
as reme dying the repre s entational shortcomings of RCC such a theory would b e a very s igni cant
step towards solving the reasoning problem Thi s i s b ecaus e any syntactically complete st order
theory must b e decidable
Recent di scover ie s by Nicholas Gotts and mys elf strongly sugge st that thi s goal cannot b e
obtaine d Sp eci cally there can b e no complete st order character i sation of the intende d domain
Thi s can b e demonstrate d by showing that if such a formalisation were given it would provide
a complete theory and deci s ion pro ce dure for st order ar ithmetic which i s known to b e b oth
undecidable and not character i sable by any axiomatic theory G odel The demonstration
involve s showing that the concepts of ar ithmetic can b e de ne d in terms of spatial prop ertie s which
are also de nable in RCC Details of the pro of are b eyond the scop e of the pre s ent the s i s
Given that RCC i s undecidable and a complete st order character i sation of spatial regions i s
imp oss ible further enquiry into RCC can pro cee d in two directions Firstly it i s almost certain that
a complete character i sation of the intende d domain can b e given by adding one or more nd order
axioms and p erhaps also further st order axioms to the theory Secondly s ince a comprehens ive
reasoning algor ithm for the domain of RCC i s imp oss ible it will b e imp ortant to identify more
re str icte d language s for expre ss ing spatial information for which e ective algor ithms or at least
deci s ion pro ce dure s can b e constructe d
The RCC theory provide s a very expre ss ive language for sp ecifying spatial information
However there are certain feature s that are problematic In thi s chapter I attempt to clar ify a
numb er of asp ects of the theory and sugge st some mo di cations to its formali sation Sp eci cally I
cons ider extens ionality and identity conditions the status of the quas i Bo olean functions the sort
theory and the null region the NTPP axiom and mo dels of the theory I then pre s ent a revi s e d
axiom s et constructe d so as to avoid some of the main problems brought to light by the analys i s
In chapter I shall give a partial deci s ion pro ce dure for the new theory
Identity and Extens ionality
In contrast with the theory of Clarke the RCC theory contains no axiom of extens ionality In
thi s s ection I cons ider whether or not such an axiom ought to b e adde d to the theory
Axiomatic theor ie s particularly thos e which s eek to character i s e a s ingle pr imitive relation
often contain some kind of axiom of extensionality Thi s i s an axiom which ass erts that the identity
of any two ob jects follows f rom the ir indi scer nibility with re sp ect to some prop erty Thus in s et
theory we have
x y z z x z y x y
Such axioms can b e regarde d as strengthene d forms of Le ibniz pr inciple of the identity of in
discernibles Thi s pr inciple i s the left to r ight comp onent of a s econd order axiom which can b e
CHAPTER ANALYSIS OF THE RCC THEORY
regarde d as de ning identity
x y x y x y
Rather than requir ing ob jects to b e indi scer nible with re sp ect to all prop ertie s we may require
only that they cannot b e di stingui she d in terms of a f amily of prop ertie s forme d by partially
instantiating some relation over the univers e of ob jects The idea b ehind thi s sp eciali sation of the
axiom i s that thi s f amily of prop ertie s i s regarde d as xing all prop ertie s expre ss ible in the theory
In the RCC calculus of regions the obvious axiom of extens ionality would b e
x y z C x z C y z x y Cext
Thi s state s that if two regions x and y cannot b e di stingui she d by some instance of C z
i e we cannot nd any region z such that C x z do e s not have the same truth value as C y z
they must b e the same region The force of thi s axiom i s to claim that C i s the de ning relation
for regions regions can only b e di stinct if they di er with re sp ect to the ir connecte dne ss with
other regions Whether thi s i s reasonable dep ends on what we take to b e the domain of regions
If regions are made up of di screte atoms then con gurations can eas ily ar i s e where two di stinct
regions are indi scr iminable in terms of the regions they are connecte d to But if every region has
a non tangential prop er part and for every pair of non identical regions there i s some region which
i s part of one but not the other then Cext must hold
In the RCC theory we can der ive something very s imilar to the axiom of extens ionality From
the de nitions of EQ x y and P x y given ab ove we can very eas ily show that
x y z C x z C y z EQ x y CEQ
However s ince the EQ symb ol i s intro duce d by de nition thi s der ive d formulae do e s not have
the force of the axiom of extens ionality b ecaus e EQ nee d not nece ssar ily have the prop ertie s of
logical equality Hence the der ivation do e s not show that an axiom of extens ionality i s re dundant
in the RCC calculus What it shows rather i s that if we take the equivalence
x y x y P x y P y x P
as an axiom rather than a de nition and assume that the symb ol i s to have its usual logical
prop ertie s then thi s formula i s equivalent to Cext and can thus s erve as an axiom of extens ionality
for the RCC theory
The Quas i Bo ole an Functions
Most of the complexity of the RCC theory ar i s e s f rom the quas i Bo olean functions In thi s s ection
I examine the role of the s e functions in the theory and sugge st how they could b e handle d in a
more preci s e and economical way
CHAPTER ANALYSIS OF THE RCC THEORY
The Status of the Function De nitions
In Randell Cui and Cohn the functions are intro duce d by means of a non standard form
of de nite de scr iption op erator For example a sum function i s character i s e d as
sum x y z w C w z C w x C w y
def
where the iota notation i s to b e interprete d as follows
x y y x means x x x
def
Thus the sum de nition can b e rewr itten as
x y w C w sum x y C w x C w y
It should b e note d that thi s formula i s not purely de nitional s ince b ecaus e all functions must
have a value the us e of the sum function carr ie s exi stential commitment In general a formula
which intro duce s a new function symb ol into a theory cannot b e regarde d as a de nition unle ss
entitie s with appropr iate prop ertie s to b e value s of the function are already guarantee d to exi st as
1
a cons equence of the axioms of the theory
It i s also imp ortant to note that the formula character i s e s the sum function only in the context
of the C pre dicate It can b e contraste d with the following explicit character i sation
x y z z sum x y w C w z C w x C w y
Thi s formula imp os e s a stronger condition on the domain of the C relation namely that given
any two regions x and y there i s exactly one region that i s connecte d to just the regions that
are connecte d e ither to x or to y It i s quite easy to s ee that the contextual sum de nition i s
logically equivalent to the left to r ight direction of the explicit sum de nition However the r ight
to left direction do e s not follow To get the r ight to left implication we also nee d the axiom of
extens ionality Cext given in the last s ection Alter natively one could replace the contextual sum
axiom with the explicit one If thi s i s done and we also stipulate that sum x x x then the axiom
of extens ionality i s imme diately der ivable
RCC without Functions or Sorts
There are two reasons for the us e of sort theory in formulating the RCC theory Firstly to accom
mo date functions which are partial with re sp ect to the domain of regions and s econdly b ecaus e
by casting a theory in sorte d logic and us ing a pro of pro ce dure de s igne d to treat sortal information
in an e cient way the e ectivene ss of automate d theorem proving can often b e greatly increas e d
Cohn However the apparatus of functions and sorts do e s re sult in a formal language which
i s rather complex b oth in its syntax and s emantics If we are pr imar ily intere ste d in inve stigating
1
This applie s whether or not we employ a sorte d logic However if we us e a sorte d logic we can allow that the
value s of functions nee d not be regions so the existential commitment nee d not a ect the theory with regard to the
propertie s of regions
CHAPTER ANALYSIS OF THE RCC THEORY
the content and cons equence s of the RCC axioms it i s p erhaps b etter to cast the theory in a
s impler language RCC can eas ily b e mo di e d so as to give a function f ree unsorte d vers ion of the
theory The axioms intro ducing the quas i Bo olean functions are replace d by exi stential statements
Where the function i s partial the exi stential statement i s ne ste d within an implication Thus the
2
axioms intro ducing u sum compl and pro d can re sp ectively b e replace d by the following
x y C x y
x y z w C w z C w x C w y
x y C x y y z C z y NTPP z x O z y P z x
x y O x y z w C w z v P v x P v y C v w
The Complement Function
Of all the axioms in the RCC theory the one that intro duce s the complement function i s the most
complex and its cons equence s the harde st to f athom In its or iginal form the axiom i s
compl x y z C z y NTPP z x O z y P z x complDef
def
and if we assume Cext thi s i s equivalent to
x y y compl x z C z y NTPP z x z O z y P z x complDef
From thi s it can readily b e prove d that
x EC x compl x
The de nition of compl s eems to b e rather more complex than i s de s irable The condition
y compl x i s ass erte d to b e equivalent to two s eparate universal constraints on x and y
Moreover the rst of the s e sp eci e s exactly what i s connecte d to y the complement of x in
terms of the NTPPs of x If the theory i s extens ional with re sp ect to C then thi s sp eci cation alone
should determine all the prop ertie s of any region s complement
However the s econd constraint sp ecifying that the things that overlap the complement of x are
exactly the things that are not part of x also app ears to b e true in the intende d interpretation
and even s eems to completely sp ecify the complementation function One might hop e that the two
conditions could b e prove d equivalent as a cons equence of the de nitions of the relations involve d
and the other functions But de spite cons iderable e ort and extens ive us e of the Otter theorem
prover McCune I have not b een able to demonstrate thi s Thus the compl axiom s eems
to contain not only exi stential commitment but also to indirectly ass ert a universal equivalence
b etween two ways of de scr ibing certain prop ertie s of regions
In view of the s e obs ervations I sugge st that it i s more p erspicuous to replace the compl axiom
by the following two axioms whos e conjunction i s equivalent to the or iginal
x y z C z y NTPP z x
2
Here x x means there is a unique entity satisfying i e x x y y y x
CHAPTER ANALYSIS OF THE RCC THEORY
z C z x NTPP z y z O z x P z y
A further worry concer ning the compl axiom i s that I was unable again de spite cons iderable
e ort to show that complementation i s a symmetr ical op eration i e that x compl y y
compl x Thi s may mean that RCC i s lacking the following clearly de s irable theorem
x y z C z x NTPP z y z C z y NTPP z x
Thi s could also b e der ive d if we adopte d the s imple formula x compl compl x x as an axiom
Relation to Ortho dox Bo ole an Algebras
Bo olean algebras are a very well understo o d class of mathematical structure s Since I will b e
making much us e of the s e algebras e sp ecially in the next chapter it will b e as well to give them
a formal de nition
A Bo olean algebra i s a structure A hS i where S i s a s et of all the elements of the
3
algebra i s a function f rom S S to S and i s a function f rom S to S The s e op erations
must sati sfy the equations given in table in which the x y op eration i s de ne d as equivalent to
x y and the null and unit elements are de ne d by x x and x x
def def
The s e equations are taken with some mo di cation of the pre s entation f rom Kuratowski
p
x y y x x y y x
x y z x y z x y z x y z
x x y x x x y x
x x x x
x y y x y x y y
x y z x y x z
Table An equational theory of Bo olean algebras
It will b e recalle d that in the RCC theory there i s no null region which would corre sp ond to
the least element in an ortho dox Bo olean algebra and thi s i s why the RCC functions are calle d
quas i Bo olean But there s eems no reason why the functions in the RCC theory should not
b e regarde d as genuine Bo olean op erators over the domain REGION NULL Thi s would x the
prop ertie s of the s e op erators by reference to a well understo o d structure However if we regard the
RCC functions in thi s way we still have the problem of axiomatically linking the Bo olean algebra
to the relational part of the theory Thi s problem i s complicate d by the sort theory
3
I shall usually wr ite the complementation operation as a pre x function but where the algebra is a
X to mean the complement of the s et X Bo olean algebra of s ets I shall often wr ite
CHAPTER ANALYSIS OF THE RCC THEORY
A Single Generator for Bo ole an Functions
A standard Bo olean algebra has the prop erty that all op erators are de nable in terms of a s ingle
pr imitive function In f act there are two p oss ible pr imitive s that can b e us e d in the terminology
of electronic circuitry they are NAND and NOR In a Bo olean algebra of regions the s e op erations
corre sp ond to complement of pro duct and complement of sum Thus us ing the rst alter native
starting with a function cp x y the more f amiliar Bo olean op erations together with null and
universal constants can b e de ne d as follows
compl x cp x x
def
sum x cp cp x x cp y y
def
pro d x y cp cp x y cp x y
def
pro d x compl x
def
u sum x compl x
def
Thi s means of intro ducing the Bo olean functions by pure de nitions f rom a s ingle function has
the great advantage that in axiomatis ing the theory we nee d only b e concer ne d with xing the
meaning of cp and its relationship with C all prop ertie s of the other functions and constants will
b e cons equence s of the ir de nitions
Intro duction of a Null Region
If we allow the null entity to b e a bona de region then the technical problems asso ciate d with the
Bo olean functions di sapp ear The functions b ecome total rather than partial and hence there i s no
4
nee d to us e a sorte d logic in order to employ the s e functions in a st order formali sm
Intro duction of a null region require s some revi s ion of the fundamental RCC axioms An
intuitive cons ideration of the notion of connection sugge sts that the null region should not b e
cons idere d as connecte d to any other region Thus we have the new axiom
x C x
Cons equently the re exivity of the connection relation must b e re str icte d so as only to hold for
non null regions Thus the Csym axiom must b e replace d with the weaker formula
x y C x y C x x
Atoms and the NTPP Axiom
Randell Cui and Cohn give an informal pro of of the imp oss ibility of having atomic regions
in a mo del of the axioms The s e putative atoms would b e regions having no prop er parts Here
we assume a theory without the NTPP axiom which of cours e explicitly rule s out such mo dels
4
Of cours e we may still wish to employ a sorte d logic for the purp os e of increasing the e ciency of automate d
de duction
CHAPTER ANALYSIS OF THE RCC THEORY
Supp os e a region r has no prop er parts It therefore has no non tangential prop er parts and
thus b ecaus e of the compl axiom it follows that every region i s connecte d to the complement of r
Thus assuming the Cext axiom compl r must b e the universal region u We can further conclude
that P r compl r and then f rom the de nition of O we s ee that O r compl r However as men
tione d ab ove f rom complDef and Cext one can der ive x EC x compl x Thus EC r compl r
But f rom the de nition of EC we must have O r compl r a contradiction
It has b een sugge ste d that there i s a cons i stent atomic mo del of the RCC axioms in which only
one non null region exi sts
Thus the NTPP axiom i s der ivable f rom the other axioms of the theory Randell Cui and Cohn
sugge st that the di culty ar i s e s b ecaus e the de nition of the part relation i s incompatible with the
exi stence of atoms Three p oss ible solutions are given
The rst i s to divide the domain of regions into three di sjoint sorts PROPER REGIONs
ATOMs and PARTICLEs All of the s e kinds of region must have NTPPs in accordance with
the NTPP axiom However the prop er parts of ATOMs are PARTICLEs and not PROPER
REGIONs It i s further require d by additional axioms that if two ATOMs overlap they must
b e equal and every PROPER REGION has a part which i s an ATOM Whilst thi s prop osal
may have some attractions as a conceptual scheme it i s f ar f rom clear whether it can really b e made
into a cons i stent theory and the adde d complexity of the sort structure would make the language
f ar more unwieldy than the bas ic RCC theory
The two further alter native treatments of atoms given by Randell Cui and Cohn involve
even more radical departure s f rom the bas ic theory One of them require s the function sum as well
as the sort ATOM to b e taken as pr imitive s in addition to the or iginal C The other require s a new
sort of POINTs to b e adde d to the domain and i s bas e d on a new pr imitive relation IN p r of
incidence holding b etween p oints and regions C i s then intro duce d as a de ne d relation The s e
alter native theor ie s are to o f ar f rom the or iginal to b e cons idere d in the pre s ent work
In summary it must b e said that the or igin of the non atomicity of regions in the RCC theory
i s not fully understo o d Each of the alter native s prop os e d by Randell Cui and Cohn
s eem more complex than i s de s irable and have not b een worke d out in detail Another plaus ible
sugge stion made in that pap er i s that the problem lie s with the de nition of P but a revi s e d
de nition was not given
Mo dels of the RCC Theory
The RCC theory was initially develop e d through a metho dology of sp ecifying intuitively correct
axioms rather than by cons ider ing mathematical mo dels of space However in order to e stabli sh
imp ortant meta theoretic re sults such as completene ss and categor icity di scuss e d further b elow
some kind of formal s emantics i s nee de d Be ing formulate d in st order logic the general purp os e
s et theoretic interpretation of that language may of cours e b e employe d but cons ideration of the
particular nature of the RCC theory sugge sts that other kinds of mo del may b e more appropr iate
CHAPTER ANALYSIS OF THE RCC THEORY
Graph Mo dels of the C relation
Mo dels of the C relation can b e repre s ente d by symmetr ic re exive digraphs or more s imply by
non directe d graphs in which each no de i s implicitly taken as b e ing connecte d to its elf If we only
require C to b e re exive and symmetr ic then all such graphs will corre sp ond to p oss ible mo dels
of the theory As we add further axioms we place constraints on admi ss ible structure s for the C
relation For examining the s e mo dels it will b e helpful to b e able to refer to the s et of all regions
5
connecte d to some region x Thus C x which may b e calle d the C set of x i s de ne d as
C x fy j C x y g
def
In terms of C s ets symmetry and re exivity corre sp ond re sp ectively to the f acts
x C x and x C y y C x
and the extens ionality axiom Cext can b e expre ss e d by
C x C y x y
Other logical prop ertie s of RCC such as thos e stemming f rom the quas i Bo olean function axioms
would corre sp ond to more subtle constraints on the domain of C s ets
Mo dels bas e d on connection graphs and or C s ets are very straightforwardly relate d to the
relational vo cabulary of the RCC theory and the ontological commitments emb o die d in such mo dels
do not go b eyond what i s implicit in the theory However they have a numb er of shortcomings
Graph mo dels are very general and can b e given for any theory bas e d on a binary relation so they
do not character i s e any prop ertie s which are particular to the spatial domain Cons equently they
do not accord well with our p erception of real s ituations In f act as a means of building a mental
picture of a s ituation de scr ib e d by some RCC formulae graph mo dels are wors e than us ele ss if
we vi suali s e two connecte d regions as two blobs joine d by an arc we thereby picture the regions
as di sconnecte d A further problem for the re s earcher i s that the graph mo dels cannot readily b e
relate d to class ical mo dels of geometry and top ology
Mo dels in Point Set Top ology
In contrast with graphs of the C relation the top ological space s of class ical p oint s et top ology
provide a well understo o d class of mathematical structure s which de spite some subtletie s
s eem to accord much b etter with our p erceptions of spatial s ituations Whilst asso ciating phys ical
b o die s with s ets of p oints i s an abstraction which require s a certain amount of imagination spatial
relationships b etween p oint s ets can b e picture d in much the same way as relationships b etween
phys ical b o die s One di erence i s that in the p oint s et mo del we can di stingui sh b etween open and
closed s ets whereas phys ical b o die s do not come in op en and clos e d var ietie s However as we so on
shall s ee it i s p oss ible to give a p oint s et interpretation of region under which no op en clos e d
di stinction ar i s e s
5
The s e s ets are also employe d in the analys e s of Biacino and Gerla and Gerla
CHAPTER ANALYSIS OF THE RCC THEORY
Advo cate s of the na ve approach to knowle dge repre s entations may ob ject to the us e of top o
logical mo dels on the grounds that the mathematical content of the s e mo dels go e s f ar b eyond the
understanding of space enjoye d by the average p erson Dogmatic adherents of the region bas e d
approach may also ob ject to the app earance of p oints in the mo dels of a theory which i s supp os e d
to avoid commitment to the exi stence of p oints Whilst I acknowle dge the motivations for the s e
ob jections I take what I regard as a more pragmatic approach to the examination of region bas e d
theor ie s and am prepare d to employ any mathematical apparatus that s eems to b e us eful I do
think that one reason why region bas e d formalisms may b e us eful i s that they are clos e to natural
ways of de scr ibing space but I do not think thi s means that in developing and inve stigating a
formal theory of regions one should b e re str icte d to employing only na ve concepts
Interpreting RCC in Point Set Top ology
To character i s e the meaning of the non logical vo cabulary of RCC in terms of p oint s et top ology
we nee d to sp ecify preci s ely how the individuals of the theory i e regions and the connection
relation are to b e interprete d by reference to a top ological space One p oss ible sp eci cation i s as
follows
Regions are identi e d with non empty op en s ets of p oints
Regions are connected if the ir closure s share at least one p oint
Thi s interpretation i s that sugge ste d for the RCC theory in Randell Cui and Cohn
If we require that the theory should sati sfy the extens ionality pr inciple Cext thi s imme diately
leads to a re str iction on the class of op en s ets that can b e cons idere d regions no two di stinct
regions can b e identi e d with op en s ets that have the same closure The most obvious way to
ensure thi s i s to sp ecify that regions corre sp ond only to regular op en s ets i e thos e which are
equal to inter iors of the ir closure s
From the top ological character i sation of C we ought to b e able to der ive interpretations in
terms of p oint s et top ology of all relations de nable in RCC Given the st order de nition of P
P x y z C z x C z y and the f act that for regular op en s ets c X c Y i
def
X Y it i s clear that the partho o d relation b etween regions corre sp onds to the subs et relation in
the p oint s et interpretation
The inters ection of two regular op en s ets i s always a regular op en s et so two op en s ets
share a p oint just in cas e they share a non empty regular op en subs et If we assume that every
non empty regular op en s et of p oints corre sp onds to some region we can say that two regions
overlap if they share a p oint and thi s will accord with the st order de nition of overlap in terms
of the C relation O x y z P z x P z y
def
Formally the C P and O relations can b e de ne d as
C x y c X c Y
def
P x y X Y def
CHAPTER ANALYSIS OF THE RCC THEORY
O x y i X i Y
def
The s e de nitions give us a r igorous formal sp eci cation of the RCC connection and overlap
relations in terms of p oint s et top ology But they make us e of a highly expre ss ive s et theoretic
language including b oth quanti cation and the element relation and hence are not very us eful
for automate d reasoning In the next chapter we shall s ee how e ss entially the same top ological
interpretation can b e expre ss e d algebraically without the us e of s et theory and quanti cation
The Bo ole an Algebra of Regular Op en Point Sets
In the last s ection we saw that the RCC regions can b e identi e d with non empty regular open
s ets in a top ological space If thi s interpretation i s to b e adequate for the full theory equipp e d
with quas i Bo olean functions we nee d to b e able to interpret the s e as functions op erating on
non empty regular op en s ets If we were s imply to us e the elementary Bo olean s et functions
complement union inters ection to mo del Bo olean functions on regions we would imme diately
run into di cultie s The problem i s that if we apply the s e op erations to regular op en s ets the
re sulting s et i s not nece ssar ily regular op en the complement of a regular op en s et i s regular clos e d
and the sum of two regular op en s ets i s op en but nee d not b e regular
Thi s problem can b e avoide d by identifying Bo olean functions on regions with the op erators in
the regular open Boolean algebra of a top ological space Given a top ological space hU O i the
elements of thi s algebra are the regular op en s ets The Bo olean constants and functions are then
de ne d as follows
U
def def
X X i
def
x y X Y x y i c X Y
def def
Thus the regular complement i s de ne d as the inter ior of the ordinary s et complement and the
regular sum i s obtaine d by taking the inter ior of the closure of the s et union Pro duct i s s imply
de ne d as inters ection It can eas ily b e ver i e d that given regular op en s ets as op erands the
re sults of the s e op erations are also regular op en s ets
A Dual Top ological Interpretation
There i s also a dual interpretation under which regions are identi e d with closed s ets the s e
are connecte d if they share a p oint and overlap if the ir interiors share a p oint As b efore the
requirements of the theory mean that the clos e d s ets corre sp onding to regions must b e non empty
and regular a regular clos e d s et i s a s et X such that X c i X The regular ity condition
ensure s that s ets corre sp onding to regions must have a non empty inter ior
CHAPTER ANALYSIS OF THE RCC THEORY
Completene s s and Categor icity
In s ection I mentione d that some re sults of Grzegorczyk have imp ortant cons equence s
regarding the prop ertie s of spatial theor ie s Thi s pap er cons iders st and higher order theor ie s of
Bo olean algebras supplemente d with additional spatial functions and or relations The st order
theory of Bo olean algebra i s decidable but Grzegorczyk shows that the intro duction of e ither a
closure op eration or an exter nal connection relation sati sfying in each cas e a small s et of algebraic
conditions re sults in a structure whos e st order theory i s undecidable
The assume d conditions on the closure op eration are just thos e given in s ection and the
conditions on the exter nal connection relation are as follows
EC x y pro d x y
pro d x y sum x x x sum y y y EC x y EC x y
EC sum x y z EC x z EC y z
The s e conditions are quite weak and one would exp ect them to b e sati s e d in any plaus ible theory
of connection Thi s means that any st order language containing Bo olean or quas i Bo olean
functions and a connection relation must b e undecidable
The que stion of what levels of expre ss ivene ss lead to undecidable language s i s of crucial im
p ortance for automate d reasoning In the following chapters we shall s ee that it i s p oss ible to
sp ecify quite expre ss ive repre s entations for spatial information which are decidable The strategy
i s to nd ways of expre ss ing spatial relationships without the nee d for a full st order language
One approach i s to us e a st order language with limite d forms of quanti cation In chapter
I shall show that in a st order theory bas e d on the C relation it i s in many cas e s p oss ible to
eliminate quanti ers by replacing quanti e d claus e s with equivalent quanti er f ree formulae An
other approach i s to us e a order repre s entation language which i s more expre ss ive than class ical
prop os itional logic Although augmenting a Bo olean algebra with additional op erators such as a
closure function may lead to an undecidable st order theory it can also greatly extend the range
of information which can b e expre ss e d in the form of algebraic equations without quanti cation
In chapter we shall s ee how a order mo dal logical language can b e us e d to reason ab out such
constraints
6
An imp ortant corollary of the undecidability re sult i s that no nitary st order theory of spatial
regions p oss e ss ing a certain minimal expre ss ivity can b e complete A theory i s complete with
re sp ect to a language L i for every formula expre ss e d in the language L e ither or
7
i s logically valid If a nitary st order theory i s complete it i s also decidable Thi s
follows f rom the s emi decidability of nitary st order logic any logically valid st order formula
i s provable in nite time so to decide whether follows f rom one can attempt to prove in parallel
6
More will be said in s ection ab out the re str iction of this re sult to nitary systems
7
Note that if this is the cas e L can contain only a xe d nite vo cabulary of non logical expre ssions constraine d
by the theory If it containe d arbitrary relations functions or constants it could not be complete
CHAPTER ANALYSIS OF THE RCC THEORY
or by alter nating f rom one pro of to the other b oth the s entence s and A pro of
of one of the s e formulae can always b e obtaine d in nite time
Any theory in which one can de ne a relation of exter nal connection sati sfying certain conditions
must b e undecidable Moreover s ince any complete st order theory i s decidable any st order
theory in which thi s relation i s de nable must b e incomplete Thi s means that there are purely the
oretical RCC formulae i e formulae not involving any arbitrary constants which are contingent
with re sp ect to the RCC axioms and indee d with re sp ect to any s ens ible s et of st order axioms
Hence RCC i s not categorical there must b e multiple non i somorphic mo dels of the theory
and cannot b e made categor ical without adding some nd order axiom Becaus e of thi s lack of
categor icity the entailments provable in the RCC theory are only thos e that hold in a very large
class of p oss ible mo dels many of which will have a very di erent structure to what i s intende d
In f act it i s readily apparent that there i s no s ingle mo del of the axioms For instance the
dimens ionality of regions i s not xe d one can interpret them as b e ing of two three or even higher
dimens ion Moreover spatial con gurations which are imp oss ible in say D may b ecome p oss ible
in or more dimens ions I have devote d cons iderable e ort to the problem of nding a categor ical
vers ion of the RCC theory and have shown how by adding extra axioms many unwante d mo dels can
b e rule d out I have concentrate d sp eci cally on character i s ing the dimens ionality of RCC regions
and on eliciting a complete s et of exi stential axioms thi s work i s rep orte d in Bennett a
However it was only towards the end of my PhD re s earch that I reali s e d that categor icity could
not b e achieve d by means of a nite st order theory
The undecidability of RCC and s imilar theor ie s means that the problem of incorp orating qual
itative spatial information into AI systems divide s into two parts i the foundational problem of
providing a su ciently r ich theory of spatial concepts with a preci s e formal s emantics and ii the
problem of constructing inference algor ithms for reasoning in terms of us eful but le ss expre ss ive
repre s entation language s
A Revi s e d Vers ion of the RCC Theory
I now pre s ent an axiom s et for an unsorte d st order theory of regions The theory di ers f rom
Clarke s s and the RCC theory in that a null element i s treate d as a rst class region Thi s means
that the Bo olean comp onent of the theory can b e axiomatis e d much more straightforwardly than in
the earlier theor ie s Apart f rom thi s the theory i s intende d to b e much the same as RCC Following
RCC rather than Clarke every non null region i s connecte d to its complement and no di stinction
can b e made b etween op en and clos e d regions
It must b e stre ss e d that although my revi s e d axiom s et avoids many of the problems with the
RCC theory that were note d earlier in thi s chapter a great deal of further work remains to b e done
on thi s theory Thi s i s b eyond the scop e of the pre s ent work In the remainder of the the s i s I shall
fo cus on alter native s to st order theor ie s that are b etter suite d to automate d reasoning
CHAPTER ANALYSIS OF THE RCC THEORY
Preliminary De nitions
To make the axioms eas ier to state we nee d the following de nitions
D P x y z C z x C z y
def
D O x y z C z z P z x P z y
def
D NTP x y z C z x O z y
def
Fundamental Axioms
A s et of fundamental axioms can now b e state d as follows
A x y C x y C x x
A x y C x y C y x
A x y z C x z C y z x y
A x y z u C z u NTP u x NTP u y
A x C x x y C y y NTP y x
Axiom i s the new re str icte d re exivity axiom which allows only the null region to b e di scon
necte d f rom its elf i s the unchange d symmetry axiom and i s the extens ionality axiom
The fourth axiom guarantee s that for any two regions x and y there i s a region z which i s
connecte d to every region which i s not a non tangential part of b oth x and y and b ecaus e of the
extens ionality axiom there can only b e one such region Under the intende d interpretation z i s
the complement of the pro duct of x and y A complement of pro duct function cp x y can now
b e de ne d as
D cp x y z u C u z NTP u x NTP u y
def
Unlike the function sp eci cations in the or iginal RCC theory thi s i s purely de nitional b ecaus e
the exi stential imp ort and uniquene ss of the function are already entaile d by the other axioms As
was explaine d in s ection the Bo olean functions and universal and null constants can all b e
eas ily de ne d in terms of the cp function Moreover b ecaus e in the new theory the null entity i s
accepte d as a true region the s e will b e prop er rather than quas i Bo olean functions
Finally axiom i s a new vers ion of the NTPP axiom mo di e d to take account of null regions
and us ing the s impler NTP in place of NTPP
Additional Axioms
The system should also sati sfy the theorems given b elow At pre s ent I take the s e as additional
axioms However it i s likely that they are not all indep endent of each other and of the fundamental
axioms in which cas e they could b e omitte d f rom the axiom s et On the other hand the obs erva
tions made in s ection mean that even with axioms AA the system b e ing str ictly st order
CHAPTER ANALYSIS OF THE RCC THEORY
cannot b e complete so one may wi sh to add still more axioms to obtain a stronger theory with a
more re str icte d s et of mo dels
AA z C z x NTP z y z O z x P z y
AA x y z C z x NTP z y z C z y NTP z x
AA The structure hR sum compli i s a Bo olean algebra where R i s the domain of regions
and sum and compl are de ne d f rom cp as sp eci e d in s ection
AA x y P x y sum x y y
AA and AA were di scuss e d in s ection and relate to de s ire d prop ertie s of the compl
function AA i s state d as a meta level prop erty but could b e replace d by a s et of st order
formulae character i s ing a Bo olean algebra in terms of the Bo olean functions of the ob ject language
One could us e equational formulae bas e d on the theory given in table It i s clear that many
and p erhaps all of the s e formulae would b e der ivable f rom the other axioms of the theory AA
ensure s that the part relation coincide s with the usual partial order ing on the elements of the
Bo olean algebra
Mo dels of the Revi s e d Theory
Poss ible mo dels of the revi s e d theory include top ological mo dels which are much the same as thos e
given ab ove for the RCC theory s ection except that the domain of individuals contains the
empty s et Thus in an op en s et interpretation the regions will corre sp ond to arbitrary regular op en
s ets of a top ological space T and C x y will hold just in cas e the closure s of the s ets corre sp onding
to x and y share at least one p oint The value of the function cp x y would then b e given by the
inter ior of the complement of the pro duct of the s ets corre sp onding to x and y and the Bo olean
algebra generate d by cp would b e the regular op en Bo olean algebra over T
It i s clear that axioms A hold in such mo dels A must also hold s ince it can b e shown
that
x y u C cp x y u NTP u x NTP u y
holds under the sp eci e d interpretation of cp
Axiom A imp os e s an additional dens ity condition on the space T Sp eci cally that every
non empty regular op en s et of T i e every s et corre sp onding to a non null region include s a
non empty regular clos e d subs et The inter ior of thi s subs et corre sp onds to a non empty NTP of
the region
Chapter
A Order Repre s entation
As in other areas of knowle dge repre s entation constructing a formali sm for repre s enting
spatial information involve s a trade o b etween expre ss ive capability and the tractability of
computing s emantic relations such as entailment b etween expre ss ions In chapter s everal
very expre ss ive theor ie s of spatial regions were de scr ib e d All of the s e addre ss e d the problem of
repre s enting spatial information by employing logical language s of st or higher order i e
language s including quanti ers But as di scuss e d in s ection reasoning in st order logic
i s not only intractable but undecidabl e so unle ss some sp ecial purp os e reasoning algor ithm i s
known such a repre s entation do e s not provide a practical mechani sm for computing inference s
In thi s chapter I demonstrate how a order quanti er f ree repre s entation which i s an
extens ion of the ordinary class ical prop os itional calculus can b e us e d to repre s ent a s igni cant
class of spatial relationship s Thi s repre s entation also yields a deci s ion pro ce dure for reasoning
ab out thi s information
Spatial Interpretation of Order Calculi
The most f amiliar interpretations of order logical calculi are as propositional logics the non
logical constants are regarde d as denoting propositions and the connective s as op erating on the ir
prop os itional arguments to form more complex prop os itions Within such a conception the
class ical connective s are interprete d as expre ss ing truth functional combinations of the ir arguments
However taking non logical constants as denoting prop os itions i s not the only way that the s e calculi
can b e interprete d which i s why I de scr ib e them as order rather than prop os itional In thi s
chapter I explain how the class ical prop os itional logic which I refer to as C can b e employe d as a
language for spatial reasoning Under thi s interpretation the non logical constants denote regions
and the connective s corre sp ond to op erations forming new regions f rom the ir arguments
Thi s interpretation i s compatible with well known mo del theoretic accounts of order calculi
in which prop os itions are asso ciate d with s ets rather than with truth value s The s e s ets are often
thought of as s ets of possible worlds in which a prop os ition i s true but they can also b e regarde d
as s ets of points or p erhaps atoms making up a spatial region Such interpretations are generally
employe d as mo dels of modal logics rather than the s imple class ical calculus whos e s emantics i s
CHAPTER A ORDER REPRESENTATION
adequately capture d by the s impler truth functional s emantics In the next chapter we shall s ee
that the non truth functional op erators of mo dal logics can also b e given a spatial interpretation
The p oss ibility of repre s enting spatial relations in class ical prop os itional logic ar i s e s b ecaus e
the logic of spatial regions include s a Bo olean algebra as a sub structure Thi s has b een known for
a long time it forms the bas i s of Venn diagrams Venn By generali s ing the pr inciple s of
Bo olean reasoning the re st of the chapter develops a rather more elab orate system in which it i s
p oss ible to repre s ent and reason ab out a much larger class of spatial relations The generali sation
involve s a meta level addition to the bas ic syntax and pro of theory of a order logic which i s
nee de d to increas e the expre ss ive p ower of the repre s entation sp eci cally it enable s negative as
well as p os itive constraints to b e repre s ente d Thi s metho d of repre s enting negative constraints in
a order logic i s as f ar as I know completely or iginal It i s also quite general and in subs equent
chapters will b e applie d to mo dal and intuitioni stic logical repre s entations
Set Semantics for the Clas s ical Calculus
The order class ical calculus henceforth C can b e given a s emantical interpretation in which the
constants denote arbitrary subs ets of some univers e U and the logical connective s corre sp ond to
elementary s et theoretic op erations Sp eci cally a mo del for the logic C i s a structure hU K i
where U i s a non empty s et K i s a denumerably in nite s et of constants and i s a denotation
function which ass igns to each constant p in K a subs et P of U The domain of i s extende d
1
to all formulae forme d f rom the constants by stipulating that
S i s the s et of all elements of U that are not elements of S For example if where for any s et S
a A b B and c C then a b c A B C Under thi s interpreta
tion it can b e shown that
Clas s ical Set Semantics Theorem CSST
A formula i s a theorem of C if and only if
for every mo del hU K i the equation U i s sati s e d
The denotation function induce s a corre sp ondence b etween formulae and terms forme d f rom
constants denoting s ets and elementary s et op erations henceforth set terms It will b e us eful to
de ne some notation to de scr ib e the relationship b etween the s e typ e s of expre ss ion
For every prop os itional constant p there i s a corre sp onding s et constant P
i i
1
With s emantic and other meta level functions such as I enclos e the arguments in square rather than round
brackets The small Greek letters and are employe d as schematic var iable s standing for arbitrary prop ositional expre ssions
CHAPTER A ORDER REPRESENTATION
I wr ite ST as a means of expre ss ing the s et term obtaine d f rom the formula by replacing
order constants p by s et constants P and the connective s and re sp ectively
i i
by and Note that ST i s a meta level syntactic op eration and not an ordinary
extens ional function
The convers e meta level function f rom a s et term to a class ical prop os itional formula
will b e wr itten CF If the empty s et symb ol o ccurs in it will b e replace d by the
f als ity constant its negation will b e wr itten as
ST
It will also b e convenient to us e the relational notation to refer to the
CF
mapping b etween class ical formulae and corre sp onding s et terms thus we can wr ite e g
ST
Q Again thi s i s a meta level relation b etween expre ss ions p q P
CF
The s et s emantics may b e regarde d as a generali sation of the usual truth functional s emantics
for C if U i s taken to b e a s ingleton s et f g then all formulae are ass igne d one of two value s
or f g Hence for any truth value ass ignment f K ft f g to the order constants there i s
a s et ass ignment K ff g g such that p f g if f p t and p if f p f
f f f
Moreover the value s of the truth functions on truth value s are mirrore d by corre sp onding value s
of the s et op erations on the two p oss ible s et value s So if the domain of i s extende d to complex
f
formulae according to the sp eci cation for the function given ab ove then f g i i s
f
given the value t under the truth functional ass ignment f and i i s ass igne d f by f
f
Pro of of CSST If i s converte d to conjunctive normal form CNF then each
conjunct will contain a pair of complementary literals l and l if and only if i s a
tautology The s et term ST can also b e converte d to an analogous normal form
intersection normal form INF by means of s imple re wr ite rule s any s et term can b e
expre ss e d as an inters ection of unions of s et constants and the ir complements Thus
can b e expre ss e d in the form
i j n nk n nl
If a s et term corre sp onds to a tautological prop os ition then when expre ss e d in INF
of a s et constant and its each union in the expre ss ion must contain some pair and
complement So whatever the ass ignment to the s et constants each union and hence
the inters ection of the s e unions will denote the universal region
On the other hand supp os e i s not a tautology then there i s a truth value ass ign
ment f p to the atomic prop os itions in such that i s f als e according to truth
i
functional s emantics Hence f rom the der ive d s et ass ignment over the univers e f g
f
as de scr ib e d ab ove we can construct a mo del hf g K i in which do e s not denote
f
the univers e
If we are only intere ste d in the pure class ical calculus s et s emantics may b e cons idere d a
re dundant generali sation of truth functional s emantics s ince CSST shows that a Bo olean term
CHAPTER A ORDER REPRESENTATION
has the value in all ass ignments to its constants over any domain if and only if it take s the value
in all ass ignments over the domain f g Hence cons ideration of a element algebra i s su cient
to determine validity of any entailment in C It i s only when as in the next chapter we intro duce
additional op erators corre sp onding to non Bo olean functions that we nee d to cons ider ass ignments
over larger domains
An Entailment Corre sp ondence
The correlation b etween class ical theorems and Bo olean terms which are universal in any mo del i s
a sp ecial cas e of a more general corre sp ondence b etween the entailment relation in class ical logic
and entailments among Bo olean equations The s e entailment relations will b e repre s ente d with the
following notation
j means that in the calculus C the formula i s entaile d by the s et of
n C
formulae f g Thus j means that i s a theorem of C
n C
j where are s et equations means that in any mo del i e ass ign
n S n
ment of s ets to the constants o ccurr ing in the equations for which the equations
n
hold the equation also holds j means that holds in every mo del
S
The s et equations we shall b e most often concer ne d with are universal i e of the form U
where U i s the univers e of whatever mo del i s under que stion Thi s pre s ents a slight notational
di culty if we want to say that a universal equation holds in all of some class of mo dels b ecaus e
the universal s et will not generally b e the same s et in each mo del For thi s purp os e I employ the
or as a sp ecial symb ol U We can regard thi s e ither as a sp ecial logical symb ol equivalent to
2
meta var iable standing for whatever s et i s the univers e under cons ideration
Us ing the s e notations the following theorem can now b e state d
Clas s ical Entailment Corre sp ondence Theorem CECT
j if and only if U U j U
n C n S
ST
for each i where
i i
CF
Pro of of CECT If j then the formula must
n C n
U must hold in every mo del b e a tautology hence the equation
n
But in any mo del sati sfying U U one must have
n n
Therefore U
On the other hand supp os e j thi s means that there i s some truth
n C
functional ass ignment f under which are all true whilst i s f als e We then
n
us e the der ive d ass ignment over the domain f g as an ass ignment to corre sp onding
f
s et constants o ccurr ing in the terms Under thi s ass ignment we shall have
i
U U and So U U j U
n n S
2
Note that if a term contains U then in CF this will be replace d by
CHAPTER A ORDER REPRESENTATION
Re asoning with Non Universal Equations
The corre sp ondence theorem CECT allows us to us e class ical prop os itional formulae to reason
ab out universal s et equations and the formula CF can b e regarde d as repre s enting the equation
U Moreover b ecaus e of the equivalence
X Y X Y U X Y if and only if
any s et equation can b e put into the universal form U
In order that we may us e order formulae to reason ab out arbitrary Bo olean s et equations
it will b e us eful to de ne a transform CFe which give s a formula repre s entative of
any equation Such a repre s entative i s provide d by the formula CF where U i s
equivalent to For a universal equation CFe U will just b e equal to CF For an
arbitrary non universal equation we could us e the de nition
CF CF CF CF CFe CF
def
but s ince it i s more convenient to de ne CFe by
CFe CF CF
def
In terms of CFe we can state the following corollary of CECT which character i s e s entailment
b etween arbitrary Bo olean s et term equations
j if and only if CFe CFe j CFe
n S C
In f act we shall almost always deal with equations which are in the universal form but even
in the s e cas e s the CFe op erator i s still a us eful notation for translating f rom Bo olean equations to
the ir repre s entative formulae
Repre s enting Top ological Relationship s in C
Table shows how four spatial relations can b e character i s e d by constraints state d in terms of the
class ical prop os itional calculus C The rst column of the table sp eci e s a spatial relation us ing
the formal vo cabulary of the RCC theory The s econd column give s an informal de scr iption of the
relation The third column again de scr ib e s the same relation in terms of an elementary s et term
equation all the equations are given in universal form Thi s character i sation i s in accord with the
interpretation of RCC regions as non empty regular op en subs ets of a top ological space given in
s ection The nal column give s a formula of C that may b e cons idere d as repre s enting the
spatial relation Thi s formula i s given by CFe where i s the s et equation of the third column
The theorem CECT tells us that entailments among elementary s et equations are f aithfully
mirrore d by entailments among corre sp onding C formulae Thus in order to reason with spatial
information expre ss ible in terms of such s et equations one can transform the equations into formulae
of C and then te st inference s us ing some metho d of prop os itional theorem proving
CHAPTER A ORDER REPRESENTATION
Relation Description Set Equation C formula
DR x y x and y are di screte X Y U x y
P x y x i s part of y X Y U x y
Pi x y y i s part of x X Y U y x
EQ x y x and y are equal X Y X Y U x y
Table De nitions of four top ological relations in C
For example the inference
DR a b P c a DR a c
dep ends on the following entailment b etween s et equations
A B U C A U j C B U
and thi s can b e shown to b e valid b ecaus e in C we have
a b c a j c b
Hence even with thi s very s imple enco ding into C some s igni cant spatial inference s can b e
determine d
Apart f rom the four relations given in table a large class of other relations can also b e
repre s ente d including x i s the univers e x x i s null x x i s the complement of y x y
the sum of x and y i s the univers e x y and x i s the sum of y and z x y z
The corre sp ondence b etween binary top ological relations among regions and the s et equations
or C formulae which can b e us e d to repre s ent them are illustrate d in gure The gure contains
ve sub diagrams showing each of ve JEPD relations that can hold b etween two regions Thi s
class i cation do e s not di stingui sh b etween connection and overlapping or b etween tangential and
non tangential parts Of the ve relations only DR and EQ can b e uniquely sp eci e d by a C formula
It i s not surpr i s ing that the di stinction b etween connection and overlapping cannot b e sp eci e d
in terms of the purely Bo olean formulae of C In the p oint s et interpretation of RCC thi s di stinction
dep ends on the top ological closure op eration but in the s imple language of Bo olean s et equations
no such op eration i s available To capture the di stinction we shall nee d to us e the more expre ss ive
repre s entation de scr ib e d in the next chapter However it i s more di sapp ointing that the relation
of partial overlap cannot b e directly repre s ente d by any formula of C and even though the part
relation corre sp onds directly to implication the proper part relation cannot b e uniquely sp eci e d
although we can eas ily say that one region i s part of another we cannot rule out the p oss ibility
that the two regions are equal
CHAPTER A ORDER REPRESENTATION
:(a ^ b)
a b = U
or a b or
:a _ :b
a b = U
DR(a b)
a b ?
PO(a b)
P(a b)
(a b)
a b a b Pi
a b= U
a b = U
PP(a b)
PPi(a b)
:a _ b a _ :b
or a b or
b ! a
a ! b
EQ(a b)
(a b) (a b) = U
a $ b
Figure Top ological relations repre s entable in C
Mo del and Entailment Constraints
As it stands our repre s entation i s very limite d many s imple spatial relations cannot b e de ne d
solely by means of universal s et equations For example we have obs erve d that the relation
PP x y x i s a proper part of y cannot b e so expre ss e d Neverthele ss informally thi s rela
tion can b e de ne d quite straightforwardly as that relation which holds whenever P x y i s true
but not EQ x y So it would s eem that we can character i s e the prop er part relation if we can nd
a way to repre s ent the abs ence of a relation which we can already de ne
We must now ask how the negations of s et equation constraints should b e repre s ente d Take for
example P x y x i s not part of y Supp os e we s imply negate the class ical formula repre s enting
X Y U P x y we would then get x y But thi s formula corre sp onds to the s et equation
or equivalently X Y U and thi s will only hold when b oth X U and Y So we s ee that the
negation of a formula do e s not corre sp ond to the abs ence of the relation enforce d by that constraint
X Y U which i s the direct negation In terms of s ets what we really wante d to repre s ent was
of the s et equation for P x y But negating the formula in the prop os itional repre s entation do e s
not give us thi s b ecaus e such a negation i s interprete d as a complement op eration on the s et term
rather than a negation of the whole equation Thi s means that the abs ence of the relations de ne d
so f ar cannot b e repre s ente d directly as C formulae
We nee d to increas e the expre ss ive capabilitie s of our repre s entation language so we can rep
re s ent s ituations in which we sp ecify not only that a numb er of universal s et equations hold but
also that certain such equations do not hold Thus we shall employ the more general constraint
CHAPTER A ORDER REPRESENTATION
language of universal s et equations and the ir negations and us e thi s to de scr ib e spatial s ituations
In order to us e class ical formulae to logically enco de the s e constraints we nee d some way of indic
ating whether the formula i s to b e interprete d as an equality or an inequality Thus a collection
of constraints will b e repre s ente d by a pair hM E i where M i s a s et of formulae corre sp onding
to equalitie s and E i s a s et of formulae corre sp onding to inequalitie s The formulae in M are
calle d model constraints b ecaus e they corre sp ond to equational constraints on p oss ible mo dels
The formulae in E are calle d entailment constraints for reasons which will b e made clear in the
next s ection The language cons i sting of pairs of s ets of C formulae will b e calle d C
Cons i stency of C Situation De scr iptions
What we now nee d i s a metho d of determining f rom a pair of formula s ets hM E i whether
the corre sp onding spatial algebraic constraints are cons i stent hM E i repre s ents a s et of
constraints of the form fm U m U e U e U g Clearly i s incons i stent if
j k
and only if the following entailment holds
m U m U j e U e U DE
j S k
Here the r h s i s a di sjunction of s et equations and as such cannot b e translate d into a union at
the level of s et terms just as negating a s et equation i s not equivalent to applying the complement
op eration to its s et term The corre sp ondence theorem CECT do e s not tell us how to interpret
di sjunctions of s et equations in C However it can b e e stabli she d that in the domain of s ets
3
entailments of thi s kind are convex in the s ens e of Opp en A class of entailments i s convex
in thi s s ens e i
whenever j then j for some i f ng
n i
The following theorem ass erts the convexity of entailments of the form of DE
Convexity of Di sjunctive Bo ole an Algebraic Entailments BEconv
U U j U U
m S n
i
U U j U for some i f ng
m S i
Pro of of BEconv Cons ider a di sjunctive entailment of the form of DE and let S
b e the s et of s et constants which it contains Supp os e none of the di sjuncts on the
r h s are entaile d by the equations on the l h s Thi s means that for each di sjunct
U
i
of subs ets of some univers e U to the e U there i s an ass ignment S
i i i
constants in S such that e U i s f als e whilst the equations m U are all true We
i i
can assume without loss of generality that the univers e s in each of the s e ass ignments
3
Note that later in this the sis I shall us e the term convex with its ordinary s ens e as a property of the surface of
a region Hopefully this will not caus e to o much confusion
CHAPTER A ORDER REPRESENTATION
S
U
such that U are di sjoint We now construct a new ass ignment S U
i
i
S
X The U s thus form di screte subspace s of U Clearly thi s and X
i i
i
ass ignment must make all the l h s equations true and each of the di sjuncts on the
r h s f als e Thus the r h s will b e entaile d if and only if at least one of its di sjuncts i s
individually entaile d by the l h s Thi s means that the class of entailments of the form
of DE i s convex
In f act BEconv may b e regarde d as an imme diate cons equence of a general cons i stency prop
erty of equational literals which I shall call ELcons By an equational literal I mean a p os itive or
negative equality relation which may contain constants function symb ols and var iable s Var iable s
are assume d to b e implicitly universally quanti e d The prop erty i s as follows
Cons i stency of Equational Literals ELcons
j
m m n n
i
j for some i f ng
m m i i
ELcons can b e e stabli she d by cons ider ing p oss ible pro ofs of incons i stency in some pro of system
for st order logic with equality which i s known to b e refutation complete One such system i s
that where the only pro of rule s are binary re solution paramo dulation and f actor ing Du y
Since we are dealing with s ets of literals i e only unit claus e s f actor ing i s not require d and a
s impli e d vers ion of paramo dulation can b e employe d The details of the rule s that are us e d do
not matter s ince ELcons can b e demonstrate d f rom quite general obs ervations The pro of i s as
follows
Pro of of ELcons Supp os e we refute a s et of equational literals by means of binary
re solution and paramo dulation Once an application of binary re solution can b e made
incons i stency i s prove d imme diately so any succe ssful refutation must cons i st of a
s er ie s of paramo dulations followe d by a s ingle binary re solution Note also that each
paramo dulation e ither involve s two p os itive literals and generate s a new p os itive literal
or it involve s a p os itive and a negative literal and generate s a new negative literal
The s e obs ervations enable us to show that any refutation make s e ss ential us e of exactly
one negative literal The key p oints are that the der ivation of a p os itive literal cannot
involve any negative literals and that no rule op erate s on more than one negative literal
Cons ider the nal step in the refutation thi s i s a re solution b etween a p os itive and
a negative literal The p os itive literal i s e ither in the or iginal s et of literals or has b een
der ive d by a s equence of paramo dulations involving only p os itive literals The negative
literal i s e ither in the or iginal s et or has b een generate d f rom a p os itive and a negative
literal In the latter cas e the p os itive literal must have b een der ive d f rom only p os itive
literals and the negative literal i s e ither in the or iginal s et or i s in tur n der ive d f rom
a p os itive and negative literal However long thi s s equence continue s it i s clear that
exactly one negative literal f rom the or iginal s et i s involve d in the pro of
CHAPTER A ORDER REPRESENTATION
The negative literals in the left hand condition of ELcons may b e move d over to the r ight to
give an equivalent entailment
j ELconv
m m n n
From ELcons it imme diately follows that entailments of the form of ELconv are convex ELconv
has the same syntactic form as DE but whereas ELconv sp eci e s a purely logical entailment
b etween equations the entailment relation j o ccurr ing in DE sp eci e s that the entailment holds
S
if the terms are interprete d in accordance with elementary s et theory In general if the j in
ELconv i s replace d by a more sp eci c entailment relation j the convexity prop erty may no
longer hold However if can b e expre ss e d as a purely equational theory entailments w r t
can b e expre ss e d as purely logical equational entailments of the form of ELconv Hence the
convexity re sult will still hold for such theor ie s In particular it holds for the relation j where
S
S i s elementary s et theory which i s just an interpretation of Bo olean algebra s ince S can b e
sp eci e d purely in terms of equations Thi s give s us an alter native pro of of BEconv
If we combine BEconv with our interpretation of C expre ss ions and then apply CECT we
imme diately get the following theorem character i s ing the cons i stency of C expre ss ions
C Cons i stency Theorem C CT
A C expre ss ion hM E i i s cons i stent if and only if
there i s no formula E such that M j
C
Thi s should make it clear why the formulae in the s et E are calle d entailment constraints
Repre s enting RCC Relations
We can now give C repre s entations for a s igni cant sub class of the RCC relations Let us rst
lo ok at how the s ituation typ e x i s a prop er part of y i s repre s ente d We can say that PP x y
holds when x i s part of y but the two regions are not equal Thi s give s us the equality X Y U
X Y X Y U Equalitie s are enco de d as mo del constraints and and the inequality
inequalitie s as entailment constraints so our prop os itional repre s entation for the relation PP x y
i s the pair
hfx y g fx y gi
Non Null Constraints
Recall that in di scuss ing top ological interpretations of RCC relations s ection I obs erve d that
p oint s ets corre sp onding to prop er non null RCC regions must b e non empty An imp ortant us e
of entailment constraints i s to ensure that regions involve d in a s ituation de scr iption are non null
If null regions are allowe d they have prop ertie s which may s eem counter intuitive for example the
null region i s b oth part of and di sconnecte d f rom any other region and many us eful and apparently
sound inference s may not hold if it i s allowe d that some of the regions involve d may b e null The
CHAPTER A ORDER REPRESENTATION
requirement that a region i s non null i s expre ss e d by the inequality X U which corre sp onds to
the entailment constraint x in the C repre s entation
Repre s entations of the RCC Relations
The C repre s entation allows us to repre s ent each of the ve top ological relations shown in g
ure The s e compr i s e a jointly exhaustive and pairwi s e di sjoint JEPD s et known as RCC
The mo del and entailment constraints including non null constraints of the C repre s entation for
each of the s e relations are shown in table
Relation Model Constraint Entailment Constraints
DR x y x y x y
PO x y x y x y y x x y
PP x y x y y x x y
PPi x y y x x y x y
EQ x y x y x y
Table The C enco ding of some RCC relations
The mo del constraint asso ciate d with a relation i s the stronge st formula which holds in all
mo dels in which the relation holds The entailment constraints s erve to exclude mo dels which
although cons i stent with the mo del constraint are incompatible with the relation Thus the en
tailment constraints asso ciate d with a relation in a JEPD s et will normally corre sp ond to mo del
constraints of other relations in that s et plus the non null constraints The relation PO has no
mo del constraint and i s de ne d by excluding all of the other relations
Certain entailment constraints which one might exp ect to b e require d can b e eliminate d or
weakene d b ecaus e they are indirectly capture d by other constraints For example in table
the entailment constraint x y which o ccurre d in the repre s entation of PP worke d out ab ove i s
replace d by the weaker formula y x s ince in the pre s ence of the mo del constraint x y y x
would imme diately entail x y
Re asoning with C
By making us e of the re sults obtaine d so f ar one can us e a class ical prop os itional theorem prover as
the bas i s of an e ective automate d spatial reasoning system For clar ity I conci s ely summar i s e the
cons i stency checking algor ithm for C Given a spatial de scr iption cons i sting of a s et of relations
of the form R where R i s one of the relations character i sable in C and and are constants
denoting regions the following s imple algor ithm will decide whether the de scr iption de scr ib e s a
p oss ible s ituation
CHAPTER A ORDER REPRESENTATION
For each relation R in the s ituation de scr iption nd the corre sp onding prop os itional
i i i
repre s entation hM E i
i i
S S
E i M Construct the overall C repre s entation h
i i
i i
S
E us e a class ical prop os itional theorem prover to determine whether For each formula
i
i
S
the entailment M j holds
i C
i
If any of the entailments determine d in the last step do e s hold then the s ituation i s imp oss ible
For example we may want to know whether the following s ituation i s p oss ible x i s a prop er
part of y y i s di sjoint with z and x i s a prop er part of z The C repre s entations of the three
spatial relations are re sp ectively
hfx y g fy x x y gi hf y z g f y z gi and hfx z g fz x x z gi
So the overall C repre s entation i s
hfx y y z x z g fy x z x x y z gi
We determine that thi s s ituation i s imp oss ible s ince
x y y z x z j x
C
Determining Entailments
Computing incons i stency of C expre ss ions i s a sp ecial cas e of determining entailments b etween
s ituation de scr iptions character i sable in C To refer to such an entailment I shall us e the notation
+
hM E i j hM E i We can expre ss the meaning of thi s as an entailment b etween s et equations
C
as follows
m U m U e U e U
h i
j
S
m U m U e U e U
j
k
If we then br ing the r h s over to the left and move the re sulting negation inwards we get
m U m U e U e U
h i
m U m U e U e U j
S
j k
To show the validity of thi s we must show that whichever of the equations in the di sjunction i s
chos en the re sulting equation s et i s incons i stent Thi s i s equivalent to showing that
for all p M we have hM E fpgi j + and for all q E we have hM fq g E i j +
C C
Another equivalent way of expre ss ing the s e which i s more convenient f rom the p oint of view of
actually calculating the entailments i s the following
CHAPTER A ORDER REPRESENTATION
C Entailment Theorem C ET
hM E i j + hM E i i
C
e ither hM E i j + or for all M hM f gi j +
C C
and for all E hM f g E i j +
C
Informally thi s means that a s equent i s valid i e ither hM E i i s its elf incons i stent or each
of the mo del constraints in M i s entaile d by the mo del constraints M and also each of the entail
ment constraints in E in conjunction with the mo del constraints M entails one of the entailment
constraints in E Determining the validity of a C entailment has thus b een re duce d to determ
ining the incons i stency of certain C expre ss ions and we already know that such an expre ss ion i s
incons i stent i one of its entailment constraints i s entaile d by its mo del constraints
Complexity of the Re asoning Algor ithm
Cons i stency checking for s ets of spatial relations repre s entable in C i s clearly NP hard and e ss en
tially the same as the cons i stency checking problem for C The meta level extens ion for handling
the entailment constraints re duce s each C cons i stency problem to n cons i stency problems of s ets
of C formulae where n i s the numb er of entailment constraints Note that all the s e n problems
could in pr inciple b e solve d in parallel
Another f actor which can s igni cantly limit the complexity of spatial reasoning us ing thi s en
co ding i s that in repre s enting the ve RCC relations given in table only formulae containing at
most two var iable s are employe d Thi s means that the complexity of reasoning with the s e relations
i s that of SAT the sati s ability problem for binary claus e s Thi s problem i s computationally
easy it can b e solve d in time prop ortional to n where n i s the numb er of claus e s involve d More
sp eci cally thi s problem i s in the class NC of problems which can b e solve d in p olylogar ithmic
time by us ing p olynomially many parallel pro ce ssors
A detaile d cons ideration of computational complexity i s b eyond the scop e of thi s the s i s A
survey of complexity class e s can b e found in Johnson
Chapter
A Mo dal Repre s entation
Us ing pr inciple s intro duce d in the last chapter thi s chapter develops a more expre ss ive
repre s entation for spatial relationshi ps bas e d on the order mo dal logic S I explain how
the Bo olean s et s emantics for class ical logics can b e generali s e d to take account of additional
non truth functional op erators We shall s ee how the top ological interior function can also b e
mo delle d in thi s way In f act cons idere d in thi s way the op erator of S ob eys exactly
the same constraints as an inter ior op erator Thi s corre sp ondence allows one to us e de duction
in S as a means for reasoning ab out equations b etween terms involving Bo olean functions
and an inter ior function We shall s ee that the s e equations can expre ss a large class of spatial
+
relations I go on to intro duce the language S which extends the expre ss ive p ower of S
+ +
in exactly the same way as C extends the C repre s entation The S repre s entation allows
many RCC relations to b e expre ss e d including all the RCC relations
The Spatial Interpretation of Mo dal Logics
In thi s chapter I develop a order repre s entation for spatial information which i s cons iderably
more expre ss ive than that given in the previous chapter The pr inciple s up on which it i s bas e d
are much the same as thos e employe d in formulating C but rather than us ing the s imple class ical
logic to enco de spatial information I shall us e mo dal logics whos e language contains additional
unary op erators
Mo dal op erators are usually regarde d as non truth functional op erators on prop os itions Many
kinds of prop os itional mo dality have b een studie d alethic mo dalitie s nece ss ity p oss ibility contin
gency prop os itional attitude s knowle dge b elief certainty etc deontic mo dalitie s obligation
p ermi ss ion However in the context of a s et s emantics under which order constants are
interprete d as s ets and Bo olean op erators as elementary s et op eration mo dal op erators can b e
regarde d as mappings b etween subs ets of some univers e of elements By thinking of the s e as s ets
of p oints within a space we imme diately get a spatial interpretation
To sp ecify the spatial interpretation of a mo dal op erator in a more concrete way we can regard
the univers e of p oints as having the structure of a top ological space As we saw in s ection the
structure of a top ological space determine s and i s determine d by certain functions on subs ets of
CHAPTER A MODAL REPRESENTATION
the space such as the interior and closure functions We shall s ee that the mo dal op erator of
the logic S can b e interprete d as an inter ior op erator on a top ological space Thi s corre sp ondence
allows one to us e de duction in S as a means for reasoning ab out equations b etween terms involving
Bo olean functions and an inter ior function The s e can b e regarde d as top ological constraints and
can b e us e d to expre ss a large class of spatial relations The connection b etween top ological space s
and the logic S has b een known s ince the work of Tarski and McKins ey but as f ar as I
know has never b een us e d as a vehicle for automate d spatial reasoning
Overview of the Approach Taken
In the next s ection I lo ok at the s emantics of mo dal logics and sp eci cally at algebraic mo dels
bas e d on mo dal algebras I prove a corre sp ondence b etween the de ducibility relation of a mo dal
logic and entailment among mo dal algebraic equations
In s ection I cons ider the algebraic interpretation of a top ological space as a closure algebra
and show how many top ological relationships can b e expre ss e d in terms of closure algebraic equa
tions and the negations of such equations I then obs erve in s ection that the mo dal algebras
asso ciate d with the logic S are e ss entially the same as closure algebras Thi s means that S can
b e us e d to reason ab out equational closure algebra constraints
Generali s ing the f ramework previously de scr ib e d in s ections and of the last chapter
s ection sp eci e s the extens ion of a mo dal language L to a more expre ss ive language L I
prove a us eful entailment convexity re sult for the s e language s I then show in s ection how all
the RCC relations and many more top ological relations can b e enco de d in the extende d mo dal
language S Thi s provide s a deci s ion pro ce dure for a s igni cant class of top ological relations
Finally in s ection I explain how in pr inciple mo dal repre s entations allow us to replace the
meta level expre ss ions of C and S by ob ject level expre ss ions in a mo dal logic incorp orating
an additional S op erator
Semantics for Order Mo dal Logics
To generali s e the spatial interpretation of C to order language s with additional op erators it i s
nece ssary to know some details of mo dal logics and the ir s emantics My pre s entation i s very conci s e
so the reader will nee d some pr ior knowle dge of the sub ject Two very go o d text b o oks on mo dal
logic are Hughe s and Cre sswell and Chellas
Mo dal Logics
A prop os itional mo dal language i s obtaine d by adding to the language of class ical prop os itional
1
logic a monadic op erator The inference rule s of the mo dal logic cons i st of all the rule s
of class ical prop os itional logic plus some additional rule s concer ning the mo dal op erator Many
1
For some purp os e s one may wish to add s everal distinct mo dal operators to the language The re sulting system
is calle d a multi modal logic
CHAPTER A MODAL REPRESENTATION
di erent s ets of rule s have b een prop os e d captur ing di erent intende d meanings and prop ertie s of
the op erator The s e give r i s e to a wide range of di stinct mo dal logics A rule common to most
logics that have b een calle d mo dal i s the rule of nece ss itation RN thi s state s that if any formula
i s a logical theorem then so i s the formula
Further logical prop ertie s of the mo dal op erator are usually pre s ente d in terms of axiom
schemata A schema sp eci e s that all formulae exhibiting a certain logical form have the status
of axioms Thus if the pro of system of the underlying class ical prop os itional logic i s pre s ente d in
axiomatic style i e as a s et of axiom schema and the rule of modus ponens then the pro of system
of a mo dal logic L i s obtaine d by s imply adding further axiom schemata and the rule RN I wr ite
n L
to mean that the formula i s de ducible f rom the s et of formulae f g in the logic L
n
For every mo dal op erator there i s a dual op erator de ne d by Cons equently
s ince negation ob eys the usual class ical pr inciple s it i s eas ily prove d that so one
can equally well take as the pr imary mo dal op erator and intro duce by de nition
The Logic S
S i s one of the s impler and b etter known mo dal logics It may also b e calle d KT s ince it i s
obtaine d f rom class ical prop os itional logic by adding the the rule of nece ss itation and the following
axiom schemas
K
T
A mo dal logic which sati s e s the schema K as well as ob eying the rule of nece ss itation i s
known as normal
Kr ipke Semantics
Currently the b e st known interpretations of mo dal logics are thos e in terms of Kripke semantics
In a Kr ipke s emantics a model cons i sts of a s et of p oss ible worlds together with an accessibility
relation a binary relation b etween worlds asso ciate d with each mo dal op erator Prop os itions
denote s ets of p oss ible worlds the s et of worlds in which they are true A Kr ipke mo del M i s
thus a structure hW R P di where W i s a s et of worlds R i s the acce ss ibility relation P i s a s et
of constants fp g and d i s a function mapping elements of P to subs ets of W
i
Such a mo del determine s the truth of each mo dal formula at each p oss ible world Class ical
formulae are interprete d as follows
Atomic formulae p are true in exactly the worlds in the s et d p
i i
Conjunctions are true in worlds where b oth and are true
CHAPTER A MODAL REPRESENTATION
Di sjunctions are true in worlds where e ither or or b oth i s true
Negations are true in worlds where i s not true
M
to mean that formula i s true at world in mo del M A mo dal op erator i s We wr ite j
then interprete d as follows in a mo del M hW R P di
M M
i j for all W s t R j
A frame i s a s et of all Kr ipke mo dels sati sfying some sp eci cation of the prop ertie s of the
acce ss ibility relation R For example the s et of all Kr ipke mo dels in which R i s re exive and
symmetr ic constitute s a f rame Finally we say that a formula i s valid in some f rame F if it i s
true at every world in every mo del in F
The logic S i s character i s e d by the f rame F cons i sting of all Kr ipke mo dels whos e acce ss
S
ibility relations are re exive and transitive R i s a quasi ordering on W Every theorem provable
according to the pro of system for S sp eci e d ab ove i s valid in F and convers ely every formula
S
valid in F i s provable in the pro of system
S
A vast sp ectrum of di erent mo dal op erators can b e sp eci e d by placing more or le ss general
2
re str ictions on the corre sp onding acce ss ibility relation Furthermore Kr ipke s emantics allows
one to sp ecify op erators whos e logic s eems to corre sp ond well with intuitive prop ertie s of mo dal
concepts employe d in natural language Indee d a numb er of logics prop os e d for natural language
mo dalitie s which were or iginally sp eci e d pro of theoretically by axiom schemata intende d to
capture intuitive prop ertie s of mo dal concepts can b e capture d very eas ily within the Kr ipke
paradigm by quite s imple re str ictions on the acce ss ibility relation
Whilst the Kr ipke approach certainly provide s a very exible approach to mo dal s emantics
its generality i s often overstate d Cons equently many re s earcher s in b oth AI and philosophical
logic tend to think of p oss ible worlds s emantics as e ss entially bas e d up on acce ss ibility relations
However although Kr ipke mo dels may b e appropr iate for certain typ e s of mo dal op erator in other
cas e s it may b e more natural to supp os e a quite di erent structur ing of p oss ible worlds or even a
s emantics that i s not bas e d on p oss ible worlds at all
Mo dal Algebras
A modal algebra i s a mathematical structure that provide s a s emantics for mo dal logics which i s
more general than a Kr ipke mo del Just as the formulae of class ical prop os itional logic can b e
interprete d as referr ing to elements of a Bo olean algebra mo dal formulae can b e interprete d as
elements of a Bo olean algebra supplemente d with an additional unary op eration ob eying certain
constraints Thi s i s a mo dal algebra Bo olean algebras with additional op erators were rst studie d
in detail by J onsson and Tarski The ir connection to mo dal logics was inve stigate d by
Lemmon a b A clear account of the e ss ential prop ertie s of mo dal algebras and the ir
2
Often such re str ictions are thought of as de ning a logic rather than an operator but this is misleading since the
p ossible worlds s emantics allows any number of di erent operators to be encompass ed in a single logical language
CHAPTER A MODAL REPRESENTATION
relation to Kr ipke s emantics i s given by Hughe s and Cre sswell Chapter and a much
more detaile d examination can b e found in Goldblatt
A mo dal algebra can b e repre s ente d by a structure M hS i where hS i i s a
Bo olean algebra and for all elements x and y of the algebra the op erator sati s e s the equation
x y x y add
3
Op erators ob eying thi s equation are known as additive A direct cons equence of additivity i s the
4
following monotonicity prop erty which will b e us eful later
if x y then x y mon
Further equational re str ictions may b e place d on the op erator Of particular imp ortance are
x x i e x x x epi s
norm
x x idem
Algebraic Mo dels
5
We can now de ne an algebraic model for a mo dal language as a structure hS P i where
hS i i s a mo dal algebra P i s the s et of constants of the language and i s a function mapping
mo dal formulae to elements of S For each constant p P p may b e any element of S Thi s
ass ignment to the constants determine s the value of all complex formulae according to the
6
following recurs ive sp eci cation
Note that under thi s interpretation the op eration of the algebra i s asso ciate d with the mo dal
rather than Thi s i s b ecaus e of the additivity of the algebraic op erator the algebraic equation
character i s ing additivity corre sp onds to the mo dal schema which i s true
in every normal mo dal logic
We say that a formula i s universal in a mo del hS P i i i e if the
mo del ass igns to the formula the unit universal element of the mo dal algebra hS i An
3
It is additive operators which are the pr imary fo cus of the inve stigations of J onsson and Tarski
4
Pro of x y x y y y x y y x y x y QED
5
I am assuming here that the language has only one mo dal operator For a multi mo dal language the mo del
would have s everal functions one for each mo dality
i
6
Speci cations for the connective s and can easily be der ive d f rom the ir de nitions in terms of
and
CHAPTER A MODAL REPRESENTATION
algebraic frame F i s a s et of all algebraic mo dels whos e algebras sati sfy some s et of equations E
E
constraining the op erator Finally we say that a formula i s valid with re sp ect to some algebraic
f rame F if it i s universal in every mo del in F
E E
In order that algebraic mo dels provide a s emantics for some mo dal logic L we must nd a s et
if and only if it of character i stic equations E such that a formula i s valid in the f rame F
L E
L
i s a theorem of L For brevity I shall denote the f rame asso ciate d with the logic L by F rather
L
For instance the f rame F i s the s et of all mo dels sati sfying the equations add epi s than F
S E
L
norm and idem It i s known that a formula i s valid with re sp ect to F i it i s a theorem of the
S
logic S Hughe s and Cre sswell Chapter
Note that if i s a theorem of some logic L then and must have the same denotation
in every algebra in F Thus s ince and are interprete d as extens ional algebraic functions
L
and must also b e theorems of L Hence any mo dal logic which can b e
given an algebraic s emantics of thi s kind will b e clos e d under the rule of equivalence if
then which I shall refer to as RE
Power Set Algebras
7
According to Stone s repre s entation theorem Stone every Bo olean algebra i s i somorphic
to a Bo olean algebra whos e elements are s ets and whos e op erators are identi e d with the usual
union inters ection and complementation op erations of elementary s et theory Moreover such an
algebra can always b e emb e dde d in a Bo olean algebra whos e elements are all the subs ets of some
universal s et W
J onsson and Tarski showe d that a s imilar theorem holds for Bo olean algebras with ad
ditional additive op erators Thi s means that every mo dal algebra can b e i somorphically emb e dde d
W
in a mo dal algebra whos e elements are all memb ers of the p ower s et of some s et W One
may think of the elements of W as p oss ible worlds and s ince each prop os ition p of the mo dal
language i s interprete d as an element in the mo dal algebra may b e regarde d as the s et of
worlds in which p i s true
Where an algebraic mo del i s bas e d on a p ower s et algebra I shall repre s ent it by a structure
hU P i where the sum op erator i s to indicate that the Bo olean op erators corre sp ond
to the op erators of elementary s et theory As in the previous chapter I us e the meta symb ol U
to denote the universal s et in whatever algebra i s b e ing cons idere d The p ower s et algebras are
repre s entative of the whole class of mo dal algebras in the s ens e that an equation which i s true in all
p ower s et algebras i s true in every mo dal algebra b ecaus e every mo dal algebra can b e emb e dde d
in a p ower s et algebra Thi s means that in character i s ing validity in terms of algebraic f rame s we
can re str ict the f rame s to contain only mo dels bas e d on p ower s et algebras In the s equel I shall
assume that we always cons ider only mo dels bas e d on p ower s ets and I shall refer to the re sulting
s emantics as algebraic set semantics A mo dal op erator in a p ower s et algebra maps every
subs et X of the univers e to another subs et X
7
A comprehensive study of this theorem can be found in Johnstone
CHAPTER A MODAL REPRESENTATION
Mapping Between Algebraic and Logical Expre s s ions
As with the class ical s et s emantics it will b e us eful to intro duce meta level notation for referr ing
to the mapping b etween mo dal formulae and mo dal algebraic terms I assume that the s e terms are
interprete d as s ets in a p ower s et algebra Thus MAT i s the mo dal algebraic term obtaine d
f rom the formulae by replacing the connective s and by the op erators and
and the order constants p by s et constants P Since the op erator i s equivalent to
i i
thi s i s replace d by the algebraic op erator The function MF i s the invers e of MAT so
MAT
that MF i s the formulae such that MAT I shall wr ite to refer to the
MF
mapping in the form of a relation
I also de ne by analogy with CFe intro duce d in s ection the transform MFe such
that MFe U MF for universal equations and MFe MF MF for
non universal equations The expre ss ion MFe refers to a mo dal formula which b ecaus e of
the corre sp ondence theorem Mcorr which will b e given in s ection may b e regarde d as
repre s entative of the mo dal algebraic equation However b ecaus e of the form of the entailment
corre sp ondence theorem S ECT also prove d in s ection one might say that an equation
constraining an S mo dal algebra i s b etter repre s ente d by MFe rather than MFe
Equations character i s ing a class of algebraic structure s a f rame will in general contain f ree
var iable s which are taken as implicitly universally quanti e d the equations hold for all elements
of the algebra Thus an equation with f ree var iable s will corre sp ond to a class of mo dal formulae
which can b e repre s ente d as a formula schema Becaus e of thi s it i s convenient to generali s e MF
so as to op erate on terms with f ree var iable s In such a cas e the re sulting expre ss ion will b e a
mo dal schema rather than a formula and schematic logical var iable s will take the place of the f ree
var iable s in the algebraic term Accordingly MFe can also b e allowe d to op erate on equations
containing f ree var iable s again the re sult will b e a schema rather than a formula
By means of MFe a s et of algebraic equations de ning a f rame F can b e translate d directly
L
into a s et of mo dal schemas which sp ecify the pro of system of the corre sp onding logic L To ensure
the pro of system i s complete it will also b e nece ssary to add the inference rule RE which i s intr ins ic
to algebraic s emantics as explaine d at the end of s ection
Entailment among Mo dal Algebraic Equations
If some entitie s of intere st in our cas e the s e will b e spatial regions are identi e d with elements in
an algebra then equations b etween algebraic terms can b e us e d to sp ecify relationships b etween
the s e entitie s One can then reason ab out the s e relations in terms of entailments among algebraic
equations Since s et algebras are repre s entative of the class of mo dal algebras the notion of entail
ment among mo dal algebraic equations can b e de ne d in terms of p oss ible s et ass ignments to a
language of mo dal algebraic terms
A s et ass ignment to a language of algebraic terms i s a structure hS U mi where S i s
U
a s et of constants U i s a universal s et S ass igns a subs et of U to each constant in
U U
S and m sp eci e s the mo dal op erator as a s et function If i s a term built f rom
CHAPTER A MODAL REPRESENTATION
the constants in S by means of Bo olean and mo dal op erators then i s the s et ass igne d
to by Thi s i s determine d by m and the usual interpretation of Bo olean op erations
on s ets If we say that satis es the equation
Entailment relations among mo dal algebraic equations can now b e sp eci e d as follows
means that for every ass ignment hS U mi j
i n n
MA
L
where S include s all the constants o ccurr ing in the terms and sati sfying the equations
i i
asso ciate d with the f rame F if sati s e s the equations it also sati s e s
L n n
the equation
Relating S Mo dal Algebraic Entailment to De ducibility
If a mo dal logic L i s character i s e d by a mo dal algebraic f rame F there i s a corre sp ondence b etween
L
de duction in the logic and entailment b etween algebraic equations in the algebras in F Becaus e
L
of thi s we can us e mo dal logics to reason ab out algebraic equations
From the de nition of an algebraic f rame for the logic L we have the following corre sp ondence
b etween universal s et equations and logical theorems
MAT
U i where Mcorr j
L
MF
MA
L
In the last chapter we saw how class ical prop os itional formulae can b e us e d to reason ab out
spatial prop ertie s that can b e state d as equations of the form U The correctne ss of reasoning
us ing thi s enco ding was justi e d by the Class ical Entailment Corre sp ondence Theorem CECT
Later in thi s chapter s ections and we shall s ee how by us ing a s imilar corre sp ondence
theorem mo dal formulae can b e us e d to reason ab out a much wider range of spatial prop ertie s
To generali s e the class ical cas e to arbitrary mo dal logics we would nee d to e stabli sh the validity of
a conjecture such as the following
General Mo dal Entailment Corre sp ondence Conjecture GMECC
U i U U j
n L n
MA
L
MAT
where
i i
MF
Note that GMECC prop os e s a corre sp ondence b etween an entailment relation and a de ducib
ility relation rather than b etween two entailment relations as was the cas e for the theorem CECT
of the last chapter CECT relate s entailments b etween Bo olean s et term equations to order
entailments under the standard truth functional s emantics for C In us ing CECT to justify the
us e of class ical theorem provers for spatial reasoning we to ok for grante d the f act that any sound
and complete pro of system for class ical order logic i s f aithful to the truth functional s emantics
In attempting to e stabli sh GMECC one i s attempting to generali s e Mcorr which relate s mo dal
algebraic identitie s directly to mo dal theoremho o d and there i s prima facie no nee d to to intro duce
another s emantics
CHAPTER A MODAL REPRESENTATION
8
An even more imp ortant thing to note ab out GMECC i s that it i s not true for many
mo dal logics there are cas e s where an entailment b etween mo dal algebraic equations holds but
the corre sp onding logical entailment b etween mo dal formulae i s invalid For example in an S
p Neverthele ss as we would exp ect S mo dal algebra P U entails P U but p
S
do e s re sp ect Mcorr applying the de duction theorem to the s equent p p yields p p
P P U and thi s i s not generally true which corre sp onds to a mo dal algebraic equation
for algebras in the f rame F The problem ar i s e s b ecaus e all algebras in the f rame F must ob ey
S S
the identity or equivalently U U Indee d thi s identity i s sati s e d by all algebras
in any f rame F where the logic L ob eys the rule of nece ss itation
L
Although S do e s not ob ey the GMECC conjecture the following corre sp ondence b etween the
entailment relation among universal s et equations constraining algebras in F and the de ducibility
S
relation of S can b e prove d
S Entailment Corre sp ondence Theorem S ECT
U i U U j
n S n
MA
S 4
MAT
where
i i
MF
Pro of of S ECT Since S i s an extens ion of class ical logic it ob eys the de duction
theorem i By combining thi s with
n S S n
Mcorr we get the more general corre sp ondence
i j U
n S n
MA
S 4
Hence
i j U
n S n
MA
S 4
From elementary s et theory and the additivity of it can eas ily b e shown that the
equation on the r h s i s equivalent to so we can e stabli sh
n
S ECT by showing that
U i U U j j
n n
MA MA
S 4 S 4
The r h s can then b e re wr itten to give
U i U j j
n n
MA MA
S 4 S 4
and thi s equivalence can b e more succinctly expre ss e d as
U y i U j j
MA MA
S 4 S 4
8
De spite the existence of simple counter examples for a long time I believe d GMECC and I even publishe d a
faulty pro of in Bennett b Fortunately the slightly weaker theorem S ECT which is provable is su cient
to s erve the purp os e to which I or iginally put GMECC
CHAPTER A MODAL REPRESENTATION
It i s quite straightforward to show that the left to r ight direction of y holds for any
normal mo dal algebra and hence any algebra in F Recall that a mo dal algebra i s
S
normal i it ob eys the equation norm Thi s means that U U Thus
if in accordance with the l h s then it i s clear that any normal algebra
sati sfying U also sati s e s U which i s what the r h s says
The r ight to left direction of y i s cons iderably harder to show I prove it by proving
the contrap os itive i e
U yy then U j if j
MA MA
S 4 S 4
Let S b e the s et of all constants o ccurr ing in the terms and If the antece dent of
y y i s true there must b e some ass ignment hS U mi sati sfying the equational
constraints of the f rame F and such that From we can
S
construct an ass ignment which ver i e s the cons equent of yy i e U but
U
Let U We de ne hS U m i by stipulating that
U for all constants S
m X U U m U U X for all s ets X U
The sp eci cation of the mo dal function m lo oks rather complicate d however it i s just
the cons equence of requir ing that for any X U the value of m X according
to should b e equal to m X under To sp ecify thi s preci s ely I de ne l to
b e the dual of m i e l X m X From thi s de nition it i s easy to s ee
that m X l X The interpretation of as s et complement i s dep endent
on the sp eci c value of the universal region U To make thi s dep endence explicit
m X can b e expre ss e d as U l U X and convers ely l X U m U X
Similarly m X U l U X where l i s the dual of m If we now stipulate that
l X l X we nd that
m X U l U X U l U X U U m U U X
By sp ecifying m in thi s way I ensure that the op erator i s interprete d as the
same function in as in except that the domain of m i s limite d to subs ets of U
It can then b e shown that for any term made up of constants in S
U We know thi s identity holds for atomic terms b ecaus e of the de nition of
so to show it inductively for all terms we nee d to show that if it holds for and it
and For the Bo olean op erators the require d identitie s must hold for
are demonstrate d by the following s equence s of equations
U U U U U U U
U U U U
U U U U
CHAPTER A MODAL REPRESENTATION
In the rst of the s e the identity U U U dep ends on the f act that
U U
For the cas e of we have U We
can now interpret the op eration as l which has b een de ne d so as to coincide
with l thus U U l U l But I have
already shown that U so U l U l U
Now s ince m i s additive its dual l di str ibute s over giving U l l U
Becaus e U and all algebras in F sati sfy idem it i s easy to show that
S
9
l U U Now s ince U U it imme diately follows that U l U
U l U Finally thi s expre ss ion can b e rewr itten to give the de s ire d re sult
U l U U U U U
We must ver ify that the algebra sp eci e d by i s a memb er of F I have e stab
S
li she d that for every term built f rom constants in S U Thi s means
that every equation sati s e d by will also b e sati s e d by Since by
hyp othe s i s must sati sfy all the f rame equations of F must also sati sfy the s e
S
f rame equations
To complete the pro of I must show that ver i e s the r h s of y y Since the
algebra generate d by i s in F it must sati sfy epi s which means that for any term
S
and cons equently We know that U
so U but U so U Recall that was chos en to ver ify
the antece dent of yy b ecaus e Thus U and f rom thi s it
follows that U $ U Hence we have $ U
As with the class ical cas e an arbitrary mo dal s et equation can b e directly transforme d into
universal form and the formula MFe can b e regarde d as repre s enting the equational constraint
The mo dal logic S can thus b e us e d to reason ab out arbitrary equations constraining algebras
in the f rame F according to the following generali sation of S ECT
S
i MFe MFe MFe j
S n
MA
S 4
The form of S ECT i s a bit awkward in that in the S de duction corre sp onding to an entailment
b etween equations we nee d to add an extra op erator to the formulae on the left of but not
S
to the formula on the r ight Thi s means that the que stion What i s the S repre s entation of
the equation do e s not have a s imple answer However it i s eas ily shown that a s equent
Thus for the purp os e i s in f act valid if and only if
n S n S
of te sting entailments it can b e said that the repre s entation of an equation i s MFe
9 0 0
If the algebra speci e d by satis e s idem x x then m m X m X Thus l U m U
0
m m m mm m U The requirement
0 0
that l U U is of particular signi cance in that it is the reason why we nee d to have on
n
1 0
S 4
the r h s of S ECT rather than the simpler but stronger condition
n
1 0
S 4
CHAPTER A MODAL REPRESENTATION
Top ological Clo sure Algebras
The purp os e of my examining the algebraic s emantics of mo dal logics culminating in the demon
stration of the theorem S ECT was that just as CECT enable d us to us e class ical order logics
to reason ab out thos e spatial relationships which are e ss entially Bo olean in character S ECT will
enable us to reason ab out a wider range of relationships by means of de duction in the logic S
Thi s will enable us to reason in terms of certain top ological prop ertie s which were not expre ss ible
in the class ical repre s entation The key link i s that mo dal algebras of the f rame F are e ss entially
S
the same as closure algebras which give an algebraic character i sation of top ological space s
Clo sure and Inter ior Algebras
The theory of top ological space s i s traditionally state d in the language of s et theory But if we are
concer ne d only with the structure of a top ological space with re sp ect to the Bo olean op erations and
the inter ior and closure op erations we can do without the full language of s et theory and give a
purely algebraic account of the space which do e s not involve any us e of the elementho o d relation
Thi s abstraction re sults in a Bo olean algebra with an additional op erator ob eying appropr iate
conditions for e ither an inter ior or a closure function In the rst comprehens ive treatment of the s e
algebras McKins ey and Tarski the closure op erator was taken as pr imitive and the re sulting
algebra calle d a closure algebra A closure algebra i s a structure hS ci where hS i
i s a Bo olean Algebra and the op erator c sati s e s the equations for a closure function given in
s ection The s e include in particular the equation c X Y c X c Y which means that c
i s an additive function In other words hS ci i s a mo dal algebra
An interior algebra i s a structure hS ii where hS i i s a Bo olean Algebra and i sati s e s
the equations character i s ing an inter ior op erator An inter ior can b e interprete d in terms of a mo dal
algebra but with i corre sp onding to the algebraic op eration
Closure or inter ior algebraic equations provide a s imple constraint language for de scr ibing
top ological relationships b etween arbitrary s ets of p oints in a top ological space Some of the more
s igni cant constraints which can b e expre ss e d are given in table
Constraint Meaning
X c X X i s clos e d
X c X X i s op en
X c c X X i s regular op en
X Y Y X i s part of Y
c X Y Y The closure of X i s part of Y
X Y X and Y are di sjoint
c X c Y The closure s of X and Y are di sjoint
X Y Z X i s the union of Y and Z
Table Some constraints expre ss ible as closure algebra equations
CHAPTER A MODAL REPRESENTATION
RCC Relations Repre s entable in Inter ior Algebra
I now cons ider how RCC relations can b e repre s ente d in inter ior algebra To do thi s I employ the
top ological interpretation of the RCC theory which was given in s ection Recall that under
thi s interpretation regions are identi e d with non empty regular open sets Two regions overlap if
the ir corre sp onding s ets share a p oint and are connecte d if the closure s of the s e s ets share a p oint
Thus the relations can b e formally de ne d in terms of top ology by
O x y X Y
def
C x y c X c Y
def
The s e de nitions give us a r igorous formal sp eci cation of the RCC connection and overlap
relations in terms of p oint s et top ology But they make us e of a highly expre ss ive s et theoretic
language including b oth quanti cation and the element relation Given that the s e relations are
intuitively very s imple one may wonder whether it i s p oss ible to give an alter native character i sation
of C and O in the much le ss expre ss ive language of inter ior algebraic equations
As it happ ens the negations of each of the s e relations can b e quite eas ily de ne d as follows
X i Y U DC x y i
def
DR x y X Y U
def
But C and O cannot thems elve s b e de ne d as inter ior algebraic equations Thi s follows f rom
the general obs ervation that purely equational constraints are always cons i stent with any purely
equational theory there must always b e at least a tr ivial one element mo del in which all constants
denote the same individual Thus if the negation of some constraint can b e expre ss e d as an
equation then the constraint its elf cannot b e equationally expre ss ible otherwi s e that constraint
would b e cons i stent with its own negation
To de ne C and O we would nee d a language containing b oth inter ior algebraic equations and
the negations of such equations Thi s extende d language will b e cons idere d later but for now
I shall cons ider only thos e top ological relations de nable with equations alone Table give s
de nitions of s even binary relations DC DR P Pi NTP NTPi and EQ Thi s s et which will b e
calle d RCC i s of particular s igni cance b ecaus e as will b e shown in the next s ection each of
the RCC relations can b e expre ss e d as a conjunction of p os itive and negative RCC relations
Note that RCC i s ne ither jointly exhaustive nor pairwi s e di sjoint if two regions partially overlap
they stand in none of the s even relations and DR b e ing the di sjunction of DC and EC can hold
of two regions which are also DC A numb er of other binary RCC relations are expre ss ible by
10
means of inter ior closure algebra equations For example EQ sum x y u can b e expre ss e d by
X Y U
10
However it appears that RCC is the complete s et of binary RCC relations expre ssible in inter ior closure
algebra which are essential ly binary in that they are not re ducible to any monadic condition and speci cation of
the relation in RCC doe s not involve reference to a third region such as u Ver ifying this would require further
examination of the class of inter ior algebraic equations
CHAPTER A MODAL REPRESENTATION
RCC Relation Interior Algebra Equation
DC x y i X i Y U
DR x y X Y U
P x y X Y U
Pi x y X Y U
NTP x y i X Y U
NTPi x y X i Y U
EQ x y X Y X Y U
Table Seven relations de ne d by inter ior algebra equations
Note that equations in table assume that the regions are op en To make thi s explicit in
an inter ior algebraic repre s entation of RCC relations one ought to include an equation of the
form X i X for each region x o ccurr ing in the s et of equations In f act in s ection I
argue d that for a str ictly correct p oint s et interpretation of RCC relations one should require that
regions should b e regular op en Thi s requirement i s also eas ily enforce d by equations of the form
X i i X
One could equally well employ the inter ior algebra f ramework to sp ecify RCC relations in
terms of the dual s et theoretic interpretation of RCC mentione d in s ection Under that
interpretation regions are taken as non empty regular closed s ets which connect i they share a
p oint and overlap i they share an inter ior p oint The RCC relations would then b e sp eci e d as
given in table Thi s enco ding which i s arguably s impler than that of table was pre s ente d by
me in Bennett b and i s the bas i s of subs equent analys i s by Renz and Neb el However
in the next chapter where I pre s ent an intuitioni stic interpretation of inter ior algebraic constraints
we shall s ee that the op en s et interpretation i s much more convenient
RCC Relation Interior Algebra Equation
DC x y X Y U
DR x y i X i Y U
P x y X Y U
Pi x y X Y U
NTP x y X i Y U
NTPi x y i X Y U
EQ x y X Y X Y U
Table Alter native de nitions for clos e d regions
CHAPTER A MODAL REPRESENTATION
Us ing Inequalitie s to Extend Expre s s ive Power
I now cons ider a more expre ss ive constraint language bas e d on inter ior algebras in which one
can sp ecify b oth inter ior algebraic equalitie s and negations of such equalitie s Since each of the
RCC relations corre sp onds to an equation in inter ior algebra the extende d language allows
straightforward repre s entation of all thos e relations which can b e expre ss e d in the form
R x y R x y R x y R x y RCC conj
j j k
where each of the relations R i s a memb er of RCC
i
I have inve stigate d the complete s et of relations repre s entable in thi s way by means of a s imple
Prolog program the co de including inline d do cumentation i s given in app endix C Since
the RCC relations are not logically indep endent many combinations of the form of RCC conj
are equivalent It i s easy to sp ecify all the entailments and incompatibilitie s among pairs of
RCC relations and negate d RCC relations which are ass erte d of the same two ob jects Any
combination which contains an incompatibility i s equivalent to the imp oss ible relation and any
combination which contains two relations one of which i s entaile d by the other i s equivalent to
the combination re sulting f rom removing the entaile d relation Every combination containing no
incompatible pair and no relation that i s entaile d by another sp eci e s a di stinct relation in its
most s imple form and can b e regarde d as its canonical repre s entation The Prolog program rst
generate s every relation sp eci cation of the form of RCC conj and identi e s which of the s e are
canonical
We have s een that whether certain combinations of relations are regarde d as p oss ible dep ends
up on whether we allow regions to b e null the null region i s b oth part of and di sconnecte d f rom every
other region but no two non null regions can stand in b oth the s e two relations If we allow that
the regions involve d may p oss ibly b e null we nd that di stinct relations can b e repre s ente d
The complete li st of the s e relations i s given in app endix C If we require that b oth regions
involve d in a relation must b e non null then of the s e relations are p oss ible The s e include
each of the RCC relations Table shows how each of the RCC relations can b e expre ss e d
RCC Rel Equivalent RCC Conjunction Algebraic Constraint s
DC x y DC x y i x i y U
EC x y DR x y DC x y x y U i x i y U
PO x y DR x y P x y Pi x y x y U x y U x y U
TPP x y P x y EQ x y NTPP x y x y U x y i x y U
TPPi x y Pi x y EQ x y NTPPi x y x y U x y x i y U
NTPP x y NTPP x y i x y U
NTPPi x y NTPPi x y x i y U
EQ x y EQ x y x y
Table The RCC relations repre s ente d as inter ior algebra constraints
CHAPTER A MODAL REPRESENTATION
as a conjunction of RCC relations and the ir negations and also give s the corre sp onding inter ior
algebraic constraints
The relations of the form RCC conj form a semi lattice with re sp ect to the conjunction op er
ation Thi s just means that conjunction i s asso ciative symmetr ical and idemp otent Clearly the
sub structure compr i s ing only thos e relations including the non null constraints on b oth argument
regions also forms a s emi lattice It i s f airly easy to show by insp ection that the RCC relations
constitute a s et of minimal elements i e atoms of thi s s emi lattice One nee ds to check that
the re sult of conjoining any RCC relations with an additional RCC constraint i s e ither the
imp oss ible relation corre sp onding to the element of the lattice or i s equivalent to the or iginal
RCC relation
Each RCC relation i s equivalent to some di sjunction of RCC relations and b ecaus e RCC
i s JEPD jointly exhaustive and pairwi s e di sjoint the negations of RCC relations also corre sp ond
to di sjunctions of RCC relations provide d that non null constraints on the arguments are in
force Thi s means that each of the relations repre s entable in thi s way i s also a di sjunction of
RCC relations Hence the language of inter ior algebraic equations and the ir negations provide s
a repre s entation for almost half of the spatial relations which are di sjunctions of the
RCC relations In particular all of the RCC relations can b e expre ss e d as well as the pr imitive
C relation
Enco ding Clo sure Algebraic Constraints in S
It was e stabli she d by Tarski and McKins ey that the S b ox op erator can b e mo delle d
algebraically by an inter ior op erator We have s een that in the s et algebra interpretation of a
order logical calculus op erators are identi e d with maps f rom subs ets to subs ets of some univers e
the class ical connective s are asso ciate d with Bo olean functions and mo dal op erators are asso ciate d
with additive functions which may b e constraine d by further equational constraints A closure
algebra i s a Bo olean algebra with an additive closure op erator and i s thus a mo dal algebra c i s the
mo dal op erator which I have hitherto denote d by Hence c can b e taken as the interpretation of
a logical mo dal op erator I now show that the de ning equations of the c op erator mean that
thi s i s an S mo dal op erator
By making us e of the meta level notation relating mo dal algebraic equations and corre sp ond
ing mo dal formulae it i s easy to state preci s ely the relationship b etween closure mo dal algebraic
equations and mo dal formulae The repre s entation of a closure mo dal algebraic equation in
mo dal logic i s the formula MFe Becaus e the equations sp ecifying prop ertie s of the closure
op eration contain f ree var iable s they will b e mapp e d to mo dal schemata rather than formulae The
character i stic equations of a closure algebra and corre sp onding mo dal schemata are as follows
CHAPTER A MODAL REPRESENTATION
Closure Axioms Modal Schemata
X c X c X
c c X c X
c
c X Y c X c Y
Table Closure Axioms and Corre sp onding Mo dal Schemata
As mo dal formali sms are more often sp eci e d in terms of the op erator the transformation
bas e d on the dual corre sp ondence b etween then inter ior op erator and yields more f amiliar
schemata
Interior Axioms Modal Schemata
i X X X T
i i X i X
i U U N
i X Y i X i Y R
Table Inter ior Axioms and Corre sp onding Mo dal Schemata
Clearly T i s equivalent to the schema T s ee s ection and given that T holds
can b e weakene d to which i s the schema Furthermore it i s well known that
the schemata N and R in conjunction with the rule RE are equivalent to the combination of schema
K and the rule of nece ss itation RN Thus sp ecifying that N R and RE hold i s an alter native way
of sp ecifying that a mo dal logic i s normal s ee Chellas chapter Recall that RE holds
in any algebraic s emantics for a mo dal op erator Hence the mo dal logic der ive d f rom an inter ior
or closure algebra by transforming equational algebraic constraints into mo dal schemata i s exactly
the logic S Cons equently in virtue of the corre sp ondence theorem S ECT de duction in S can
b e us e d to reason ab out closure algebraic equations such as thos e given in table s and
RCC Relations Repre s entable in S
Since the S mo dality can b e interprete d as an inter ior function over a top ological space we can us e
thi s interpretation to enco de top ological relations as S formulae The bas i s of thi s repre s entation
i s exactly the same as for the C repre s entation but by us e of the additional mo dal op erator it i s
p oss ible to make a di stinction b etween connection and overlapping which cannot b e expre ss e d in C
Table shows the S formula corre sp onding to each of the RCC relations The middle column
shows the algebraic s et equation asso ciate d with the relation We s ee that if the inter ior op erator
i i s identi e d with the corre sp onding mo dal algebra op erator then the inter ior algebraic
equation i s repre s ente d by the S formula MFe
CHAPTER A MODAL REPRESENTATION
RCC Relation Interior Algebra Equation S formula MFe
DC x y i X i Y U x y
x y DR x y X Y U
x y P x y X Y U
x y Pi x y X Y U
x y NTP x y i X Y U
NTPi x y X i Y U x y
EQ x y X Y X X U x y x y
Table Seven relations de ne d by inter ior algebra equations and corre sp onding S formulae
I now illustrate how the corre sp ondence theorem S ECT enable s de duction in S to b e us e d
to reason ab out entailment among certain RCC relations Cons ider the following argument
NTP a b DR b c j DC a c
Thi s corre sp onds to the following entailment b etween inter ior algebraic equations
i A B U B C U A i A B i B C i C j i A i C U
11
Here the equations of the form i constrain the regions to corre sp ond to op en s ets By
app ealing to S ECT thi s can b e shown to b e valid b ecaus e we have
a c a b b c a a b b c c
S
The S repre s entation i s quite expre ss ive but do e s have s er ious limitations For instance
although b oth di sconnection DC x y and di scretene ss DR x y can b e repre s ente d it i s still not
p oss ible to sp ecify the relation of exter nal connection EC x y We have also s een that although
the ir negations can b e repre s ente d the fundamental relations C and O cannot b e repre s ente d In
order to overcome the s e de ciencie s we nee d a language in which one can expre ss closure algebraic
inequalitie s as well as equalitie s
Extende d Mo dal Logics L
In order to increas e the expre ss ive p ower of S so that we can repre s ent b oth p os itive and negative
algebraic constraints I us e the same metho d that was applie d to C in the last chapter Given a
order mo dal logic L we can de ne an augmente d repre s entation language L whos e expre ss ions
are pairs hM E i where M and E are formulae of L and are calle d re sp ectively model and entailment
expre ss ion hM E i i s cons i stent if and only if no formula in constraints We stipulate that an L
E i s entaile d by the s et M according to the logic L
11
In general to be faithful to RCC one should ensure that regions are regular open by adding the stronger
constraint i i but the inference in this example is valid for any open regions
CHAPTER A MODAL REPRESENTATION
Thi s kind of augmentation could in pr inciple b e applie d to any logical language whatso ever
However if it i s to b e us eful one must have some de nite interpretation of the meanings of L
expre ss ions or at least some of the s e expre ss ions Just as C expre ss ions may b e interprete d
as s ets of p os itive and negative equational constraints on Bo olean algebras L expre ss ions can
b e interprete d as s ets of p os itive and negative equational constraints on mo dal algebras De ning
such an interpretation require s a syntactic mapping b etween mo dal formulas and equations A
straightforward transform i s given by MFe and its invers e but we shall s ee that for reasoning
ab out most mo dal algebras slightly more complex mappings must b e employe d To us e L to te st
cons i stency of arbitrary p os itive and negative equational constraints every s et of such constraints
must b e repre s entable in L However it i s not nece ssary that every L expre ss ion b e interpretable
as a s et of constraints thi s interpretation may b e applicable only to a sub language of L For
instance we shall s ee that in the cas e of S only thos e expre ss ions where all mo del constraints
have as the pr imary op erator can b e coherently interprete d as constraints on S algebras
To show that an interpretation of thi s kind i s sati sf actory one must show that for all L ex
pre ss ions that are interpretable as s ets of algebraic constraints the stipulate d cons i stency checking
metho d for L expre ss ions i s sound and complete with re sp ect to cons i stency of the corre sp onding
constraints on mo dal algebras As in the cas e of C thi s task can b e divide d into two parts e stab
li shing a convexity re sult for entailment among mo dal algebraic constraints and then exploiting
an appropr iate corre sp ondence theorem relating entailments in the mo dal logic and entailments
b etween mo dal algebraic equations We shall s ee that b ecaus e of the f ailure of GMECC for most
mo dal logics the s econd step do e s not s eem to b e achievable in a uniform way a corre sp ondence
12
theorem if one exi sts must b e e stabli she d s eparately for each given mo dal logic
Convexity of Mo dal Algebras
In s ection we saw that the theory of equational constraints on Bo olean algebras i s convex in the
s ens e that a conjunction of equational constraints can only entail a di sjunction of such constraints
if it entails at least one di sjunct of that di sjunction Cons equently a s et of p os itive and negative
equational constraints i s cons i stent if and only if the contrary of one of the negative constraints i s
entaile d by the conjunction of the p os itive constraints The same re sult can b e prove d for mo dal
algebras i e Bo olean algebras supplemente d with additional additive op erators Since all mo dal
algebraic equations can b e put in the form U thi s i s guarantee d by the following theorem
Convexity of Di sjunctive Mo dal Algebraic Entailments MEconv
U U U U j
n m
MA
L
i
U for some i f ng U U j
i m
MA
L
Like BEconv MEconv i s clos ely relate d to ELcons By app ealing to ELconv and the
12
However in s ection I shall give an alternative metho d of extending mo dal logics by adding an extra operator
This metho d doe s yield a general corre sp ondence theorem for the extende d language s
CHAPTER A MODAL REPRESENTATION
f act that mo dal algebras can b e sp eci e d by purely equational theor ie s we can de duce that any
mo dal algebra i s convex w r t entailments of the form of DE In app endix B I give an alter native
mo del theoretic pro of of MEconv which may prove helpful to further study of the prop ertie s of
entailments among mo dal algebraic constraints Thi s pro of relie s only on the additivity of the
mo dal op erator and do e s not require that its algebraic prop ertie s b e sp eci able just in terms of
equations Neverthele ss b ecaus e mo dal schemata corre sp ond directly to universal equations in
the algebraic s emantics any mo dal op erator whos e logical prop ertie s are sp eci able in terms of
schemata will corre sp ond to an algebraic function which i s equationally sp eci able
A Corre sp ondence Theorem for S
The convexity theorem MEconv means that checking cons i stency of s ets of p os itive and negative
mo dal algebraic constraints re duce s to the problem of determining entailments among p os itive
constraints Thus if cons i stency of L expre ss ions i s to b e f aithful to cons i stency of the asso ciate d
algebraic constraints we only nee d to show that entailments b etween p os itive algebraic constraints
hold just in cas e the corre sp onding entailments in L are valid Thi s require s a corre sp ondence
theorem such as S ECT
By combining MEconv with S ECT a corre sp ondence b etween the cons i stency of mo dal
algebraic equations and inequalitie s and cons i stency of certain S expre ss ions i s imme diately
obtaine d Also b ecaus e of the interpretation of inter ior algebras as S mo dal algebras S can b e
us e d to te st cons i stency of top ological constraints The s e re sults are encapsulate d in the following
theorem which tie s together the main corre sp ondence theorems of thi s chapter
S Corre sp ondence Theorem S CT
The following three conditions are equivalent
The s et f g of S mo dal algebraic
j j k k
equations and inequalitie s i s cons i stent i e i s sati s e d by some algebra in
the f rame F
S
The corre sp onding s et of inter ior algebraic equations and inequalitie s re sulting
f rom replacing in the s et of constraints given in all o ccurrence s of by i
i s cons i stent i e i s sati s e d by some top ological space
The S expre ss ion hM E i given by
hf MFe MFe g fMFe MFe gi
j j k k
i s cons i stent i e there i s no formula E such that M
S
S CT enable s one to te st the cons i stency of s ets of spatial relationships repre s entable in
terms of inter ior algebra equations and inequalitie s by carrying out a s er ie s of pro of checks in
the logic S The de nition of cons i stency for S expre ss ions also yields cr iter ia for determining
CHAPTER A MODAL REPRESENTATION
entailment b etween S expre ss ions which i s exactly analogous to that given for C in s ection
S Entailment Theorem S ET
+
hM E i i hM E i j
S
+ +
e ither hM E i j or for all M hM f gi j
S S
and for all E hM f g E i j +
S
Thi s enable s a s imple generali sation of S CT to give a corre sp ondence b etween entailments
b etween s ets of mo dal or top ological algebraic constraints and entailments b etween S expre s
s ions One s et of constraints entails another i the entailment holds b etween the corre sp onding
pair of S expre ss ions
Repre s enting RCC Relations in S
Since S can repre s ent b oth equations and inequalitie s b etween terms made up of Bo olean op era
tions and an inter ior op erator it can expre ss a very large class of spatial relationships In particular
it can repre s ent all thos e RCC relations which can b e expre ss e d in the form RCC conj i e
as a conjunction of p os itive and negative RCC relations s ee s ection ab ove In the S
repre s entation the p os itive relations R R will corre sp ond to mo del constraints and the
j
negate d relations R R will corre sp ond to entailment constraints
j k
Relation Model Constraint Entailment Constraints
x y DC x y x y
x y x y x y EC x y
PO x y x y x y y x x y
x y x y y x x y TPP x y
y x y x x y x y TPPi x y
x y NTPP x y y x x y
NTPPi x y x y x y y x
EQ x y x y x y
x y x y C x y
x y z EQ x sum y z x y
Table The S enco ding of some RCC relations
The repre s entations of the RCC relations are given in table The way they are obtaine d can
b e summar i s e d as follows expre ss the RCC relations in terms of RCC relations and interpret
the s e as equational constraints on inter ior algebras as given in table Then translate the s e
constraints into S according to table The formulae corre sp onding to p os itive RCC relations
b ecome mo del constraints in the S repre s entation and thos e corre sp onding to negate d RCC
CHAPTER A MODAL REPRESENTATION
relations b ecome entailment constraints Note that the S corre sp ondence theorem require s
that mo del constraints have an extra initial adde d to the re sult of applying MFe to the mo dal
algebraic equation but thi s i s not require d in the entailment constraints Thi s asymmetry stems
f rom S ECT
Let us now cons ider how the S repre s entation can b e us e d to te st the cons i stency of a s imple
s et of spatial relations Take for example the following conjunction of RCC relations
TPP a b DC b c PO a c
Translating into S according to table we get the following repre s entation
hf a b b c g f a b b a a c a c c a a b cgi
Thi s i s an ordere d pair cons i sting of two s ets of S formulae the rst s et b e ing mo del constraints
and the s econd entailment constraints App ealing to part of S CT we determine that the
relations are incons i stent b ecaus e
a b b c a c
S
13
i e one of the entailment constraints i s entaile d by the mo del constraints
As mentione d in s ection one can also repre s ent RCC relations in inter ior algebra in terms
of the dual clos e d s et interpretation of RCC s ee s ection The re sult of enco ding thi s in
S i s given in table
Relation Model Constraint Entailment Constraints
DC x y x y x y
EC x y x y x y x y
PO x y x y x y y x x y
x y TPP x y x y y x x y
y x TPPi x y y x x y x y
x y NTPP x y y x x y
y x NTPPi x y x y x y
x y EQ x y x y
C x y x y x y
x y z EQ x sum y z x y
Table S enco ding bas e d on the clos e d s et interpretation of RCC
Regular ity and Bo ole an Combination of Regions
In the top ological interpretation of RCC given in s ection it was argue d that regions of the
RCC theory should b e identi e d only with non empty regular open subs ets of a top ological space
13
Str ictly speaking one should add extra mo del constraints of the form x for each region involve d
in the de scr iption s ee the following s ection However the s e additional formulae are not relevant to the example
CHAPTER A MODAL REPRESENTATION
Recall that a region x i s regular op en i i c x x If our mo dal enco ding i s to b e f aithful
to the intende d meaning of RCC relations we nee d to enforce thi s regular ity condition Happily
regular ity can eas ily b e expre ss e d in S as follows
p p p p or equivalently
It must b e note d that thi s condition i s not a general schema such that every instance must b e true
It i s rather an additional mo del constraint that should b e imp os e d on all the atomic constants us e d
in de scr ibing a s ituation b ecaus e the s e are intende d to b e identi e d with regular s ets
The regular ity of RCC regions i s also relevant to the enco ding of Bo olean functions of regions
In s ection I explaine d how if the regions of the RCC theory are to b e interprete d as regular
op en s ets then the Bo olean op erations sum pro d and compl of the theory corre sp ond to op erations
within a regular op en algebra rather than to the elementary Bo olean s et op erators In thi s algebra
inters ection corre sp onds to ordinary s et inters ection but the regular op en complement of a s et i s
the inter ior of its ordinary s et complement and the regular op en sum of two s ets i s the inter ior of
the closure of the ir union The s e op erations can eas ily b e repre s ente d in S and S pro d x y
i s translate d as x y compl x as x and sum x y as x y
Eliminating Entailment Constraints
The pro ce dure s for cons i stency checking and determining entailments for a logic L of the kind
de scr ib e d ab ove rely on the us e of s imple meta level reasoning In thi s s ection I explain how
by intro ducing a further additional mo dal op erator into the underlying logic L reasoning can b e
conducte d at the ob ject level of thi s enr iche d language which will b e de s ignate d L
In reasoning with an extende d order language L the meanings of the two typ e s of constraint
are handle d at the meta level determining entailments in the s e language s involve s checking a
numb er of di erent ob ject level entailments in the logic L A s et of algebraic constraints enco de d in
an L expre ss ion hM E i i s cons i stent if and only if none of its entailment constraints in E i s entaile d
by the s et of all mo del constraints in M A natural que stion regarding the s e repre s entations i s
whether it might b e p oss ible to extend the calculi involve d so that the s emantics of the two typ e s
of constraint was built directly into the ob ject language Thi s would mean that computation of
entailments could b e carr ie d out entirely at the ob ject level
In terms of algebraic s emantics it i s quite easy to intro duce a new mo dal op erator by means
of which the mo del entailment constraint di stinction can b e made at the ob ject level If i s
the algebraic denotation of a formula we de ne by
U i U
i U
14
Thi s op erator i s an S mo dal op erator s ince a formula i s true in a mo del i the formula
14
S is the mo dal logic obtaine d by adding the schema to the schematic speci cation of S In
terms of Kr ipke s emantics S is character is e d by the f rame of all Kr ipke mo dels whos e acce ssibility relations are
equivalence relations See e g Chellas for further information on S
CHAPTER A MODAL REPRESENTATION
i s true at every p oint world in the mo del I shall call it a strong S op erator b ecaus e it do e s not
allow the p oss ibility ar i s ing in the slightly weaker Kr ipke character i sation of S that there are
worlds p oints which are not relevant to evaluating the at a particular world b ecaus e the s et of
15
worlds i s partitione d into clusters which are not acce ss ible to each other Given the de nition of
we have U i U Thus negations of universal s et equations and hence all equations
can b e converte d into p os itive equations Thi s obviate s the nee d for entailment constraints s ince
a mo del constraint has the same meaning as taken as an entailment constraint More
sp eci cally the translation of an L expre ss ion
hf g f gi
j k
into L i s the formula
j k
Cons equently any expre ss ion of L can b e repre s ente d by a s imple ob ject level formula in the
multi mo dal language L
An Example of an Entailment Enco de d in C
Let us lo ok at a s imple example of spatial reasoning carr ie d out in C i e the class ical
order calculus supplemente d with a strong S b ox op erator Exactly the same pr inciple s apply to
reasoning in S but us ing C make s for a s impler and clearer example We shall cons ider the
trans itivity of the prop er part relation PP
PP a b PP b c j PP a c
x y U but y x U We also require that x and y are non null PP x y holds when
Non null constraints on regions can now b e expre ss e d as x for any region X Thus the mo dal
repre s entation of PP a b i s
a b b a a b
Hence the trans itivity of PP corre sp onds to the entailment
a b b a b c c b a b c
j a c c a a c
In te sting the validity of thi s entailment it i s natural to pro cee d as follows Since the r h s i s
a conjunction the s equent i s valid i each of the four s equents with the same l h s but just one
conjunct on the r h s i s valid Of the s e four s equents the two with a and c on the r h s
are tr ivially valid b ecaus e the s e formulae also o ccur on the l h s To prove the validity of the other
15
In most circumstance s the strong and weak S operators cannot be distinguishe d at the ob ject level But the
di erence may sometime s be signi cant For example a multi modal logic may contain s everal distinct weak S
mo dalitie s but only one strong S operator
CHAPTER A MODAL REPRESENTATION
two it i s convenient to move all conjuncts on the l h s which have an initial negation over to the
r ight We shall then have the following two s equents
a b b c j a c b a c b a b c
a b b c c a j b a c b a b c
We can ver ify the s e pro of theoretically by the application of just one mo dal rule together with
ordinary class ical reasoning Thi s i s the rule RK which holds in any normal mo dal logic
n
RK
n
Thi s rule together with the de duction theorem means that
if j then j
n n
Application of thi s pr inciple validate s b oth of our s equents s ince
a b b c j a c and a b b c c a j b a
The Utility of L as Compare d with L
Intro duction of the new b ox op erator to enable p os itive and negative constraints to b e di stingui she d
give s us a more uniform repre s entation Whereas previously the meaning of an expre ss ion was
tie d up e ss entially with the reasoning metho ds employe d in the new language expre ss ions have a
clear algebraic interpretation We nee d no longer concer n ours elve s with the di stinction b etween
mo del and entailment constraints but can now de scr ib e spatial s ituations s imply by a s et of mo dal
formulae and can reason ab out cons i stency and entailment directly in thi s ob ject language
On the other hand it i s not clear that thi s enr iche d language i s more de s irable f rom the com
putational p oint of view Intro duction of the new op erator make s the language f ar more expre ss ive
and cons equently much harder to reason with However we have s een that as long as the new mo dal
op erator i s only us e d to expre ss what was previously expre ss e d by means of the mo del entailment
constraint di stinction then all op erators will only o ccur e ither up f ront or negate d up f ront in
the s et of formulae de scr ibing a s ituation and it s eems likely that the optimal approach to reas
oning with such formula s ets i s to mimic the S cons i stency checking algor ithm de scr ib e d ab ove
Sp eci cally thi s means rewr iting the s equents according to s imple class ical pr inciple s to obtain
s ets of s equents in which all formulae have a s ingle at the f ront the l h s i s a conjunction and the
r h s a di sjunction of such formulae Once the s equents are in thi s form it i s easy to s ee that the
s equents which corre sp ond to entailments ver i able by the extende d order reasoning algor ithm
can all b e prove d us ing only the mo dal rule RK together with class ical reasoning
Since we know that the cons i stency checking metho d for S i s correct we can conclude that
only the rule RK i s nee de d to prove all entailments in L involving formulae in which the o ccurs
e ither up f ront or negate d up f ront Since the logic of S ob eys RK it follows that if S i s us e d
CHAPTER A MODAL REPRESENTATION
only to expre ss the mo del and entailment constraints of S one can in f act treat as if it were
just another S op erator Neverthele ss the more intuitive interpretation of the mo dal op erator
in thi s context i s as the strong S op erator In s ection I shall us e a mo dal repre s entation in
which strong S op erators are employe d within complex formulae In such contexts cannot b e
treate d as an S op erator
Chapter
An Intuitioni stic Repre s entation
and its Complexity
In the last chapter I showe d how spatial interpretation of the mo dal logic S enable s a wide
range of spatial relationship s to b e enco de d Thi s means that entailments among the s e relations
can b e determine d by means of an S theorem prover In thi s chapter I give an alter native
enco ding of spatial relations into the order intuitionistic calculus I also examine examine the
complexity of reasoning us ing the intuitioni stic repre s entation We shall s ee that the problem
of determining entailments i s in the p olynomial complexity class known as NC
The Top ological Interpretation of I
One of the most s igni cant early applications of s emantic metho ds to the inve stigation of logical
1
systems i s the top ological interpretation of the intuitioni stic calculus Tarski gave a s e
mantics for order intuitioni stic logic henceforth I which like that just given for S make s us e
of an inter ior op erator Under Tarski s s emantics a mo del for I i s a structure hU i P i where
now ass igns to each constant p P an open subs et of U a s et X such that i X X The
i
domain of i s then extende d to all I formulae as follows
i
i
Thi s denotation function i s such that all intuitioni stic theorems denote U under any ass ignment of
2
open s ets to non logical constants Note that I us e di erent symb ols and for negation
1
Mostowski Lecture give s an intere sting account of the early work in this area
2
In fact a more uniform pre s entation could be obtaine d by simply putting the s et de nition of the classical
connective s within the scope of an inter ior operator but in the cas e of the conjunction and disjunction connective s
the extra i operation would be re dundant since the unions or inters ection of two open s ets is always open
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
and implication in I f rom thos e us e d in C but for conjunction and di sjunction I us e the same
symb ols s ince the ir interpretations are the same in b oth systems The top ological interpretation of
I means that each formula of I can b e correlate d with an inter ior algebraic term which I shall
refer to by the meta notation IAT Thi s term i s obtaine d f rom by replacing prop os itional
and by i constants by s et constants by by by i
The algebraic s emantics for the intuitioni stic calculus sugge sts a quite straightforward spatial
interpretation which enable s one to understand clearly why certain theorems that are class ically
valid do not hold intuitioni stically Cons ider the inf amous law of the exclude d middle p p
The constant p will b e identi e d with an op en s et which we can think of as the s et of inter ior
p oints of some b ounde d region Intuitioni stic negation i s asso ciate d with the op eration of taking
the inter ior of the complement of a region in other words where p i s identi e d with the p oints
within a b oundary p i s identi e d with the p oints outs ide the b oundary The s et asso ciate d with
p p i s the union of the s ets asso ciate d with p and with p Clearly thi s contains all p oints
within our imagine d b oundary and all p oints outs ide the b oundary but do e s not contain any of
the p oints lying on the b oundary Hence the s et asso ciate d with p p do e s not nece ssar ily
contain all p oints in the univers e so formulae of thi s form are not in general theorems in f act a
formula of the form p p i s an intuitioni stic theorem if and only if e ither p or p i s a theorem
So although it may b e argue d that such top ological interpretations are not really in the spir it of
intuitioni sm the spatial interpretation can s erve to demystify and give a clearer understanding of
the intuitioni stic calculus
One drawback of thi s repre s entation i s that no logical op erator corre sp onding to the inter ior
function app ears explicitly in the language the function o ccurs in the interpretations of intu
itioni stic negation and implication and i s only referre d to indirectly in logical formulae us e d to
repre s ent spatial constraints Becaus e of thi s the I repre s entations of spatial relations are le ss
p erspicuous than thos e of the S enco ding where the mo dal op erator corre sp onds directly to the
inter ior function
Relation b etween I and S
In order to understand the relationship b etween spatial repre s entation in terms of I and the rep
re s entation in terms of S develop e d in the last chapter it will b e us eful to know something ab out
how the s e two logical language s are thems elve s relate d It has long b een known s ee Fitting
that formulae of the intuitioni stic prop os itional calculus can b e translate d into mo dal formulae in
such a way that an intuitioni stic formula i s a theorem if and only if the re sulting mo dal formula i s
valid in the logic S The translation can b e sp eci e d in terms of a recurs ive meta level function
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
trans as follows
trans p p where p i s a constant
i i i
trans trans
trans trans trans
trans trans trans
trans trans trans
Algebraic s et s emantics br ings out very clearly the a nity b etween I and S I can b e regarde d
as a sub language of S b ecaus e the algebraic terms asso ciate d with I formulae form a sub class
of the terms asso ciate d with S formulae Actually thi s i s not quite true b ecaus e whereas atomic
formulae in S may denote arbitrary subs ets of a top ologically structure d univers e thos e of I
denote only open subs ets of the space Thus in expre ss ing an I formulae in S every atom p must
b e replace d by the formulae p s ince corre sp onds to the inter ior function p will now denote
an arbitrary op en s et So intuitioni stic formulae corre sp ond only to a subs et of necessary S
formulae An intuitioni stic negation i s s emantically equivalent to the S op eration
i s equivalent to or Conjunction and di sjunction have and
the same interpretation in the s emantics of b oth logics and so are unchange d in the translation
to S
Corre sp ondence Theorem for I
Tarski s Second Pr incipal Theorem in the pap er Sentential Calculus and Topology Tarski
e stabli she s that a prop os itional formula i s a theorem of I if and only if the corre sp onding s et
term denote s the univers e in any top ological space under any ass ignment of op en s ets to the s et
constants o ccurr ing in the term The pro of of thi s i s f airly involve d and i s not reconstructe d here
I us e the notation to denote entailment in I and j to denote top ological entailment i e
I T
entailment b etween s et equations which may contain the inter ior op erator i Tarski s theorem can
3
then b e wr itten formally as
Intuitionistic Corre sp ondence Theorem Icorr
if and only if j IAT U
I T
In us ing I to repre s ent spatial relations we shall exploit very s imilar corre sp ondence relations
to thos e holding for C and S In order to s ecure the corre sp ondence b etween entailment in I and
entailment b etween s et equations in the top ological algebra of s ets we nee d to generali s e Tarski s
re sult to a corre sp ondence b etween entailments
3
This theorem holds for any top ology whatsoever Adding conditions to the top ology would mean the corre s
p onding logic would be stronger The limiting cas e is the discrete top ology corre sp onding to classical logic
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
Intuitionistic Entailment Corre sp ondence Theorem IECT
n I
if and only if
U U j U
n T
where IAT
i i
Pro of of IECT The p os itive half i s s imple an I entailment p p p holds
n I
i p p p so by Icorr we have j i U But
I n T n
if a s et has U as its inter ior then it must b e equal to U Cons equently the equation
U must hold in every mo del Thus whenever U for i
n i
n we must also have U in other words U U j U
n T
Supp os e on the other hand p p p Becaus e of Icorr thi s means that
n I
j i U so there i s some mo del M hU i P i in which there
T n
i s at least one element of which i s not an element of On the bas i s
n
of thi s mo del we now construct a mo del M hU i P i whos e univers e U i s the
s et denote d by in M We s et i X i X for all X U and for all
n
prop os itional constants p we s et p p U It i s easy to s ee that if hU ii i s a
i i i
top ological space then so i s hU i i s ee s ection
I now show that the new ass ignment i s such that for any formula U
Thi s condition i s clearly sati s e d by atomic formulae so it can b e prove d by induction
for all formulae if we can show that whenever formulae and sati sfy the condition it i s
also sati s e d by and The rst two cas e s are straightforward
U U U U
U U U U
The cas e of i s slightly harder to show
i U i U U i U
Then s ince U U we have i U i U U and i di str ibute s over
giving i U i U But U i s an inters ection of the s ets which are the
i
denotations of formulae So s ince all formulae denote op en s ets U must also b e
i
op en Hence i U U So we have
i U i U i U U U
Now cons ider
i U i U U U
Since U U it i s easy to show that thi s last term i s equivalent to
i U U
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
and b ecaus e i di str ibute s over inters ections and i U U thi s i s equivalent to
i U U
Finally we have
i U U U
Thus U for any formula So in particular for each i n
U U U i e in the new mo del all antece dent formulae
i i i
denote the univers e We also have U U Furthermore we
know that there i s at least one element of U which i s not an element of Thi s means
that U so M provide s a counter example to the entailment Thi s conclude s
the pro of of IECT
Intuitionistic Repre s entation of RCC Relations
The top ological interpretation of I enable s one to us e intuitioni stic logic in much the same way
as S to reason ab out spatial relationships Paralleling the approach of the previous chapter I
character i s e RCC relations as equational constraints in inter ior algebra and then rely on the cor
re sp ondence theorem to reason ab out the s e constraints us ing a theorem prover for the intuitioni stic
logic As note d ab ove the corre sp ondence b etween terms in an inter ior algebra and formulae of I i s
more indirect than the corre sp ondence with S formulae b ecaus e in the interpretation of I unlike
that of S no logical connective corre sp onds e ither to the inter ior or to the complement op erator of
the algebra However the enco ding of many top ological relations i s still straightforward Table
shows enco dings into I of each of the RCC relations intro duce d in s ection
RCC Set Equation I formula
x y DC x y i x i y U
x y DR x y x y U
P x y x y U x y
Pi x y x y U y x
x y NTP x y i x y U
NTPi x y x i y U x y
EQ x y x y x y U x y
Table Repre s entation of the RCC relations in I
In virtue of the theorem IECT an entailment among RCC relations holds if and only if the
corre sp onding intuitioni stic entailment holds Thus we can determine that the argument
NTP a b DR b c j DC a c
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
i s valid b ecaus e it corre sp onds to the following intuitioni stically valid s equent
a c a b b c
I
The I Enco ding
The language I extends the expre ss ive p ower of I in just the same way that C and S augment
the language s C and S Thus it enable s the sp eci cation of negative as well as p os itive equational
constraints on inter ior algebras Table shows how each of the RCC relations can b e repre s en
te d by s ets of mo del and entailment constraints sp eci e d by means of I formulae As with S the
repre s entations can b e obtaine d by rst analys ing the RCC relations into conjunctions of RCC
relations and the ir negations The I formulae corre sp onding to the p os itive RCC conjuncts then
b ecome mo del constraints and thos e corre sp onding to negative conjuncts b ecome entailment con
straints in the S repre s entation The table also shows how the fundamental relation C of the
RCC theory can b e repre s ente d as well as the quas i Bo olean function sum s ee s ection b elow
Relation Model Constraint Entailment Constraints
x y x y DC x y
x y x y x y EC x y
x y x y y x x y PO x y
x y y x x y TPP x y x y
y x x y x y TPPi x y y x
x y NTPP x y y x x y
y x NTPPi x y x y x y
x y EQ x y x y
x y x y C x y
x y z EQ x sum y z x y z
Table Some RCC relations de ne d in I including the RCC relations
Let us cons ider for example the repre s entations of the relations DC x y and EC x y If two
regions share no p oints they cannot overlap although they may b e connecte d In such a cas e
the equation i X Y U must hold thi s can b e repre s ente d by the I formula x y In I
unlike C thi s formula i s not equivalent to x y The latter corre sp onds to the s et equation
X i Y U which says that the union of the exter iors of two regions exhaust the space If i
the regions touch at one or more p oints then the s e p oints of contact will not b e in the exter ior of
e ither region so thi s equation will not hold Hence the s econd stronger formula can b e employe d
as a mo del constraint to de scr ib e the relation DC x y If the relation EC x y holds then the
weaker constraint x y holds but x y must not hold so thi s stronger formula i s an
entailment constraint
Cons i stency of I expre ss ions i s determine d analogously to C and S expre ss ions an I
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
expre ss ion hM E i i s incons i stent i there i s some E such that M Again the f act that
I
each of the negative constraints can b e cons idere d s eparately i s due to a convexity prop erty of the
class of constraints which are repre s ente d by thi s formali sm Since the theory of the top ological
inter ior op erator i s purely equational and the constraints corre sp onding to the mo del and entail
ment constraints are thems elve s also equations thi s convexity prop erty i s a direct cons equence of
ELcons which was prove d in s ection
In s ection I explaine d how the incons i stency of the de scr iption TPP a b DC b c
PO a c could b e demonstrate d by means of the S repre s entation The corre sp onding I
4
repre s entation according to table i s
hfa b b cg f a b b a a c c a a c a b cgi
Thi s I expre ss ion i s incons i stent b ecaus e
a c a b b c
I
i e one of the entailment constraints in entaile d by the mo del constraints
The Regular ity Constraint and Bo ole an Functions Co de d in I
In s ection I explaine d how regions could b e constraine d to b e regular by means of an S
mo del constraint In I thi s constraint can b e enforce d by the mo del constraint formula
p p
In the top ological s emantics thi s corre sp onds to the condition i i P P or equivalently
i c P P The condition P i c P nee d not b e explicitly adde d b ecaus e p p i s already
a theorem of I It i s intere sting to note that the intuitioni stic formulae ass igne d regular s ets by
the top ological s emantics are thos e for which the class ical law of double negation holds
As argue d in s ection the most coherent top ological interpretation of the RCC theory i s
to identify the RCC regions with regular op en s ets or alter natively regular clos e d s ets Thi s
to repre s ent RCC relations as well as adding mo del constraints means that in employing I
ensur ing regular ity of the regions explicitly mentione d one should also ensure that all Bo olean
combinations of the s e regions also corre sp ond to regular s ets To ensure thi s the s e op erations can
b e repre s ente d in I as follows pro d x y i s translate d as x y compl x as x and sum x y as
x y Given the top ological interpretations of the connective s involve d it i s easy to s ee that
if its argument regions are regular the regions denote d by any Bo olean function will b e regular
4
For full generality one ought to add extra mo del constraints constraining the regions to be regular as explaine d
+ +
in s ection Note that unlike S in I all regions are automatically constraine d to be open
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
E cient Top ological Re asoning Us ing I
The implementation of the spatial reasoning algor ithm de scr ib e d in Bennett b us e d a s equent
calculus pro of system for intuitioni stic logic which containe d certain optimi sations making it more
e cient in te sting the s equents require d by the top ological reasoning algor ithm and render ing it
incomplete for the full intuitioni stic logic Following the complexity analys i s of Neb el a it
b ecame apparent that a f ar more e ective sp ecial purp os e pro of pro ce dure could b e constructe d
Thi s s ection examine s the the pro of theory of the re str icte d class of s equents that nee d to b e te ste d
and shows how thi s analys i s yields an e cient clearly p olynomial pro of metho d
Sequent Calculus for I
To formalis e the pro of theory I us e a Gentzen style Gentzen s equent calculus for I which
i s e ss entially the same as that given by Dummett The pro of rule s of the calculus can b e
5
sp eci e d as follows
Axioms P P f C
Re wr ite P P f
def
P and Q P Q C
P Q C P Q Rule s
P C and Q C P or Q
P Q C P Q
P Q P Q P and Q C
P Q C P Q
When applie d in the top to b ottom direction the rule s pre s erve provability and generate all
valid s equents When us e d to prove a s equent the rule s are applie d b ottom to top in an attempt
to show that the s equent i s der ivable f rom axioms However not all rule s pre s erve provability
when applie d upwards so the pro of s earch i s non determini stic Rule s which pre s erve provability
in b oth directions are calle d invertible All the rule s are invertible except and
From the computational p oint of view the most s er ious defect of thi s rule s et i s that in applying
the rule proving a s equent i s re duce d to proving two s equents one of which may b e more
complex than the initial s equent In a depth rst s earch for a pro of thi s may lead to in nite
lo ops whos e detection i s computationally exp ens ive on the other hand a breadth rst s earch i s
extremely exp ens ive in terms of space
5
Roman capital letters denote arbitrary formulae Roman small letters denote atomic formulae and Greek capitals
denote arbitrary s ets of formulae The left hand side of a s equent is regarde d as a s et of formulae rather than a
s equence so the order of formulae on the left doe s not matter
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
Theorem proving in the I s equent calculus i s more complex than that of C in C all connective s
can b e eliminate d determini stically b ecaus e the rule s pro duce Bo olean combinations of s equents
which are logically equivalent to the or iginal s equent so all rule s are invertible Thus whereas
theorem proving in C i s NP complete so assuming P NP require s time which i s exp onential
in the s ize of formula to b e te ste d checking I theorems i s probably even more di cult it i s
b elieve d that it require s O n log n space as well as exp onential time Hudelmaier
Hudelmaier s Rule s
The problem ar i s ing f rom the rule has recently b een solve d by Hudelmaier The idea i s
to replace the rule by four more sp eci c rule s the applicability of which dep ends on the structure
of the antece dent of the formula Hudelmaier s rule s are
P Q R C
a P C
MP
P Q R C a a P C
P R Q R C Q R P Q and R C
P Q R C P Q R C
Each of the s e except i s invertible As indicate d by the us e of the small a the modus
ponens rule MP nee d only b e applie d when the antece dent of the implication i s atomic In
upwards application of each of the s e rule s the re sulting s equents can b e shown to decreas e in
complexity according to a sp ecially constructe d measure of s equent complexity
Spatial Re asoning Us ing Hudelmaier s Rule s
We have s een that cons i stency of spatial relations which are instance s of the RCC s et can b e
determine d by te sting the validity of certain I s equents Moreover if we are dealing only with the
RCC relations the s e s equents only contain formulae of the forms shown in table
a a b a b a b a b
In the remainder of thi s s ection I shall show how given the limite d range of formulae and the
completene ss of the Hudelmaier s equent rule s an e ective cons i stency checking pro ce dure for s ets
of RCC relations can b e constructe d
The s equent rule s assume that negation i s handle d by replacing each negate d formula by the
equivalent formula f Thi s can b e implemente d as a s imple determini stic re wr ite rule After
eliminating negations in thi s way another s impli cation can b e made by applying Hudelmaier s
rule Thi s means that formulae of the form a b are re wr itten rst to a b f and
then to a b f The re sulting s equents will contain only formulae of the forms
a f a b a b f a f b a f b f Iforms
Note that amongst the s e formulae the antece dents of all implications are atomic so us ing the
Hudelmaier rule s et the only rule applicable to implications i s MP Apart f rom implications the
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
only other typ e s of formulae are atomic prop os itions and two forms of di sjunction The di sjunctions
can b e handle d by the normal and rule s Both the s e rule s give r i s e to a branch in the
s earch space and b ecaus e there may b e any numb er of di sjunctions amongst the premi ss e s the
s earch space i s exp onential Neverthele ss b ecaus e the non determini stic rule i s not nee de d
thi s pro of pro ce dure can b e us e d to te st cons i stency of quite large s equents in reasonable time
The Prolog program given in app endix C i s bas e d on the metho d just de scr ib e d A slight
di erence i s that rather than re wr iting formulae of the form a b f to a b f and then
applying the normal MP rule I implemente d the following var iation of MP
a b P C
MP
a b a b P C
I also adde d a pruning rule to delete re dundant implications whos e conclus ion was already
amongst the premi ss e s Small optimisations such as thi s which are logically tr ivial can often yield
a marke d improvement in the p erformance of an automate d theorem prover In the next s ection
we shall s ee that pruning rule s play a key part in the sp eci cation of a p olynomial time pro of
pro ce dure for the s e s equents
Further Optimi sation
In s ection I shall pre s ent the mo del theoretic analys i s given by Ber nhard Neb el of the I
s equents ar i s ing f rom the RCC enco ding Thi s analys i s enable d Neb el to show that cons i stency
checking of s ets of RCC relations can b e p erforme d in p olynomial time Inspire d by thi s re sult I
inve stigate d how s equent calculus pro ofs of the relevant s equents could b e optimis e d As exp ecte d
pro ofs in the s equent calculus can also b e carr ie d out in p olynomial time In the re st of thi s s ection
I pre s ent a s er ie s of s equent re wr iting rule s which achieve s thi s end I assume that all formulae
in the s equents have b een re duce d to the forms Iforms as explaine d in the previous s ection
Eliminating Di sjunctions without Branching
Di sjunctions would normally b e eliminate d by applying the rule s and The s e create a
branch in the pro of we attempt to ver ify each of the s equents obtaine d by replacing the di sjunction
by one of its di sjuncts Clearly thi s pro ce dure leads to a s earch space which i s exp onential in the
numb er of di sjunctions which i s approximately prop ortional to the numb er of top ological relations
whos e cons i stency i s b e ing te ste d Thi s s ituation i s made wors e b ecaus e the rule must
b e applie d non determini stically However given the limite d class of formulae app ear ing in the
s equents rather than carrying out thi s split we can work out the p otential e ects without actually
applying a branching rule
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
The plan will b e rst to take account of the di sjunctive content of premi ss e s and conclus ion by
6
applying certain pruning rule s the s imple st of which take the following forms
Q P R Q P
Pr Pr
P Q P P Q P R
After carrying out all p oss ible applications of the s e rule s we will have an equivalent s equent in
which none of the di sjunctive premi ss e s have a di sjunct which i s the same as the conclus ion or a
di sjunct of the conclus ion Such di sjuncts will b e calle d un prunable
Another kind of pruning rule can b e applie d to implicative premi ss e s
P f Q
Pr
P Q Q
We notice that thi s rule applicable where the cons equent of an implication i s the same as the
conclus ion do e s not generali s e to the cas e of a di sjunctive conclus ion it i s not sound to re duce a
pro of of p q q r to that of p f q r and thi s i s preci s ely the re sp ect
in which an intuitioni stic implication P Q i s logically weaker than the di sjunction P Q
Although when the cons equent of an implication i s a di sjunct of the conclus ion we cannot prune
the implication its elf it may b e that thi s circumstance justi e s the pruning of some di sjunctive
7
premi ss in accordance with the following rule which has two var iants
P q r r r r r r S S T
n n n
Pr
P q q r r r r r r S S T
n n n
Thi s generali s e s Pr by taking account of chains of implication leading f rom a di sjunct of a
premi ss to a di sjunct of the conclus ion
In implementing thi s pruning rule it i s more convenient rst to compute the trans itive closure of
all formulae of the form p Q o ccurr ing in the s equent Once thi s i s done chains of implication
nee d not b e cons idere d so the pruning rule i s s imply applie d to s equents of the form P q q
S S T
Re ducing Di sjunctions to Implications
I now show that in the s equents in que stion the pruning rule s fully take account of the extent
to which the inferential p ower of di sjunctions excee ds that of corre sp onding implications Be
caus e of thi s after applying the pruning rule s we can replace di sjunctions with implications and
determini stically apply the rule Hence te sting validity i s re duce d to a Hor n like problem
Let us cons ider the inferential p otential of the remaining un prunable di sjunctive premi ss e s
The only rule that can directly b e applie d to the s e i s the rule However thi s rule cannot
directly yield the conclus ion or a di sjunct of it b ecaus e otherwi s e one of the di sjuncts would have
6
Tr ivial var iants of the s e rule s must also be applie d The s e are obtaine d by replacing P Q by Q P
and or replacing P R by R P in the rule s given ab ove
7
S T may be replace d by T S
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
b een prunable Thi s means that if the s equent i s valid at least one of the di sjuncts must b e capable
of taking part in a subs equent MP rule application Becaus e of the limite d range of formulae
in the s equents we can anticipate all the forms of p otential mo dus p onens applications and give
rule s which yield the same cons equence s but bypass the rule s Moreover all the s e rule s are
invertible
We have three rule s where the implication of the MP i s der ive d f rom a di sjunction
p q C
MP
p p f q C
q p f C p q f C
MP a MP b
p p f q f C q p f q f C
and three more rule s where it i s the antece dent that come s f rom the di sjunction
p f q f C
MP
p f q f p C
r f p p q r f q C
MP
r f p p q C
p f q q r f p f r f C
MP
p f q q r f C
Finally we have a numb er of rule s such as the following in which b oth the implication and its
antece dent are der ive d f rom di sjunctions
p f q q f r p f r C
MP a
p f q q f r C
It can now b e s een that the pro of p oss ibilitie s a orde d by the s e rule s are retaine d when formulae
of the form p f q are replace d by p q and formulae of the form p f q f by
the two formulae p q f and q p f
The re sult of applying rule s MP MP and MP can equally b e achieve d by applying
MP after thi s replacement
Rule s MP MP and MP all pro duce new di sjunctions but pr ior application of the prun
ing rule s ensure s that the s e cannot contain as a di sjunct e ither the conclus ion or a di sjunct of the
conclus ion Hence the s e new di sjuncts can only participate in a pro of by means of further applic
ation of one of the MP rule s Moreover if a chain of such applications i s us eful in constructing a
pro of it must eventually lead to an application of one of the rule s MP MP or MP which
yield a new non disjunctive formula Examination of the MP rule s will reveal that if di sjunctions
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
are replace d by implications as sp eci e d ab ove the re sult of any such s equence of rule s can b e
der ive d by a corre sp onding s equence of MP rule s
Given that thi s translation of di sjunctions to implications pre s erve s provability and noting that
the formulae q p f and p q f are logically equivalent it follows that provability i s
also pre s erve d if formulae p f q f are replace d by the s ingle formula p q f
Completion of the Pro of Pro ce dure
Having eliminate d all di sjunctive premi ss e s we are left with a s equent containing on the left hand
s ide only atomic prop os itions and implications with atomic antece dents and on the r ight hand
s ide a formula of one of the forms p f p q p q f p f q and p f q f
We pro cee d as follows
Cas e a For the non di sjunctive conclus ions we can imme diately apply the rule twice in
the cas e of a conclus ion of the form p q f so that the conclus ion i s re duce d to a s ingle atom
In the re sulting s equent the only p oss ible further rule applications are of Mo dus Ponens Thi s rule
i s applie d until e ither the atomic conclus ion i s der ive d in which cas e the s equent i s valid or els e
no p oss ible applications remain in which cas e the s equent i s invalid
Cas e b If the conclus ion i s a di sjunction we rst make all p oss ible applications of Mo dus
Ponens and attempt to der ive a di sjunct of the conclus ion If thi s f ails we then apply the rule
splitting the pro of into two branche s For each branch we pro cee d as for cas e a
Complexity of the Improve d Algor ithm
The numb er of formulae of a given typ e o ccurr ing in a s equent generate d by the RCC reasoning
algor ithm i s b ounde d by the s ize n of the s et of top ological relations to b e te ste d Checking
for applications of the Pr and Pr rule s i s clearly linear in n Determining applications of the
Pr rule involve s determining the closure of the trans itive relation of implication Thi s can b e
compute d in order n time Once thi s closure has b een compute d application of all p oss ible Pr
inference s b ecome s n s ince it involve s checking pairs of formulae f rom the l h s of the s equent
The other non tr ivial part of the pro of algor ithm i s the application of Mo dus Ponens rule s
Since the rule involve s two formulae one pass of MP applications i s order n Becaus e the
8
trans itive closure of implications has already b een compute d and b ecaus e the maximum numb er
of antece dents in a formula i s two a maximum of two pass e s are require d to exhaust all p oss ible
MP applications
So the pro of metho d de scr ib e d provide s an order n time algor ithm for checking cons i stency
of thos e I s equents which ar i s e in the top ological cons i stency checking algor ithm as compare d
with Hudelmaier s O n log n space algor ithm for arbitrary s equents The numb er of such s equents
which must b e checke d to determine the cons i stency of a s et of RCC relations i s equal to the numb er
8
This will also have to be re compute d after disjunctions are replace d by implications alternatively as in the
current implementation all the implications der ive d f rom disjunctions can be adde d at the beginning of the decision
pro ce dure
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
of entailment constraints in the I repre s entation of the relations which i s its elf approximately
prop ortional to the numb er of relations Thi s means that in terms of the numb er of top ological
relations whos e cons i stency i s to b e checke d the new algor ithm i s of order n
Implementation and Performance Re sults
The improve d algor ithm has b een conci s ely prototyp e d in SICStus Prolog The co de i s given
in app endix C Preliminary te sts indicate that the algor ithm can determine cons i stency of very
large s ets of top ological relations in an acceptable time The pro ce dure p erforms particularly well
if a databas e i s accumulate d incrementally so that at each stage computation of the closure of
implications i s linear in the numb er of implicative formulae already store d
Figure A spatial reasoner implemente d in Prolog us ing I
To te st the e ectivene ss of the algor ithm a cons i stent databas e of n top ological relations holding
amongst r regions was randomly generate d The relations were generate d by picking pairs of regions
at random and a random relation f rom the RCC relation s et with all regions require d to b e non
null If the randomly generate d relation was cons i stent with the databas e it was adde d otherwi s e
it was rejecte d Thi s was rep eate d until n cons i stent relations had b een adde d The random
databas e was then us e d to te st query re sp ons e time random RCC relations were generate d and
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
the I reasoner exploiting the improve d algor ithm was us e d to determine whether the s e relations
were nece ssary incons i stent or contingent with re sp ect to the databas e
The incremental addition of cons i stent relations holding among regions to ok on average
s econds and dur ing thi s construction on average incons i stent relations were rejecte d
The average query time for the re sulting databas e was s econds Further analys i s and revi s ion
of the program will b e nece ssary in order to enhance its p erformance Thi s i s b eyond the scop e of
the pre s ent work but it s eems very likely that an order of magnitude sp ee d up could b e obtaine d
9
quite eas ily
Neb el s Complexity Analys i s
I conclude thi s chapter with a lo ok at Ber nhard Neb el s mo del theoretic analys i s of the s equents
ar i s ing f rom the I enco ding of the RCC relations Thi s analys i s leads to an alter native pro of
that cons i stency of RCC relations can b e determine d in p olynomial time It also reformulate s
the problem within the f ramework of class ical constraints which has rece ive d much attention
f rom computer scienti sts Mackworth To understand thi s s ection fully it will probably b e
nece ssary to refer to Neb el a and to have some knowle dge of intuitioni stic mo del theory and
pro of theory s ee e g Kr ipke and Nero de By examining the intuitioni stic s equents
which are nee de d to reason with my I enco ding of the RCC relations Neb el a has shown
that the cons i stency of s ets of RCC relations can b e compute d in p olynomial time
Neb el s re sults are obtaine d by analys ing a tableau bas e d pro of pro ce dure for intuitioni stic logic
as de scr ib e d by Nero de when it i s applie d to the re str icte d range of formula typ e s
us e d in enco ding the RCC relations He showe d that the cons i stency problem for the s e s equents
can in f act b e de scr ib e d in terms of a f airly s imple s et of class ical constraints Thi s i s b ecaus e
for any invalid s equent involving only the formulae require d to repre s ent the RCC relations it i s
always p oss ible to construct a Kr ipke mo del Kr ipke containing exactly three worlds which
will b e calle d v w and w that provide s a counter example to the entailment Neb el s enco ding
s imply de scr ib e s the s e mo dels in class ical pre dicate logic by means of a binary relation F w a
which ass erts that the atomic formula a i s force d i e true at the world w
More sp eci cally each world of the Kr ipke mo del i s identi e d with a s et of constants which
are force d at that world The worlds are partially ordere d by the subs et order ing on the s e s ets
Whether a complex formulae i s force d at a world w dep ends on whether its constituents are force d
at w and also in the cas e of negation and implication whether they are force d at any larger
world
i s force d at w i b oth and are force d at w
i s force d at w i e ither or i s force d at w
9
One of the fundamental operations us e d in the Prolog program the assert pre dicate is known to be extremely
slow and pro le analysis of the program s run time showe d that over of the execution time was spent carrying
out this operation By re de signing the data structure s us e d by the algor ithm the us e of assert could be avoide d
and the performance greatly enhance d A lower level implementation e g in C would clearly be much faster
still
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
i s force d at w i i s not force d at w nor at any world larger than w
i s force d at w i at w and at any larger world wherever i s force d so i s
The counter mo dels identi e d by Neb el always sati sfy the order ing conditions v w and
v w so every formula force d by v i s force d by w and w In the s e counter mo dels the
world v i s constraine d so as to demonstrate the invalidity of a s equent v force s all the premi ss
formulae of the s equent but not its conclus ion The conditions under which a binary formula of
I i s force d at v in Neb el s counter mo dels can b e sp eci e d class ically as given in table To
te st if a s equent i s valid we cons ider a s et of constraints cons i sting of the forcing constraint for
each premi ss formula the negation of the forcing constraint of the conclus ion formula and also all
instance s of F v x F w x F w x where x i s a any constant o ccurr ing in the s equent
which ar i s e f rom the order ing conditions on the worlds Thi s s et of class ical formulae i s cons i stent
10
if and only if the or iginal intuitioni stic s equent i s invalid
Formula Forcing constraint
x F w x F w x
x y F w x F w x F w y F w y
x y F w x F w y F w x F w y
F w x F w x F v y x y
x y F v x F v y F w x F w y F w x F w y
Table Class ical de scr iption of intuitioni stic binary claus e entailment
Remarkably all the formulae in Neb el s class ical enco ding of the re str icte d I entailment problem
11
are re ducible to CNF form which means that the problem can b e re duce d to a SAT problem
Thus the cons i stency s ets of RCC relations can b e compute d in p olynomial time More preci s ely
SAT problems lie in the class NC which means that they can b e compute d in p olylogar ithmic
time on p olynomially many pro ce ssors so parallel pro ce ss ing can b e e ectively exploite d to sp ee d
up computation Thi s complexity re sult applie s also to the larger class of relations expre ss ible
in terms of conjunctions of the RCC relations and the ir negations all such relations can b e
repre s ente d us ing the I formulae covere d by Neb el s analys i s Thi s include s almost half thos e
relations that are di sjunctions over the RCC relations the full s et i s given in app endix C
It i s evident that applying paralleli sation can improve the p erformance of almost any algor ithm
that exploits my I enco ding Thi s i s b ecaus e each te st of whether an entailment constraint i s
der ivable f rom the mo del constraints can b e carr ie d out indep endently so all the s e te sts could b e
conducte d s imultaneously
The forcing constraint analys i s can also b e us e d to identify class e s of di sjunctive relations over
10
Note that Nebel s analysis doe s not cover the regular ity condition on regions Whether this can be repre s ente d
within a tractable system is a matter for further re s earch
11
A CNF formula is a conjunction each of whos e conjuncts is e ither a p ositive or negative literal or a disjunction
of two p ositive or negative literals
CHAPTER AN INTUITIONISTIC REPRESENTATION AND ITS COMPLEXITY
RCC for which cons i stency checking of constraint networks i s tractable Clearly the complete
ne ss of comp os itional inference applie s to any conjunction of RCC relations and the ir negations
and many di sjunctions of RCC relations are expre ss ible in thi s way In a network containing
di sjunctive relations it i s p oss ible to der ive new information by comp os ition without imme diately
getting a contradiction So showing incons i stency may require rep eate d application of comp os ition
Neverthele ss thi s pro ce dure still leads to an algor ithm which i s p olynomial in the numb er of no de s
of the network Neb el a In recent work by Renz and Neb el an analys i s very s imilar
to the forcing constraint interpretation of I i s applie d to the S enco ding of RCC relations given
in chapter Thi s enable s a maximal tractable class of di sjunctive RCC relations to b e
identi e d
Chapter
Quanti er Elimination
Thi s chapter explore s the p oss ibili ty of applying Quanti er Elimination transformations to
RCC formulae Such transformations provide a deci s ion pro ce dure for a large class of formulae
in the st order RCC language
Quanti er Elimination Pro ce dure s
The undecidability of a logical system i s very often asso ciate d with quanti cation General st order
logic i s only s emi decidable but by re str icting the forms of quanti cation p ermitte d in formulae
a var iety of decidable sub language s can b e found Dreb en and Goldf arb B orger Gr adel
and Gurevich Most of the b etter known order i e quanti er f ree logical formali sms are
1
also decidable The s e decidability re sults provide the bas i s for the metho d of constructing deci s ion
pro ce dure s by means of quanti er elimination Supp os e we have a st order language which i s in
some way re str icte d it may have re str icte d syntax or a limite d vo cabulary constraine d to ob ey
axioms of some theory If we can show that every formula of thi s language can b e converte d via
a s er ie s of transformations to a formula in a decidable language which i s cons i stent just in cas e
the or iginal formula were cons i stent then we have a deci s ion pro ce dure for the or iginal language
Typically the target language of such a convers ion will b e one with no or limite d quanti cation
so the e ect of transformation will b e to eliminate quanti ers
The metho d of quanti er elimination has b een us e d to remarkable e ect by Tarski to
provide a deci s ion pro ce dure for st order formulae comp os e d by applying the Bo olean connective s
and quanti cation to prop os itions which are arbitrary p olynomial equations and inequalitie s over
2
the real numb ers
1
A notable exception is general Relation Algebra which will be discuss e d in chapter
2
i e the non logical vo cabulary consists of the constants and the binary functions and and the
relations and The quanti ers range over the real numbers
CHAPTER QUANTIFIER ELIMINATION
Quanti er Elimination in RCC
In thi s s ection I prove certain equivalence s b etween RCC formulae which can b e us e d to eliminate
quanti ers in many contexts In f act the pro ofs of theorems us e d in the quanti er theorem will
b e given in a way which i s theory indep endent I s imply state every non logical assumption which
i s us e d in the pro ofs Thi s means that the elimination i s valid in any system in which the s e
assumptions are theorems All the s e assumptions are I b elieve provable f rom the RCC theory
However the treatment of Bo olean functions assume s a complete Bo olean algebra with a null
element so the ir form might have to b e altere d to t in with the sort structure of the RCC theory
All the assumptions are also theorems of my revi s e d theory given in s ection and s ince thi s
theory incorp orate s a null region assumptions involve d in the treatment of the Bo olean op erations
can b e expre ss e d directly in the theory
Cons ider the de nition of the part relation in terms of C
P x y z C z x C z y Pdef
def
If thi s de nition i s taken as a rewr ite rule applie d f rom r ight to left it can b e s een to achieve a
quanti er elimination a universally quanti e d expre ss ion involving Ci s replace d by an unquanti e d
expre ss ion in terms of P
Thi s elimination can b e generali s e d to remove a universal quanti er op erating on an arbit
rary truth functional combination of C relations First the truth functional matr ix i s converte d to
clausal normal form so that we have a conjunction of di sjunctions of C literals Since the universal
quanti er di str ibute s over conjunction it can b e move d inwards to obtain a conjunction of univer
sally quanti e d claus e s The quanti er can then b e eliminate d f rom each claus e in virtue of the
following equivalence
C claus e Quanti er Elimination Theorem CQE
x C x a C x a C x b C x b
m n
P a sumfb b g P a sumfb b g
n m n
The left hand quanti e d formula state s that if any region x i s connecte d to each of the regions
a a then x must also b e connecte d to one of the regions b b CQE state s that thi s
m n
3
i s equivalent to the condition that one of a a i s part of sumfb b g
m n
Pro of of CQE The equivalence of CQE i s demonstrate d by the following s er ie s of
formula transformations
x C x a C x a C x b C x b
m n
x C x a C x a C x sumfb b g
m n
x C x a C x a C x sumfb b g
m n
3
I wr ite sumfb b g as an abbreviation for a term of the form sum b sum b sum b b
n n
1 1 2 n 1
CHAPTER QUANTIFIER ELIMINATION
x C x a C x sumfb b g x C x a C x sumfb b g
n m n
x C a x C x sumfb b g x C a x C x sumfb b g
n m n
P a sumfb b g P a sumfb b g
n m n
The equivalence b etween and dep ends only on the de nition of sum and that b etween
and only on the de nition of P Steps and are purely logical equivalence s
and the entailment of by i s also purely logical That entails i s shown by the
following de duction s equence by means of which if we substitute sumfb b g for
n
the arbitrary term the negation of can b e der ive d f rom the negation of
x C x a C x x C x a C x
m
C k a C k C k a C k
m m m
C sumfk k g a C sumfk k g a C sumfk k g
m m m m
x C x a C x a C x
m
Sp ecial cas e s of the re duction apply when e ither the left or r ight s ide of the quanti e d C claus e
i s empty If the r h s i s empty then the claus e i s incons i stent s ince at least the universal region
must connect with all of any s et of regions If the l h s i s empty then the claus e s imply state s that
the sum of all the regions mentione d on the r h s i s equal to the univers e In terms of P thi s can b e
wr itten as P u sum fb b g We can thus eliminate the innermost universal quanti er s of
n
4
any pure C formula and in doing so end up with a formula containing only P and C relations the
remaining C relations are thos e not or iginally within the scop e of one of the innermost quanti ers
and the sum op erator
Extending the Pro ce dure
To continue the pro ce dure we would like to eliminate the innermost quanti ers of the re sulting
transforme d formulae Unfortunately the s e formulae are no longer pure they may contain other
non logical symb ols apart f rom C so the general cas e of further re duction i s more complicate d
than CQE The additional complexity take s the following forms
The quanti e d var iable may o ccur within the scop e of a sum op erator
The P pre dicate i s not symmetr ic so can act on a var iable in two logically di stinct ways
Both P and C relations may b e pre s ent
4
An RCC formula is a pure C formula i C is the only non logical symb ol o ccurr ing in it Other symb ols may
always be eliminate d by means of the ir de nitions The existential imp ort of the functions must then be taken care
of by suitable additional axioms for C
CHAPTER QUANTIFIER ELIMINATION
Quanti er Elimination for P Claus e s
My extens ion of the quanti er elimination pro ce dure only addre ss e s the rst two of the problems
just mentione d I give a pro ce dure which eliminate s quanti ers f rom a claus e containing only
P literals the arguments of the s e literals may b e arbitrary Bo olean functions and the quanti e d
var iable can o ccur anywhere within the s e complex arguments Such claus e s will b e calle d P claus e s
The decidability of P claus e s i s an intuitive cons equence of the decidability of the st order theory of
Bo olean algebras which i s well known the P relation can b e correlate d with the usual order ing on
Bo olean terms so that P can b e identi e d with the Bo olean equation However
it will b e instructive to demonstrate the decidability of P claus e s by quanti cation elimination
within the language of RCC Thi s will exp os e exactly which mereological pr inciple s are e ss ential
to a deci s ion pro ce dure
In the RCC language the re dundancy of quanti cation over truth functions of P relations i s
in many cas e s obvious For instance x P x a P x b P a b follows imme diately f rom
re exivity and trans itivity of P However for a general quanti er elimination pro ce dure we shall
nee d to eliminate quanti ers f rom all forms of P claus e To s implify thi s problem I rst convert
arbitrary P claus e s into a more re str icte d normal form In virtue of axioms AA and AA any
P literal P x x involving some var iable x where and are any quas i Bo olean functions
of constants and or var iable s can b e regarde d as a Bo olean inequality of the form x x
By applying appropr iate and well known Bo olean identitie s such an inequality can always e ither
b e shown to b e nece ssar ily true or otherwi s e b e transforme d into a conjunction of inequalitie s of
the forms x and x where x app ears alone and only on one s ide of the symb ol Thus
any P literal involving x i s e ither nece ssar ily true or equivalent to a conjunction of P literals of the
form
P x P x P x P x
i j
After applying thi s normali sation quanti ers can b e eliminate d f rom an arbitrary claus e made
up of P literals in virtue of the following equivalence
P claus e Quanti er Elimination Theorem PQE
x P x a P x a P b x P b x
i j
P x c P x c P d x P d x
k l
P sumfb b g pro d fa a g
j i
P pro d fa a g c P pro d fa a g c
i i k
P d sumfb b g P d sumfb b g
j l j
Pro of of PQE To s ee why thi s equivalence holds rst note that the left hand s ide i s
CHAPTER QUANTIFIER ELIMINATION
equivalent to
x P x pro d fa a g P sumfb b g x
i j
P x c P x c P d x P d x
k l
To make the pro of more conci s e I henceforth refer to pro d fa a g by and
i
sumfb b g by PQE then b ecome s
j
x P x P x P x c P x c P d x P d x
k l
P P c P c P d P d
k l
Becaus e the universal condition i s hard to vi suali s e I now transform it into an
exi stential If we negate b oth s ide s of thi s and then move the negations inwards we get
x P x P x P x c P x c P d x P d x
k l
P P c P c P d P d PQE
k l
The left to r ight direction i s relatively straightforward to demonstrate It can eas ily
b e der ive d by making us e of the following three pr inciple s de scr ibing prop ertie s of the
P relation
x P x P x P Ppr in
x P x P x P Ppr in
x P x P x P Ppr in
The r ight to left direction i s more di cult We must show that if the conditions
on the r ight are sati s e d there must b e some region sati sfying all the conditions of
the exi stentially quanti e d pre dicate on the left It i s clear that its elf sati s e s the
conditions P x and P x as well as all the conditions P d x However it do e s
n
not nece ssar ily sati sfy the conditions P x c To construct a region sati sfying all
n
the s e conditions we nee d to add extra bits to in such a way that the re sulting region
cannot b e part of any of the c s and we must furthermore ensure that after thi s addition
it still do e s not contain any of the d s as a part
By applying the pr inciple
P x P x O x POpr in
to the literal P c we get x P x O x c We let e b e some region
sati sfying thi s condition e i s di sjoint f rom c so clearly if we add it to then
P sum e c But we must construct a region that cannot violate any of the
n
conditions P d x sum e would violate one of the s e conditions if e containe d
n n
CHAPTER QUANTIFIER ELIMINATION
that part of d not containe d in i e if P di d e Thus rather than just
n n
adding e to we add a part of e der ive d by means of the following pr inciple
x y z P z x P y z Ppr in
which says that given any two regions there i s always some region which i s part of
the rst and do e s not contain the s econd as a part If we instantiate thi s with e and
di d we get z P z e P di d z
Let e denote a region which i s an instance of thi s exi stential statement e i s
cannot contain d clearly di sjoint f rom c s ince it i s part of e Moreover sum e
However it could still b e the cas e that sum e contains one of the other d s Thus
we recurs ively apply Ppr in to e to get a part of e which do e s not contain di d
l
as a part Thi s will b e calle d e Continuing thi s pro ce ss we nally end up with e
and we can b e sure that if thi s i s adde d to the re sulting region will not include any
l
of the d s Also s ince e must b e di sjoint f rom c the re sult will not b e a part of c
l
We let sum e
i s part of do e s not contain any of the d s and i s not part of c To complete the
pro of we nee d to succe ss ively extend to der ive a region which i s de nitely not part
of any of the c s and also do e s not contain any of the d s Thus to construct we rst
identify a region e which i s part of but not part of c we then form the s equence of
l l
regions c c where c i s di sjoint f rom c and do e s not contain any of the regions
l
di d i s then equal to sum e After rep eating thi s pro ce ss for each of the
n
c s we nally reach and thi s region sati s e s all the literals in the exi stential formula
k
on the left of PQE so thi s formula i s prove d Hence the equivalent formula PQE i s
also a theorem
Limitations and Us e s of the Pro ce dure
We would like to iterate quanti er elimination transforms to obtain a quanti er f ree formula but
a problem ar i s e s when we encounter in the cours e of the re duction a matr ix containing b oth C
and P relations s ince we have no way of eliminating a quanti er f rom a mixe d claus e of thi s kind
Indee d the undecidability of RCC means that no general quanti er elimination pro ce dure could
exi st I have studie d p oss ible ways of eliminating quanti ers in var ious re str icte d forms of mixe d
claus e In some cas e s the elimination i s straightforward but in other cas e s there s eems to b e no way
to get an equivalent quanti er f ree formula except by intro ducing additional relational vo cabulary
Thi s i s not in its elf a problem but it means that for succe ss ive iterations of quanti er elimination
claus e s containing an increas ingly extende d vo cabulary of relations must b e cons idere d
De spite its limitations the partial quanti er elimination pro ce dure de scr ib e d in thi s chapter
can b e us e d to extend the range of formulae that can b e handle d by a deci s ion pro ce dure which
employs one of the order enco ding technique s de scr ib e d in chapters and Sp eci cally one can
provide a deci s ion pro ce dure for a language which as well as allowing one to sp ecify the wide range
CHAPTER QUANTIFIER ELIMINATION
of spatial relationships that can b e enco de d directly into S or I also allows the ass ertion of
certain kinds of quanti e d claus e whos e quanti ers can b e eliminate d by applying the equivalence s
CQE and PQE as re wr ite rule s pr ior to translating into the order enco ding
Chapter
Convexity
In thi s chapter I inve stigate how the repre s entations de scr ib e d so f ar may b e extende d to
handle concepts relate d to convexity I rst pre s ent a st order axiomati sation of a convex
hul l op erator I then cons ider how the logical prop ertie s of the op erator can b e enco de d into
intuitioni sti c and mo dal repre s entations
Beyond Top ology
Hitherto I have cons idere d only prop ertie s of regions that are purely top ological in nature i e
prop ertie s that are invar iant under continuous deformations Whilst the s e prop ertie s are funda
mental they cannot provide the bas i s for a fully comprehens ive spatial de scr iption language A
fully expre ss ive spatial language would b e capable of expre ss ing metr ical information at least of a
relative kind if we intro duce an absolute metr ic unit we then have a language with the expre ss ive
1
p ower of ar ithmetic and which i s not completely axiomatisable The language of elementary p oint
geometry with a relative but not an absolute metr ic i s completely axiomatisable See e g Tarski
2
and app endix A but computing inference s within thi s language i s highly intractable
The value of a repre s entation language for AI dep ends on its expre ss ive p ower and its tractab
ility We saw in chapter that it i s p oss ible to reason e ectively with certain top ological relations
An obvious que stion i s whether one can nd a more expre ss ive language which i s still tractable and
more sp eci cally whether one can nd a tractable language capable of expre ss ing non top ological
spatial concepts Such a language would contain one or more pr imitive concepts that are not
top ological in character
Interme diate in expre ss ive p ower b etween top ology and metr ical geometr ie s such as Euclidean
geometry i s a ne geometry An a ne geometry articulate s the concept of betweenness but cannot
expre ss orthogonality or say anything ab out angular relationships b etween ob jects In thi s chapter
I cons ider a ne geometry f rom the p oint of view of reasoning in a region bas e d theory
1
Tarski b demonstrate s that a formal geometr ical language containing a congruence relation and a unit
element as pr imitive s is in some s ens e maximal ly expre ssive
2
In fact s everal distinct complete geometr ie s can be formulated s ee e g Trudeau
CHAPTER CONVEXITY
B A
B
Figure Illustration of convex hulls in dimens ions
The Convex Hull Op erator conv
The relation of b etweenne ss i s intimately connecte d with convexity a region i s convex if it i s clos e d
with re sp ect to b etweenne ss i e if every p oint lying b etween two p oints in the region i s its elf in
the region Thi s the s i s i s pr imar ily concer ne d with expre ss ing spatial prop ertie s of regions rather
3
than p oints and in a region bas e d theory it can b e argue d that convexity i s a more pr imitive notion
than b etweenne ss to decide whether one region lie s b etween two others one must cho os e b etween
a var iety of stronger or weaker notions of b etweenne ss can the regions overlap must al l p oints
of the inner region b e b etween the outer regions but the prop erty of a region s b e ing convex i s
not so ambiguous Of cours e a r igorous s emantical de nition of convexity require s regions to b e
cons idere d as subspace s of some a ne space so the class of convex regions will b e dep endent on
the prop ertie s of thi s space In the purely region bas e d analys i s of convexity carr ie d out in thi s
chapter I assume that the axiomatis e d prop erty of convexity i s intende d to b e cons i stent with an
interpretation in Euclidean space
Following Randell Cui and Cohn I take the convex hul l op erator conv as a pr imitive
function mapping regions to the ir convex hulls By the convex hull of a region i s meant the smalle st
convex region of which it i s a part If one were to stretch an elastic membrane round a region then
4
the convex hull would b e the whole of the region enclos e d Figure shows convex hulls of two
regions in dimens ions region B i s a two piece region
The conv function and a pre dicate CONV true of convex regions are inter de nable
CONV x x conv x
def
conv x y CONV y z CONV z P x z P y z
def
There are many p oss ible ways in which a ter nary relation Between x y z read y i s b etween x
and z could b e de ne d in terms of conv The s e capture di erent preci s e s ens e s of the somewhat
ambiguous natural concept of b etweenne ss Most of the ambiguity of b etweenne ss ar i s e s in
5
connection with its application to extende d b o die s rather than p oints A very weak de nition of
3
For an axiomatic and algebraic analysis of computing convex hulls of s ets of p oints s ee Knuth
4
One might say the term is slightly inappropr iate since hull normally refers to an outer shell rather than a
volume or area
5
The intuitive meaning of the betweenne ss relation on p oints leave s little scope for ambiguity except that we
CHAPTER CONVEXITY
b etweenne ss i s the following
Between x y z P y conv sum x z W
def
Between could its elf b e taken as pr imitive and CONV could then b e de ne d by W
CONV x y v w P v x P w x W Between v y w P y x
def
W Between do e s not really capture the intuitive notion of b etweenne ss b ecaus e it allows cas e s
such as where y i s in a cavity of x which i s on the opp os ite s ide of x to that f acing z It also allows
y to overlap or even b e part of e ither x or z Before giving a b etter de nition we nee d to b e clear
ab out the main asp ects of ambiguity in the concept One source of ambiguity concer ns whether
the regions involve d may overlap Probably the most natural way to s ettle thi s i s to require that
y cannot overlap e ither x or z but allow that x and z may p oss ibly overlap A s econd source of
ambiguity i s whether y must b e completely b etween x and z or may b e only partly b etween them
Both s ens e s are easy to de ne but it s eems most straightforward to de ne partial b etweenne ss
rst
Between x y z O x y O z y P
def
x z P x x P z z CONV x CONV z O y conv sum x z
We can then say that y i s completely b etween x and z if every part of y i s partially b etween
them
Between x y z y P y y P Between x y z
def
Intere stingly CONV can b e de ne d f rom Between in exactly the same way that it i s de ne d f rom
Between W
Containment Relations De nable with conv
A large numb er of new binary relations can b e de ne d in terms of the conv together with other
RCC relations For example Randell Cui and Cohn give the following de nitions of three
6
p oss ible containment relations which form a di sjoint and exhaustive partition of the DR relation
INSIDE x y DR x y P x conv y
def
P INSIDE x y DR x y PO x conv y
def
OUTSIDE x y DR x conv y
def
may wish to distinguish between strict betweenne ss where y may not be equal to x or z and the weaker version
us e d in Tarski s Elementary Geometry s ee appendix A which doe s allow this p ossibility
6
It may be argue d that for many purp os e s relations involving convex hulls are most informative when we are
consider ing non overlapping regions Such regions can corre sp ond to discrete physical b o die s regarding which we
will often be intere ste d in spatial propertie s that are much more complex than simply whether or not the regions
touch
CHAPTER CONVEXITY
OUTSIDE_INSIDEi_EC INSIDE_OUTSIDEi_EC OUTSIDE_OUTSIDEi_EC P-INSIDE_P-INSIDEi_EC INSIDE_INSIDEi_EC
OUTSIDE_P-INSIDEi_EC P-INSIDE_OUTSIDEi_EC
P-INSIDE_INSIDEi_EC INSIDE_P-INSIDEi_EC
Figure Nine re nements of EC
Randell Cui and Cohn us e the s e relations to di erentiate a s et of relations which
can hold among any two regions The relationship b etween overlapping regions i s s imply de scr ib e d
in terms of one of s ix p oss ible RCC relations For EC and DC regions we additionally sp ecify
the two containment relations R x y and R y x where R R fINSIDE P INSIDE OUTSIDEg
For DR regions each of the re sulting nine combinations of R and R i s p oss ible the EC cas e s are
illustrate d in gure Thi s yields a s et cons i sting of JEPD relations However
if two regions are nite and mutually INSIDE each other then b ecaus e of axiom they cannot
b e DC so in the cas e where the regions are require d to b e nite only of the s e relations are
p oss ible Thi s s et i s known as RCC
Following Randell Cui and Cohn I repre s ent the RCC relations that are sp eciali sa
tions of EC and DC by expre ss ions of the form x y where i s e ither I P or O
according as e ither INSIDE x y P INSIDE x y or OUTSIDE x y refers to the corre sp onding
invers e relation i e one of the s e relations but with the arguments revers e d and i s e ither D
or E according to whether the regions are completely di sconnecte d or exter nally connecte d Thus
for example P I E x y means that P INSIDE x y INSIDE y x and EC x y
More generally by combining bas ic RCC relations with the conv op erator we can sp ecify a
large numb er of relations by means of expre ss ions of the form
R x y R x conv y R conv x y R conv x conv y
Although there are di erent expre ss ions of thi s form the logical prop ertie s of convexity
mean that many of the s e are equivalent indee d many are equivalent to the empty imp oss ible
relation x y The numb er of di stinct relations expre ss ible in thi s way has not yet b een de
termine d but de spite the equivalence s it i s clearly quite large
st Order axioms for conv
In order to construct a logical language in which the op eration of forming the convex hull of a region
i s incorp orate d into the vo cabulary it i s nece ssary to understand and formali s e the logical prop ertie s
of the new op erator An obvious starting p oint i s to sp ecify fundamental prop ertie s of the convex
7
hull op erator in st order logic I give s even axioms sp ecifying imp ortant prop ertie s of conv
For readability I make us e of the CONV pre dicate de ne d ab ove s ection I also intro duce a
7
Earlier versions of the axioms can be found in Randell Cui and Cohn Bennett b Cohn
CHAPTER CONVEXITY
pre dicate Fin x to ass ert that x i s nite Thi s i s nee de d to expre ss a prop erty of convexity that
8
only holds for nite regions I shall not assume any sp eci c s et theoretic interpretation of regions
My intention i s that the axioms should b e compatible with any of the p oss ible interpretations of
RCC de scr ib e d in s ection
x TP x conv x x u
x y P x y P conv x conv y
x y Fin x conv x conv y C x y
x y conv x conv y conv x y
x y CONV conv x conv y
x y DC x y CONV x y
x y NTPP x y conv x u CONV y x
Axiom state s the obvious f act that a region must b e a tangential part of its convex hull An
exception to thi s requirement i s the universal region u if u i s convex then it will b e equal to its
own convex hull but TP u u i s f als e at least under the de nition of TP given by Randell Cui
and Cohn If u i s not convex then conv cannot b e a total function Axiom expre ss e s a
monotonicity prop erty taking convex hulls pre s erve s partho o d relationships Axiom ensure s
9
that any two nite regions having the same convex hull must b e connecte d The next three axioms
connect the prop ertie s of convexity to the Bo olean functions Axiom says that if we take the
convex hull of a sum then any convex hull op erators on the summands are re dundant Axiom
ass erts that the inters ection of any two convex regions must its elf b e convex Axiom expre ss e s
the obvious f act that the sum of two DC regions cannot b e convex Axiom expre ss e s a s imilar
prop erty that shap e s with inter ior hole s cannot b e convex The condition conv x u rule s
out anomalous counter example s where the complement of a convex region i s subtracte d f rom uto
yield a convex region
Thi s li st i s not guarantee d to b e a complete axiomatisation of the conv op erator It i s very
di cult to b e sure that a s et of axioms fully capture s a concept unle ss we have a formal mo del
or s et of mo dels within which the concept i s de ne d and show that the axioms are sound and
complete with re sp ect to that mo del thos e mo dels Inve stigating such mo dels i s the sub ject of
ongoing work Short of proving completene ss we can gain con dence in our axiom s et by showing
that exp ecte d prop ertie s of convexity can b e der ive d f rom our axiom s et For instance the following
8
Intro duction of the Fin pre dicate is metho dologically dubious since nitude is not st order axiomatisable
Neverthele ss for pre s ent purp os e s it is convenient to assume Fin as pr imitive in order to state one of the propertie s
require d of the convex hull function under its intende d interpretation
9
It might be imagine d that certain nite but in nitely complex regions could have the same convex hull and
yet not be connecte d However I have not been able to nd a reasonable s et theoretic interpretation in which this can o ccur
CHAPTER CONVEXITY
10
theorems are quite easy to prove
x conv conv x conv x f rom
x y P conv x conv y conv x y f rom
The rst of the s e expre ss e s the s imple f act that applying the convex hull op erator a s econd time
in succe ss ion i s re dundant and the s econd ass erts the di str ibutivity of conv and with re sp ect to
P Since the s e prop ertie s are very s imple we or iginally include d them in our axiom s et It may
still b e the cas e that one or more of our current axioms i s der ivable f rom the re st
Apart f rom the implicit exi stential imp ort of the conv function its elf all the conv axioms given
so f ar are universal in nature However one might exp ect there to b e other exi stential axioms
involving convexity Indee d s ince the domain of regions in the RCC theory i s atomle ss it s eems
reasonable to require that every region has b oth convex non tangential prop er parts and convex
tangential prop er parts
x y NTPP y x CONV y and x y TPP y x CONV y
Enco ding conv x in I
In Bennett b I de scr ib e d a metho d of reasoning ab out convexity by means of a meta level
extens ion of the intuitioni stic enco ding de scr ib e d in chapter The language I i s extende d to a
language I in which as well as having ordinary constant symb ols c denoting regions one can
i
conv
also employ terms conv c to refer to the convex hull of the region c Here conv i s to b e regarde d
i i
as a meta level syntactic device rather than a real function symb ol the I reasoning algor ithm
s imply treats conv c as an atomic constant The meaning of conv i s then character i s e d by an
i
additional meta level reasoning mechani sm which enforce s constraints asso ciate d with convexity
The constraints enforce d in my or iginal system corre sp ond to the following axiom s et
x conv conv x conv x
x TP x conv x
x y P x y P conv x conv y
x y conv x conv y C x y
Thi s s et amende d and slightly extende d a previous axiom s et that had b een given in Randell Cui
and Cohn however as we saw in s ection it i s now clear that further axioms are nee de d
to adequately character i s e conv It i s also known that the last of the s e axioms only applie s to nite
regions Neverthele ss it i s worth de scr ibing how the limite d axiom system can b e enforce d and
cons ider ing how thi s approach could b e extende d to take account of additional prop ertie s of conv
Obs erve that none of the axioms contains any Bo olean op erators and also that in our exten
de d I the conv ps eudo op erator can only b e applie d to an atomic constant Cons equently the
10
Thanks to Stephano Borgo
11
Convers ely can be der ive d f rom I prefer to take the former as an axiom since it doe s not involve any
Bo olean operators
CHAPTER CONVEXITY
relationships p oss ible b etween Bo olean combinations of region constants and or the ir convex hulls
12
are not in any way constraine d by the limite d axiom s et Moreover s ince all the axioms are
universal apart f rom the implicit exi stential imp ort of the conv function they are equivalent to
the s ets of all the ir ground instance s In determining whether a s et of spatial f acts state d in I
conv
i s cons i stent with the axioms the only instance s of the axioms which can b e relevant are thos e
where the var iable s are replace d by constants o ccurr ing in the f acts We thus treat the st order
axioms as schemas and instantiate them in all p oss ible ways us ing the region constants o ccurr ing
in the spatial f acts under cons ideration Thi s will re sult in a nite numb er of ground constraints
We must now cons ider how to te st whether the f acts are cons i stent with the additional convexity
constraints Axiom can have no e ect on cons i stency s ince expre ss ions of the form conv conv x
do not o ccur in I indee d the axiom tells us that there i s no reason why we should nee d to
conv
employ such expre ss ions The constraints ar i s ing f rom axiom can imme diately b e translate d into
I formulae just as any other TP relation Instance s of axioms and are of most intere st and
illustrate a general metho d by which I could b e extende d We s ee that each of the s e i s a s imple
Bo olean combination of top ological constraints P and C that can b e directly repre s ente d in I
The s e Bo olean combinations of I expre ss ible constraints can b e interprete d at the meta level
in terms of Bo olean combinations of I cons i stency problems For example if we have a s et of
f acts expre ss ible in I and add to the s e a f act such that where b oth and
are expre ss ible in I then the s et of f acts f g i s cons i stent if and only if e ither f g i s
cons i stent or f g i s cons i stent However it i s clear that the numb er of I cons i stency checks
require d to te st cons i stency of a spatial s ituation de scr iption involving Bo olean combinations of
I expre ss ible conditions i s exp onential in the numb er of di sjunctions o ccurr ing in the s e Bo olean
combinations Moreover s ince enforcing axioms such as the conv axioms require s one to cons ider
all p oss ible instantiations over the regions mentione d in the s ituation de scr iption the numb er of
di sjunctive constraints may b e quite large
Treatment of axioms and i s encompass e d by a general pro ce dure which enable s enforcement
of all axioms of the form
x x x x x x
n n n
where x x and x x sp ecify s ituations which can b e de scr ib e d by means of I
n n
To te st whether a given I s ituation de scr iption sati s e s such an axiom an iterative xe d p oint
metho d can b e us e d
Te st the I de scr iption for cons i stency If it i s incons i stent stop
Check whether any instance of the antece dent i s entaile d by the I de scr iption Thi s involve s
translating into I and substituting all combinations of constants o ccurr ing in the
de scr iption for the f ree var iable s If any such instance i s entaile d add the corre sp onding I
repre s entation of under the same substitution to the de scr iption
12
In a more complete s et we would have axioms such as which relate s conv to the Bo olean sum operator
CHAPTER CONVEXITY
Check whether any instance of the cons equent i s incons i stent with the I de scr iption
i e translate into I and substitute all combinations of constants o ccurr ing in the
de scr iption for the f ree var iable s If any such i s incons i stent add the corre sp onding
I repre s entation of the negation of under the same substitution to the de scr iption
If no new information was adde d by steps and stop the s ituation i s cons i stent with the
axiom Otherwi s e go to to te st the new extende d I de scr iption
Thi s pro ce ss must terminate and if the nal s ituation de scr iption i s still cons i stent then the
axiom i s sati s able s ince for all substitutions e ither the antece dent i s not entaile d by the de scr iption
or the cons equent has b een explicitly adde d and the cons equent i s e ither cons i stent with the
de scr iption or the negation of the antece dent has b een adde d Clearly the convex hull axioms
and are of the form which can b e capture d in thi s manner In f act s ince the ir antece dents are
s imple they can b e enforce d quite e ciently
In s ection I shall pre s ent a table of compositions of the RCC relations which was
compute d us ing the I reasoning algor ithm given in chapter augmente d with the meta level
reasoning for conv which has just b een de scr ib e d A full di scuss ion of relational comp os ition can
b e found in the next chapter
Mo dal Repre s entation of Convexity
We have s een how the top ological inter ior function corre sp onds to the S mo dal b ox op erator
Such a corre sp ondence may sugge st that other us eful functions of spatial regions can b e capture d
by mo dal op erators in a order calculus In the remainder of thi s chapter I sp ecify a multi mo dal
13
language with a convex hull op erator Thi s language contains usual class ical connective s which
will b e interprete d algebraically in accordance with s ection plus three mo dal op erators
an inter ior op erator constraine d to b ehave exactly as the S mo dality
the strong S op erator
the convexity op erator whos e prop ertie s are to b e sp eci e d
To x the meaning of the new op erator we nee d to nd order axiom schemata or rule schemata
to enforce the de s ire d prop ertie s of The s e schemata will corre sp ond to the st order axioms
given ab ove I do not know of a general metho d for p erforming thi s kind of transformation and
it s eems unlikely that such a metho d exi sts However in each cas e we can s ee that under the
algebraic interpretations of the logical op erators the schemata are equivalent to the axioms
The schema corre sp onding to axiom i s very s imple
X X S ch
13
This mater ial is a slight revision of what I pre s ente d in Bennett b
CHAPTER CONVEXITY
Axiom i s a little harder to repre s ent as a mo dal schema TP X Y means that X i s a
tangential part of X Thi s holds if e ither X i s a tangential proper part of Y or X i s equal to Y
Thus to repre s ent thi s we us e the enco ding for TPP X Y given in table but drop the s econd
entailment constraint Y X which would ensure that X and Y are non equal Hence us ing the
strong S rather than the mo del entailment constraint di stinction axiom can b e repre s ente d
by the schema
X S ch X X X
which says that all regions are part of the ir convex hull but not part of the inter ior of the ir convex
hull We may note that the initial in the rst conjunct i s re dundant s ince it i s implicit in mo dal
axiom schemata that they are true in all p oss ible worlds or in the context of algebraic s emantics
that the ir denotation i s U
Axiom which state s that if X i s part of Y then X i s part of Y can b e repre s ente d by
X Y X Y S ch
Thi s require s some explanation In general where we have a st order axiom of the form p q
thi s will b e translate d by p q where i s the repre s entation of which ensure s that
if p U then q U Note that we do not nee d p q b ecaus e the antece dent
must e ither denote in which cas e the schema i s tr ivially sati s e d or it denote s U in which the
cons equent must also denote U If we were to wr ite s imply p q thi s would repre s ent the
stronger requirement that p i s always a subs et of q whether or not p U
Us ing a s imilar transformation axiom can b e straightforwardly repre s ente d by
X Y X Y S ch
X Y corre sp onds to the entailment constraint repre s enting C X Y and ass erts that X
and Y share at least one p oint
Finally axiom can b e straightforwardly capture d by
X Y X Y S ch
It should b e note d that the strong S op erator i s not nee de d if we sp ecify the logic by means
of rule schemata rather than only axiom schemata For example Sch b ecome s
X Y
Mon
X Y
which tells us that i s monotonic with re sp ect to the part relation i e
The s econd conjunct of Sch would corre sp ond to the rule
X X
TP
CHAPTER CONVEXITY
and Sch to the rule
X Y X Y
C
Practicality of the Mo dal Repre s entation
The p oss ibility of sp ecifying convex hull as a mo dal op erator illustrate s the p otential expre ss ive
p ower of multi modal formali sms as repre s entations for spatial information However whether
such logics could actually b e us e d as vehicle s for e ective reasoning remains debatable As in the
cas e of the s impler S and I repre s entations of purely top ological relations it i s likely that
by limiting the range of formulae that can b e employe d to s imple syntactic forms one might b e
able to construct e ective deci s ion pro ce dure s for some sub language of thi s multi modal language
of convexity The crucial que stion i s whether us eful expre ss ive p ower can b e provide d within a
tractable repre s entation
Chapter
Comp o s ition Bas e d Re asoning
Or iginating in Allen s analys i s of temp oral relations the us e of Composition Tables has b ecome
a key technique in providing an e cient inference mechani sm for a wide class of theor ie s In
thi s chapter I examine comp os itional reasoning in general and its us e in spatial reasoning I
pre s ent comp os ition table s for s everal imp ortant s ets of RCC relations including the RCC
relations intro duce d in s ection Thi s table was compute d us ing the intuitioni stic
enco ding de scr ib e d in chapter together with the meta level enco ding of convexity axioms
sp eci e d in s ection Finally I lo ok at the formali sm of Relation Algebra and show how
it allows algebraic de nition of the RCC relations in terms of the pr imitive connecte dne ss
relation
Comp o s ition Table s
A compositional inference i s a de duction f rom two relational f acts of the forms R a b and S b c
of a relational f act of the form T a c involving only a and c Such inference s may b e us eful in
the ir own r ight or may b e employe d as part of a larger inference mechani sm such as a cons i stency
checking pro ce dure for s ets of relational f acts In e ither cas e one will normally want to de duce the
stronge st relation T a c that i s entaile d by R a b S b c and which i s expre ss ible in whatever
formali sm i s b e ing employe d
In many cas e s the validity of a comp os itional inference do e s not dep end on the particular
constants involve d but only on logical prop ertie s of the relations R S and T Where thi s i s so it
make s s ens e f rom a computational p oint of view to record the comp os itions of pairs of relations
so that the re sult of a comp os itional inference can s imply b e lo oke d up when require d Thi s
technique i s particularly appropr iate where we are dealing with relational information involving a
xe d s et of relations One can then store the re sult of comp os ing any pair f rom a s et of n relations
in an n n composition table The s implicity of thi s idea make s it very attractive as a p otential
means of achieving e ective capabilitie s for reasoning ab out any domain within which s igni cant
information can b e repre s ente d by a limite d s et of binary relations Since the ir intro duction by
CHAPTER COMPOSITION BASED REASONING
1
Allen comp os ition table s have rece ive d cons iderable attention f rom re s earchers in AI and
relate d di scipline s Vilain and Kautz Egenhofer and Franzosa Freksa a Randell
Cohn and Cui a R ohr ig Cohn Go o day and Bennett Schlie der
Given a s et Rels of binary relations a comp os ition table can b e identi e d with a mapping
Rels
CT Rels Rels i e if R and R are elements of Rels then the value of CT R R
i s a subs et of Rels which i s the comp os ition table entry for the pair hR R i The s et Rels will
b e calle d the basis s et of CT Clearly if there are n relations in Rels then the comp os ition table
for Rels can b e repre s ente d by an n n array or table In f act b ecaus e of the nature of relational
comp os ition such an array i s a very ine cient way to store thi s information I de scr ib e d the
re dundancy inherent in comp os ition table s in Bennett a and an abbreviate d vers ion of thi s
mater ial i s include d as app endix D of thi s the s i s
For many purp os e s a comp os ition table entry i s asso ciate d with a di sjunctive relation Becaus e
of thi s it i s convenient to b e able to wr ite a s et of relation name s as if it were the name of a
di sjunctive relation Thus
fR R g a b means xy R x y R x y a b
n n
It i s usual to assume that the elements of Rels form a JEPD partition of the p oss ible relations which
can hold b etween pairs of ob jects in the domain under cons ideration i e every pair of ob jects in
the domain i s relate d by exactly one of the memb ers of Rels Under the s e conditions any Bo olean
combination of relations i s equivalent to a di sjunction of memb ers of Rels
The preci s e meaning of a comp os ition table dep ends to some extent on the context in which it
i s employe d Sometime s it i s a record of certain kinds of cons equence of some underlying theory
which may already b e fully or partially formalis e d Alter natively the sp eci cation of a comp os ition
table may prece de the development of a formal theory of the relations involve d and i s an initial
step in sp ecifying the theory of some s et of intuitively understo o d relations In e ither cas e the
fundamental mo de of reasoning enco de d in a comp os ition table i s to te st cons i stency of tr iads of
relations of the forms R a b S b c T a c where R S T Rels such a tr iad i s cons i stent if
and only if T CT R S
Comp os itional reasoning can b e generali s e d to the cas e where one comp os e s relations which
are thems elve s di sjunctions Here it i s usually assume d that the comp os ition of two di sjunctive
relations R a b and S b c i s s imply the di sjunction of all p oss ible comp os itions R a b and
i
S b c where R and S are re sp ectively di sjuncts of R and S Thus the domain of the function
j i j
CT can b e extende d to di sjunctive relations as follows
CT R S CT R S
i j def
ij
If R a b S b c and T a c are di sjunctive relations then by computing the generali s e d com
p os ition of R and S it may b e found that some of the di sjuncts of T are not p oss ible Eliminating
1
In fact Allen calle d his table a transitivity table but comp osition table is arguably more appropr iate and it
s eems that this is becoming the standard term
CHAPTER COMPOSITION BASED REASONING
such di sjuncts can b e regarde d as a generali sation of the s imple tr iad cons i stency checking pro ce d
ure for non di sjunctive relations The more general comp os ition rule for di sjunctive relations can
b e formally sp eci e d by the following inference rule schema
R a b S b c T a c
Comp
CT R S T a c
If T CT R S no new information i s generate d otherwi s e T a c i s replace d by the stronger
relation CT R S T a c If CT R S i s di sjoint f rom T then an incons i stency has b een
detecte d
Rep eate d application of the inference rule Comp i s known as comp os itional constraint propaga
tion It i s clear that given any s et of instance s of the di sjunctive relations over Rels after rep eate d
application of Comp one will e ither generate an incons i stency or reach a state where no new in
formation can b e generate d by Comp If an incons i stency has b een detecte d we can say that the
relation s et i s incons i stent with re sp ect to CT otherwi s e it i s cons i stent with re sp ect to CT
Soundne s s and Completene s s of a Comp o s ition Table
The i ssue of the conditions under which a comp os ition table can provide a complete cons i stency
checking pro ce dure for relational f acts was rai s e d and di scuss e d by Bennett Isli and Cohn
The notions of soundne ss and completene ss of a comp os ition table app eal to some underlying theory
or intuition of the meanings of the relations involve d To say that a comp os ition table i s sound i s
to say that whenever a s et of relations i s determine d by that comp os ition table to b e incons i stent
then that s et of relations i s indee d incons i stent with the underlying theory or intuition Likewi s e a
comp os ition table i s complete p erhaps one should say refutation complete if whenever a s et of
relations i s incons i stent with the background theory intuitions thi s can b e detecte d by reference
to the comp os ition table
The s e ideas nee d to b e made more preci s e I stipulate that
A comp os ition table CT for a relation s et Rels i s sound w r t some p oss ibly unformali s e d
theory if whenever we nd among some s et of instance s of Rels a tr iad R a b S b c
T a c such that T CT R S then thi s s et of instance s i s incons i stent with
To make the completene ss prop erty fully preci s e we rst nee d the following de nition a s et of
relation instance s i s total if every pair of constants o ccurr ing in the s e instance s o ccur together in
2
exactly one instance i e every pair of constants are uniquely relate d I then say that
A comp os ition table CT for a relation s et Rels i s refutation complete w r t some p oss ibly
unformali s e d theory if whenever some total s et S of instance s of Rels i s incons i stent with
we can nd relations R a b S b c T a c S s t T CT R S
2
If a s et of relation instance s is not total this means that some pair of constants are not constraine d by any
relation Any pair of unconstraine d constants are implicitly relate d by the universal relation x y When we
are dealing with a JEPD relation s et Rels the universal relation is just the disjunction of all relations in Rels This
means that a non complete s et of relation instance s contains implicit disjunctive relations The requirement that
the relation s et is total can then be s een as part of the requirement that the relation s et is non disjunctive
CHAPTER COMPOSITION BASED REASONING
Supp os e a comp os ition table i s sound and complete with re sp ect to for non di sjunctive
relations do e s thi s mean that by employing comp os itional constraint propagation i e rep eate d
application of Comp we get a cons i stency checking pro ce dure which i s sound and complete w r t
for di sjunctive relations It i s quite easy to show that comp os itional constraint propagation
must b e sound if tr iad cons i stency checking for non di sjunctive relations i s sound Thi s i s b ecaus e
given any relations R a b S b c and T a c the rule Comp only eliminate s thos e di sjuncts
of T that are incons i stent with any p oss ible non di sjunctive strengthening of R and S However
comp os itional constraint propagation i s not in general complete The problem i s that although each
tr iad of di sjunctive relations b etween three constants may b e cons i stent there may b e no s ingle
non di sjunctive sp eciali sation of all the di sjunctive relations such that every tr iad i s cons i stent
On the other hand if a comp os ition table i s complete for non di sjunctive relations thi s do e s
always yield a complete refutation pro ce dure for di sjunctive relations by us e of a back tracking
s earch algor ithm Clearly a s et of di sjunctive relation instance s i s cons i stent just in cas e there i s
some non di sjunctive strengthening of the s e instance s which i s its elf cons i stent Thi s can always
b e found by exhaustive s earch of all p oss ible combinations of non di sjunctive sp eciali sations of
the di sjunctive relations Computationally thi s metho d require s time which i s exp onential in the
numb er of di sjunctions whereas the application of comp os itional constraint propagation require s
only O n time where n i s the numb er of constants o ccurr ing in the s et of relations to b e te ste d
Cons equently there has b een much intere st in di scover ing sp eci c s ets of di sjunctive relations for
which the comp os itional constraint propagation metho d i s indee d complete Vilain and Kautz
Neb el a Neb el b Renz and Neb el In the re st of thi s chapter I shall not b e
much concer ne d with the tractability of reasoning with di sjunctive relation s ets so my attention
will b e largely con ne d to total s ets of non di sjunctive relations
Formal Theor ie s and Comp o s ition Table s
In the previous s ection the prop ertie s of soundne ss and completene ss of a comp os ition table were
de ne d on the assumption that one has some metho d of te sting cons i stency of s ets of ground
relations If the bas i s relations are de ne d in some formal theory then thi s can b e te ste d by means
of some refutation pro of pro ce dure for the logical language in which the theory i s formulate d I
shall now lo ok in more detail at how a comp os ition table can b e compute d f rom a formal theory
and what the table means in terms of the theory We shall s ee that the p oss ibility of sp ecifying a
sound and complete comp os ition table for a s et of relations with re sp ect to some theory dep ends
up on certain prop ertie s of that theory
Although the de nitions of comp os ition table soundne ss and completene ss in terms of cons i st
ency s eem at rst s ight to b e very straightforward when we try to de scr ib e exactly how comp os ition
table entr ie s should b e logically de duce d f rom a formal theory certain di cultie s ar i s e At the
heart of the s e problems i s the way in which the comp os itional prop ertie s of relations should b e
abstracte d f rom prop ertie s of ground instance s of the s e relations Whilst comp os itional reason
ing and its soundne ss and completene ss are character i s e d in terms of ground instance s the table
CHAPTER COMPOSITION BASED REASONING
its elf contains only relation name s Becaus e of thi s if a comp os ition table i s to b e coherent the
logic of relational comp os ition must b e in some s ens e homogeneous with re sp ect to the domain of
individuals
Let us assume that the e ss ential character i stic of a comp os ition table i s its ability to di scr iminate
b etween cons i stent and incons i stent tr iads of relations Thi s leads to the following stipulation for
the comp os ition function
CTdef Given a theory in which a s et Rels of bas e relations i s de ne d the comp os ition
CT R S where R S Rels i s the s et of all relations T Rels for which the formula
i
x y z R x y S y z T x z i s cons i stent with
i
Here I have us e d exi stential quanti cation to indicate that if the combination R x y
S y z T x z i s p oss ible for any three individuals in the domain then T must b e include d
i i
in the comp os ition of R and S Thi s ensure s soundne ss of the comp os ition table s ince only tr iads
that are imp oss ible under any instantiation are rule d out by the comp os ition table
However it i s not at all clear that thi s de nition give s r i s e to a complete comp os ition table
One p oss ible problem o ccurs if we cons ider a language containing constants denoting entitie s with
sp ecial logical prop ertie s e g the universal region denote d by u in the RCC theory if the f acts
R x y and S x y involve one of the s e constants certain p oss ibilitie s for the relation T x z
might in thi s cas e b e imp oss ible and in such sp ecial cas e s the comp os itional inference justi e d
by the comp os ition table would b e to o weak to ensure completene ss Even if our language do e s
not contain sp ecial constants it i s still by no means obvious that comp os itional reasoning provide s
a complete refutation pro ce dure It may b e that there are theor ie s and relation s ets for which one
may have a total network of relation instance s which i s incons i stent even though every tr iad of
the s e instance s i s cons i stent with the theory
Neverthele ss COMPdef must surely b e the correct de nition of the CT function any stronger
de nition would b e unsound b ecaus e it would tell us that some tr iad of relations i s imp oss ible when
in f act there i s at least one instantiation for which it i s p oss ible Cons equently we must identify
conditions under which COMPdef yields a comp os ition table which i s complete with re sp ect to
To thi s end I intro duce the concept of k compactness applicable to a relation s et relative to a theory
3
within which the relations are de ne d
A relation s et Rels i s k compact w r t a theory i for any total network of instance s
of Rels the network i s incons i stent with i it include s a sub network of s ize k or le ss
which i s incons i stent with
For some s ets of relations we may nd that there can b e arbitrar ily large incons i stent total
networks all of whos e sub networks are cons i stent We say that the s e are not nitely compact If
there can b e an in nite incons i stent network with no nite incons i stent sub network the relation
4
s et i s not compact at all
3
This concept was rst intro duce d in Bennett et al
4
My notion of compactne ss is directly analogous to that which is applie d to logical language s such a language
is compact if every inconsistent s et of formulae has a nite inconsistent subs et
CHAPTER COMPOSITION BASED REASONING
From the de nition of k compactne ss it imme diately follows that a comp os ition table for a s et
of relations Rels can b e complete only if Rels i s compact with re sp ect to Furthermore if we are
concer ne d with a language in which all individual constants are arbitrary i e we have no constants
referr ing to particular individuals with sp ecial prop ertie s then if Rels i s compact with re sp ect
to the comp os ition table for Rels constructe d according to COMPdef must b e complete with
re sp ect to
Not all relation s ets are compact cons ider a theory in which individuals have the prop ertie s
of equal s ize d di scs in the plain and a s et of relations including the relation of external connection
The theory require s that any given circle can b e exter nally connecte d to a maximum of s ix other
circle s thi s could b e sp eci e d directly as an axiom of the theory or could b e a cons equence of
the axiom s et Hence a s ituation in which s even regions are all mutually exter nally connecte d i s
5
incons i stent but thi s cannot b e detecte d by checking any tr iad of relations b etween three regions
Hence no s et of relations including a relation of exter nal connection can b e compact with re sp ect
to thi s theory
The Extens ional De nition of Comp o s ition
The notion of compactne ss yields a preci s e sp eci cation of what relationship i s nece ssary b etween
a s et of relations and a theory in order that one might construct a complete comp os ition table for
that relation s et However b e ing state d in terms of the relationship b etween lo cal and overall con
s i stency thi s sp eci cation i s e ss entially meta theoretic Establi shing compactne ss will typically
involve rst showing that some class of mo dels i s canonical for the theory i e every cons i stent
s et of relational constraints has a mo del in thi s class which i s cons i stent with the theory and
then demonstrating by reasoning ab out the s e mo dels that if there i s a mo del which i s lo cally
cons i stent with every tr iad of relational constraints there must also b e a mo del which i s cons i stent
with the whole s et of constraints Such pro ofs are often di cult and very much dep endent on
the sp eci c relational theory under cons ideration Hence it would b e very de s irable to have some
general cr iter ia for compactne ss that could b e state d in terms of the theory in que stion It s eems
plaus ible that one might b e able to demonstrate that given a s et of relations and a theory the
relations are compact with re sp ect the theory just in cas e certain formulae are theorems of that
theory
A promi s ing approach to thi s problem i s to try to cast the requirements of compactne ss and
hence comp os ition table completene ss in terms of the op eration of extensional composition which
i s de nable within any st order theory Thi s op eration i s bas e d on the following de nition of the
comp os ition of two relations which i s standard in s et theory
EXCOMPdef Let R b e a relation f rom A to B and R b e a relation f rom B to C i e
A B and C are s ets R A B and R B C Then the composition of R with R
R R i s the s et of all ordere d pairs ha ci A C such that for some b B ha bi R
and hb ci R
5
This example is de scr ibe d in Cui Cohn and Randell
CHAPTER COMPOSITION BASED REASONING
In st order logic the extens ional comp os ition op erator can b e de ne d by
x y R S x y z R x z S z y ExComp
Thi s de nition i s str ictly stronger than the cons i stency bas e d de nition not only do e s it ensure
that whenever R a b and S b c hold R S a c must also hold it also require s that whenever
R S a c holds there must exi st some region say b s t R a b and S b c In f act the infer
ence f rom R a b and S b c to R S a c must b e the stronge st comp os itional inference that i s
valid for any arbitrary constants a b and c Since b i s arbitrary our premi ss e s are equivalent to
z R a z S z c and we can instantiate ExComp to get R S a c z R a z S z c
Hence the conclus ion R S a c i s logically equivalent to the premi ss e s and any inference to a
stronger relation T a c would b e unsound
If a comp os ition table CT sati s e s the cons i stency bas e d de nition of comp os ition CTdef it
i s easy to show that the extens ional comp os ition R S always denote s a relation whos e extens ion
i s a subs et of that of CT R S Thi s means that for each comp os ition table entry the following
formula i s provable
x y R S x y CT R S x y
Unlike CT R S the relation R S nee d not nece ssar ily b e equivalent to some di sjunction of a
xe d s et of bas e relations If not then CT R S must b e str ictly weaker than R S Neverthele ss
for a particular theory and s et of relations it may b e that cons i stency bas e d comp os ition coincide s
with the extens ional de nition i e
R S Rels x y CT R S x y R S x y
Since CT R S i s always s imply a di sjunction of relations taken f rom Rels thi s formula can only
b e true if the s et of di sjunctive relations over Rels i s clos e d under the extens ional comp os ition
op erator
In Bennett et al it was sugge ste d that if CT i s not extens ional i e CT R S i s weaker
than R S for certain relations then thi s must mean that information i s lost when cons i stency
bas e d comp os itions are compute d via CT and cons equently that if cons i stency of a network
i s te ste d solely by propagation of constraints imp os e d by a non extens ional comp os ition table
we may nd that it s eems to b e cons i stent when it i s actually incons i stent Thi s conjecture i s
supp orte d by the f act that R S give s the stronge st p oss ible comp os itional inference that i s sound
for arbitrary arguments However the conditions under which extens ional comp os ition provide s a
6
refutation complete pro of pro ce dure have thems elve s not b een e stabli she d nor i s it certain that
there cannot b e s ets of relations for which a weaker form of comp os itional inference might b e
refutation complete Until the s e i ssue s have b een re solve d the connection b etween extens ional
comp os ition and comp os ition table completene ss i s not clear
To clar ify the prece ding remarks it may b e helpful to cons ider the cas e of the Allen relations
In hi s or iginal pre s entation of a comp os ition table for temp oral relations Allen app ears
6
That the basic relations are JEPD and that they include equality are conditions that s eem likely to be imp ortant
CHAPTER COMPOSITION BASED REASONING
to employ a cons i stency bas e d interpretation of comp os ition table entr ie s However a st order
theory of temp oral intervals was later given by Allen and Haye s and thi s theory justi e s
extens ional interpretation of the Allen comp os ition table Ladkin showe d that the s e axioms
are also f aithful to the intende d interpretation in that the ir mo dels are i somorphic to structure s of
intervals over an unb ounde d linear order The compactne ss can then b e e stabli she d by analys ing
7
the s e mo dels in the light of Helly s theorem What i s not clear i s whether there i s a connection
b etween the f act that the Allen relations are compact and the f act that the di sjunctive Allen
relations are clos e d under extens ional comp os ition
Comp o s ition Table s and CSPs
A f ramework for problem solving that has rece ive d a great deal of attention f rom AI re s earchers i s
that of Constraint Satisfaction Problems CSPs Mackworth Tsang A CSP cons i sts
of a s et of var iable s and a s et of constraints on p oss ible value s of the s e var iable s The s e constraints
can b e regarde d as a s et of tuple s of p oss ible ass ignments p erhaps not explicitly given but checke d
on demand by some pro ce dure or as sp eci e d by some theory The typ e of reasoning involve d
in solving a CSP has much in common with that employe d in cons i stency checking by means of
comp os itional reasoning Although constraints of time and space p ermit only a very br ief lo ok
at CSPs to b e include d in the current the s i s they may prove to b e a p owerful to ol for spatial
reasoning
There are two ways in which the notion of a comp os ition table can b e ass imilate d into the
f ramework of CSPs One i s to treat the comp os ition table as a s et of ter nary constraints on
var iable s ranging over relation name s s ee e g Gr igni Papadias and Papadimitr iou Thus
for each ordere d pair of ob jects hx y i the CSP has one var iable v x y whos e domain i s the
s et Rels A comp os ition table CT i s then interprete d as a s et of constraints which can b e sp eci e d
as all instance s of formulae of the form
v x y R v y z S v x z CT R S
Thi s approach i s applicable to any comp os ition table and do e s not tell us anything ab out the
relations involve d
A more illuminating approach i s to regard the relations in a bas i s s et Rels as thems elve s con
stituting the constraints of a CSP Thi s require s further analys i s of the logical structure of the
relations involve d In the cas e of the Allen relations a natural interpretation i s to identify the
relations with order constraints on the end p oints of temp oral intervals and to take the s e end
p oints as elements of an ordere d linear eld such as the real or rational numb ers Vilain and Kautz
Neb el b In s ections and we saw how many top ological RCC relations can b e
repre s ente d by equational and di s equational constraints over inter ior algebras
7
Helly s theorem state s that the inters ection of a s et of convex subspace s of a space of dimension n has a
non empty inters ection just in cas e every n members of that s et have a non empty inters ection Thus a s et
of linear intervals has a common inters ection i every subs et of three intervals has a non empty inters ection By
character ising the Allen relations in terms of non emptine ss conditions one can then show that a total network of
the s e relations is consistent i every tr iad is consistent
CHAPTER COMPOSITION BASED REASONING
Comp o s ition Table s for RCC Relations
I now pre s ent comp os ition table s for three of the most s igni cant s ets of RCC relations The s e
table s are constructe d in accordance with the cons i stency bas e d sp eci cation of comp os ition given
by CTdef Later in s ection I shall cons ider the p oss ibility of an extens ional interpretation
of the RCC table
RCC
Recall that RCC i s the relation s et fDR PO EQ PP PPig re sulting f rom ignor ing the di erence s
b etween connection and overlapping and b etween tangential and non tangential parts which are
made by the RCC relations As we saw in chapter each RCC relation can b e de scr ib e d by
means of p os itive and negative Bo olean equations and cons equently RCC reasoning can b e enco de d
in terms of class ical mo del and entailment constraints within the order language C In f act
given the limite d expre ss ive p ower of C I have not implemente d a purely class ical reasoner but
have concentrate d on reasoners for the more expre ss ive language I which can expre ss the more
di scr iminating RCC relation s et Hence table was actually obtaine d by merging entr ie s in the
RCC comp os ition table given in the next s ection Note that the symb ol refers to the universal
relation which means that no bas e relation i s exclude d
H
R(b c)
H
H
R(a b)
DR PO EQ PP PPi H
H
DR DR PO PP DR DR PO PP DR
PO DR PO PPi PO PO PP DR PO PPi
EQ DR PO EQ PP PPi
PP DR DR PO PP PP PP
PPi DR PO PPi PO PPi PPi O PPi
Table Comp os ition table for the RCC Relations
RCC
The RCC comp os ition table was generate d us ing the I enco ding of the relations by means of
my rst implementation of an optimi s e d I theorem prover The co de i s given in app endix C The
entry for relations R and R was compute d by te sting the cons i stency of the spatial con guration
R a b R b c R a c where R i s each of the RCC relations Running on a Sparc
i i
8
workstation the program generate d the full composition table for RCC in under s econds
In s ection I shall show that the RCC relations are compact with re sp ect to the ir
interpretation in the theory of inter ior algebras Thi s means that the comp os ition table provide s a
refutation complete pro of pro ce dure for s ets of RCC relational f acts
8
By exploiting the re sults of appendix D concerning re dundancy in comp osition table s the table could have been
compute d in approximately one sixth of this time
CHAPTER COMPOSITION BASED REASONING
.
H
R b c
H
R a b DC EC PO TPP NTPPi NTPP TPPi EQ
H
H
DC DR PO PP DR PO PP DR PO PP DC DR PO PP DC > DC
DR PO
EC EC PO PP DR PO PPi DR PO PP PO PP DR EC DC
TPP TPi
DR PO
PO PP DR PO PPi DR PO PPi PO PP PO PO > DR PO PPi
PPi
DR PO DR PO
PP TPP DC DR DR PO PP NTPP TPP
TPP TPi PPi
NTPP NTPP DC DC DR PO PP NTPP DR PO PP NTPP >
DR PO PPi EC PO PPi PO PPi PO TPP TPi PO PP PPi TPPi TPPi NTPPi
PO PPi NTPPi DR PO PPi PO PPi PO PPi O NTPPi NTPPi NTPPi
EC TPP DC EQ PO NTPP TPPi EQ NTPPi
Table Comp os ition table for the RCC relations
RCC
In s ection we saw how var ious containment relations can b e de ne d by means of the exten
de d RCC theory with a convex hull op erator In particular the JEPD relation s et RCC was
intro duce d in which the EC and DC relations of RCC are further analys e d in order to sp ecify
the relation holding b etween each region and the convex hull of the other Table give s the
full comp os ition table for the RCC relations If R a b and R b c where R i s the relation
sp eci e d in the left hand column and R i s sp eci e d along the top the corre sp onding table entry
enco de s the p oss ible value s of the relation R a c
Becaus e each table entry i s some subs et of p oss ible bas e relations there i s not enough space
to give the actual relation name s Hence in order to pre s ent the table on a s ingle page a sp ecially
conci s e notation was employe d Each of the relations i s repre s ente d by one of the two symb ols
and at a certain p os ition in a matr ix The s e repre s entations are shown in the s econd
column Table entr ie s are constructe d by sup er imp os ing the repre s entations for each of the p oss ible
relations Where and should b oth b e pre s ent in the same p os ition the symb ol i s us e d
The table was generate d us ing the meta level enforcement of the conv axioms in the I repre s
entation as de scr ib e d in s ection Us ing an augmente d vers ion of the I reasoning program
given in app endix C the table was pro duce d in h m on a Sparc workstation It was sub
s equently publi she d in Bennett b The task of generating thi s table had b een prop os e d two
years earlier as a challenge for comp os ition in Randell Cohn and Cui a Cohn Randell
Cui and Bennett contains a s imilar table constructe d us ing a mo del building approach but
it has subs equently b een found that the table given there i s to o str ict in that it rule s out certain
con gurations which are in f act p oss ible for D spatial regions My table has not b een found to
contain any f als e entr ie s
It i s intere sting to note that generation of thi s table was in f act one of the very rst re sults on
spatial reasoning that I obtaine d dur ing my PhD re s earch The idea of the program was inspire d
by an account of Tarski s top ological interpretation of I given by Mostowski After a p er io d
of intens ive co ding and exp er imentation I found mys elf with a program that s eeme d to generate
CHAPTER COMPOSITION BASED REASONING
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
? ? ?
? ? ?
Rel Rep
? ? ?
? ? ?
? ? ? ?
? ? ?
1
NTPP
?
? ? ? ? ?
2
TPP
? ? ? ? ? ? ? ? ? ? ? ? ? ?
3
PO
? ? ? ?
4 ? ? ?
? ? ?
EQ
? ? ?
? ? ? ?
? ?
5
NTPP
?
? ?? ? ??? ?
6
? ?? ??? ? ?? ??? ? ?? ???
TPP
? ?? ???
7 ? ?? ? ? ? ?
I IE
8 ? ? ? ? ? ? ?
IPE
9 ? ? ?
IPD
10 ? ?? ? ? ?
IOE
11 ? ?
IOD
12 ?? ? ? ? ? ?
? ?? ?
PIE
13 ? ? ? ? ? ? ?
PID
14 ? ? ? ? ? ?
? ? ?
PPE
15 ? ? ? ? ?
PPD
16 ?? ? ? ? ?
? ?? ?
POE
17 ? ? ? ?
POD
18 ?? ? ? ? ? ?
??
OIE
? ?? ?
19 ? ? ? ? ? ? ?
OID
20 ? ? ? ? ? ?
?
OPE
? ? ?
21 ? ? ? ? ?
OPD
22 ?? ? ? ? ?
??
OOE
? ?? ?
23 ? ? ? ?
OOD
Table Comp os ition table for the RCC relations
CHAPTER COMPOSITION BASED REASONING
the correct comp os ition table Much of the re st of the work done dur ing my PhD re s earch was
concer ne d with di scover ing exactly how thi s program worke d
Compactne s s of RCC
By analys ing Neb el s class ical enco ding of the RCC relations de scr ib e d in s ection I
shall now show that the RCC relations are compact with re sp ect to the cons i stency checking
pro ce dure provide d by the I repre s entation Becaus e of Tarski s top ological interpretation of I
it follows that the RCC relations are compact with re sp ect to the general theory of top ological
space s within which the s e relations are character i s e d as sp eci e d in table
Under the forcing constraint interpretation each constant region a i s identi e d with three
class ical literals F v a F w a and F w a and each RCC relation R a b i s sp eci e d by
a s et of binary claus e s involving the literals asso ciate d with a and b Amongst the s e claus e s I
include thos e ar i s ing f rom the order ing condition on the worlds as well as directly f rom the mo del
and entailment constraints For thi s repre s entation it i s clear that binary re solution provide s a
refutation complete pro of pro ce dure
The forcing constraint claus e s are cons i stent if and only if the I constraints f rom which they
are der ive d are cons i stent and the s e in tur n are cons i stent if and only if the corre sp onding inter ior
9
algebraic constraints are sati s able in some top ological space Thus inference s among RCC
relations are mirrore d by logical der ivations among the corre sp onding class ical forcing constraints
Sp eci cally the forcing constraint claus e s corre sp onding to the comp os ition of two RCC relations
R a b and R b c are all thos e re solvents involving only a and c literals generate d by applying
binary re solution to the combine d s ets of forcing constraint claus e s asso ciate d with the two relations
Thi s s et contains all der ivable forcing constraints on a and c Thus an RCC comp os ition
i s asso ciate d with a s et of binary re solutions among CNF claus e s Convers ely every binary
re solution among forcing constraint claus e s i s correlate d with the comp os ition of a pair of RCC
relations Becaus e binary re solution i s refutation complete for class ical claus e s it follows that an
RCC network can b e shown to b e incons i stent by means of comp os itional inference if and only
if it i s incons i stent with re sp ect to the theory of inter ior algebras
What I have just shown i s not quite su cient to conclude that the RCC relation s et i s
compact with re sp ect to the theory of inter ior algebras It could b e that showing incons i stency by
comp os itional inference might require a chain of s everal such inference s whereas if a relation s et
i s compact then any incons i stent network contains an incons i stent tr iad of relational f acts which
can b e detecte d by a s ingle comp os itional inference Happily as I shall now show it tur ns out that
incons i stency of a total network of RCC relations as interprete d in inter ior algebra can always
b e detecte d by a s ingle comp os itional inference
In the I enco ding the detection of an incons i stent tr iad corre sp onds to the di scovery of two
9
An intr iguing que stion regarding this corre sp ondence is whether there is an intuitive top ological interpretation
of the forcing constraints and the three worlds asso ciate d with each region
CHAPTER COMPOSITION BASED REASONING
y z z y y z y z y z z y
x y x z x z
x y x z x z
x z x z x z x z x y
y x z x z x
x z x z x z x y x z
z x y x z x
Table Comp os itional inference s among I formulae
10
mo del constraint formulae x y and y z that entail some entailment constraint x z
x y and y z can each take one of s even p oss ible forms Table give s for each such
combination the stronge st entaile d formula involving only x and z Where the entry i s thi s
means that the only der ivable formulae involving just x and z are theorems of I Thi s can b e
ver i e d by noting that for each of the s e combinations e ither by supp os ing that y i s a theorem of
I or by supp os ing that y i s incons i stent one can der ive b oth x y and y z for arbitrary
instantiations of x and z Thus ass erting x y and y z do e s not logically constrain the
value s of x and z In all other table entr ie s we s ee that the stronge st der ivable formula i s its elf
one of the s even mo del constraints So binary comp os ition of mo del constraint formulae e ither
pro duce s no new information or a new mo del constraint formula
If one then cons iders the mo del and entailment constraints asso ciate d with each of the RCC
relations one nds that each relation i s saturate d with re sp ect to to mo del constraint formulae
in the s ens e that each of the s even p oss ible mo del constraints i s e ither entaile d by the mo del
constraint asso ciate d with the relation or entails one of the entailment constraints asso ciate d with
that relation Thi s means that if we add a new mo del constraint formula to the I repre s entation
of a total RCC network it i s e ither re dundant or make s the network incons i stent It follows that
whenever a total RCC network can b e shown to b e incons i stent by binary comp os ition of I mo del
constraints thi s can b e shown by a s ingle application of thi s typ e of inference Moreover s ince
comp os itional inference has b een shown to b e complete for te sting incons i stency with re sp ect to
the interpretation in the theory of inter ior algebras it must also follow that the RCC relations
are compact with re sp ect to thi s theory
The compactne ss of RCC with re sp ect to inter ior algebra can b e contraste d with a re sult
of Gr igni et al concer ning the realisability of a s et of RCC relational f acts by a s et of
s imply connecte d planar regions Drawing on re sults of Krato chv l ab out the recognition of
reali sable str ing graphs Gr igni et al conclude that te sting whether a s et of such f acts has a
mo del in which the constants refer to regions in the plane that are b ounde d by Jordan curve s i s
NP hard Thi s means that the RCC relations cannot b e nitely compact with re sp ect to a theory
which constrains the regions in thi s way Cons equently no comp os ition table can b e complete for
te sting cons i stency of RCC relations in thi s re str icte d planar domain
10
A non null entailment constraint x is equivalent to x x and can be treate d as be ing of the form x y
CHAPTER COMPOSITION BASED REASONING
Exi stential Imp ort in RCC Comp o s itions
Examination of the comp os ition table for RCC table reveals that an extens ional interpret
ation i s not compatible with the st order RCC theory Cons ider the entry for CT DC DC which
i s given as the universal relation Interprete d extens ionally thi s would mean that
x y z DC x z DC z y x y
which i s equivalent to
x y z DC x z DC z y
Thi s says that given any two regions x and y there i s a region z di sconnecte d f rom b oth of them
But thi s contradicts the RCC theory which allows that the sum of x and y may b e the univers e
in which cas e no region would b e di sconnecte d f rom b oth the s e regions
Another slightly more complex example i s provide d by the comp os ition of EC and TPP which
i s given as fEC PO TPP NTPPg corre sp onding to an extens ional comp os ition de scr ib e d by
x y z EC x z TPP z y
EC x y PO x y TPP x y NTPP x y
Thi s says that whenever regions a b are relate d by e ither of EC PO TPP or NTPP there must b e a
third region c such that EC a c TPP c b Situations sati sfying the s e conditions are illustrate d
in Figure As long as b i s an ordinary b ounde d region a region c sati sfying the appropr iate
conditions can always b e found However if a i s an ordinary region and b u then NTPP a b
but no region c can b e found which i s a TPP of b the univers e has no tangential prop er parts
c c c a a c b a b a b b
EC(a,b) PO(a,b) TPP(a,b) NTPP(a,b)
Figure Comp os ition of EC and TPP i s not fully extens ional
There are a numb er of ways that one might b e able to avoid such problems and hence construct
an extens ional comp os ition table The most obvious i s to remove the universal region u f rom the
domain of p oss ible referents of the region constants All the exceptions to extens ional comp os ition
that I am aware of involve u so it s eems that an extens ional interpretation could b e achieve d with
re sp ect to a mo di e d theory without a universal region The domain of thi s new theory would then
b e more homogeneous and more s imilar to that of the Allen relations where intervals are always
b ounde d Alter natively it might b e p oss ible to retain u by re ning the s et of relations so as to
di erentiate relations involving u f rom thos e among ordinary regions It s eems plaus ible that by
adding thi s additional expre ss ive p ower to the bas e relations one could arr ive at an extens ional
comp os ition table Of cours e the bas i s of the table would cons i st of cons iderably more than e ight
relations
CHAPTER COMPOSITION BASED REASONING
Relation Algebras
A formali sm that has b een valuable in the analys i s of comp os ition bas e d reasoning algor ithms for
temp oral relations Ladkin and Maddux i s Relation Algebra in which relations are cons idere d
as elements of a Bo olean algebra augmente d with composition and converse op erators ob eying
axioms rst sp eci e d by Tarski and later inve stigate d in great detail in Tarski and Givant
Although I have so f ar obtaine d only preliminary re sults concer ning the character i sation of
RCC relations within thi s formali sm I think it i s appropr iate to include the s e here I b elieve that
Relation Algebra may tur n out to provide a very p owerful language for automate d reasoning
A Relation Algebra i s a Bo olean algebra which has in addition to the usual sum product
and complement two additional op erators a binary composition op erator and a unary
converse op erator It also has constants denoting the identity relation and the universal
relation thi s i s not e ss ential s ince it i s de nable by The ob jects of a Relation
Algebra are intende d to b e binary relations conce ive d of as s ets of pairs However it tur ns out
that thi s standard interpretation i s not p oss ible for every Relation Algebra
Under the intende d interpretation and repre s ent thos e op erators which in a st order
theory of relations could b e schematically de ne d as follows
R S x y z R x z S z y
def
R x y R y x
def
x y x y
def
But in a Relation Algebra relations are bas ic entitie s and the op erators are given an algebraic
character i sation so that they can b e studie d in a order f ramework Hence a Relation Algebra
must ob ey in addition to some axiom s et character i s ing a Bo olean algebra the identitie s given in
table which x the meanings of and
x y z x y z x y x y
x y z x z y z x y y x
x x x x y y y
x x
Table Equational axioms for a Relation Algebra
It i s known that in general reasoning in a Relation Algebra i s undecidable and thi s counts
against the p otential us efulne ss of the s e algebras in automate d reasoning However for many
11
sp eci c algebras cons i stency checking i s decidable and may even b e p olynomial Indee d if an
extens ionally interprete d comp os ition table can b e given for a vo cabulary of bas ic relations in a
Relation Algebra thi s can b e us e d to eliminate the comp os ition op eration f rom complex algebraic
11
Some complexity re sults for reasoning with relation algebras are given by Ladkin and Maddux
CHAPTER COMPOSITION BASED REASONING
terms and thi s will lead directly to a deci s ion pro ce dure I b elieve that the viability of Relation
Algebra as a formali sm for automate d reasoning de s erve s further exploration
De ning Spatial Relations
The Relation Algebra formali sm can b e us e d to sp ecify a spatial Relation Algebra which de scr ib e s
the same domain as the RCC theory As in the st order RCC theory I start with a connecte dne ss
relation which i s axiomatis e d to b e symmetr ic and re exive I now denote relations by the same
letters as the ir RCC counterparts but in lower cas e Be ing symmetr ic and re exive c must ob ey
the axioms
c c symmetry and c c re exivity
In terms of c one can eas ily de ne some of the more s igni cant relations found in the RCC
theory
p c c o p p
def def
pp p tp p c o
def def
In f act making us e of the relations just de ne d we can go on to de ne all the RCC relations as
follows
dc c ntpp pp tp
def def
ec c o tppi tpp
def def
po o p p ntppi ntpp
def def
tpp pp tp eq
def def
It app ears that many if not all relations de nable in the RCC theory can b e de ne d as
relation algebraic expre ss ions forme d f rom the s ingle pr imitive relation c The re sulting algebra
can b e obtaine d by f actor ing with re sp ect to the symmetry and re exivity identitie s the f ree
relation algebra generate d by a s ingle relation However it i s likely that additional axioms would
b e nee de d to capture adequately the exi stential prop ertie s of the domain of spatial regions For
example if there i s a universal region which connects with every region in the domain then the
identity c c must hold
Chapter
Further Work and Conclus ions
In thi s nal chapter I summar i s e the main re sults of the the s i s and p oint to areas that would
b ene t f rom further work I also lo ok at how logical spatial reasoning technique s t into the
wider context of AI and computer science in general
What has b een Achieve d
In the cours e of thi s the s i s a large numb er of p oss ible spatial repre s entations have b een cons idere d
The intro ductory chapter gave an overview of the or igins and developments of var ious approache s
to reasoning with spatial information Chapter surveye d some of the more imp ortant axiomatic
theor ie s of spatial regions including p oint s et top ology Le sniewski s Mereology Tarski s Geometry
of solids Clarke s theory of spatial regions and the RCC theory In chapter the RCC theory of
spatial regions was examine d in some detail and a numb er of mo di cations were sugge ste d The key
meta theoretic prop ertie s of completene ss categor icity decidability were also cons idere d Chapters
develop e d a new approach to qualitative reasoning bas e d on enco dings of spatial concepts
into order logics Becaus e they are decidable the s e repre s entations are much b etter suite d to
computational applications than st order formali sms The next two chapters de scr ib e d di erent
ways in which the expre ss ive p ower of the order repre s entations might b e extende d in chapter
I examine d the us e of quanti er elimination in RCC like st order spatial theor ie s and showe d that
there are many class e s of quanti e d expre ss ion whos e quanti ers can b e eliminate d by syntactic
transformation to logically equivalent quanti er f ree forms chapter was concer ne d with extending
the expre ss ive p ower of order repre s entations b eyond purely top ological prop ertie s Finally in
chapter I examine d the application of comp os itional reasoning to spatial relationships
I hop e that f rom amongst the plethora of repre s entational formali sms and the var iety of reason
ing metho ds that have b een cons idere d certain general pr inciple s have emerge d Pr imary among
the s e i s the trade o b etween expre ss ive p ower and tractability which conf ronts the attempt to
tur n theory into practice in all areas of AI Whilst the intractability of reasoning within a given
formal language i s e ss entially indefeas ible I think that the ndings of thi s the s i s illustrate f ruitful
ways in which it can b e circumvente d The key obs ervation i s that a language L within which a
CHAPTER FURTHER WORK AND CONCLUSIONS
s et of concepts C are eas ily expre ss e d i s not nece ssar ily a go o d language for reasoning ab out thos e
concepts Indee d if L i s highly expre ss ive then it will b e able to expre ss logical connections in all
manner of conceptual domains but thi s generality means that L i s over expre ss ive with re sp ect to
the problem of reasoning ab out the concepts in C To achieve computational tractability what we
must lo ok for i s the minimally complex language capable of expre ss ing the concepts of C i e L
should have just enough expre ss ive p ower and no more Cons equently the enco ding of the logic of
the concepts C into a tractable language L may b e complex and indirect captur ing the s e concepts
stretche s the language to its limits
In applying thi s pr inciple of minimality a wide var iety of p oss ible logical repre s entations should
b e cons idere d In traditional logic and also in knowle dge repre s entation within the eld of AI a
f airly limite d range of formalisms have b een employe d Sp eci cally the range of available language s
has often b een s een as b e ing re str icte d to order prop os itional logic st order pre dicate logic
p oss ibly with some limitations on the syntactic forms which can b e employe d and higher order
logics Since prop os itional logic i s extremely limite d in expre ss ive p ower and logics of nd or higher
order do not have complete pro of pro ce dure s some form of st order logic has b een the f avour ite
language for repre s enting f actual information and expre ss ing logical connections b etween concepts
The us e of more expre ss ive forms of order logic has generally b een con ne d to the character i sation
of prop os itional mo di ers such as nece ss ity and b elief by means of mo dal op erators Perhaps the
most novel asp ect of the work rep orte d in thi s the s i s i s the us e of the s e more expre ss ive order
formali sms to capture the logic of purely extens ional relational expre ss ions The us e of mo dal
and intuitioni stic logic for repre s enting spatial relations illustrate s new p otential us e s in knowle dge
repre s entation of logics whos e expre ss ive p ower i s interme diate b etween the s imple Bo olean order
logic and quanti cational logics The s e augmente d order logics may prove to b e applicable in
many other conceptual domains
The enco ding of top ological relations into I provide s further supp ort for the idea that if e ective
reasoning i s to b e achieve d expre ss ive p ower should b e limite d as much as p oss ible even at the
exp ens e of making the repre s entation le ss natural As I explaine d in s ection the logic I
can b e regarde d as an alter native syntax for a certain sub language of S While the re str icte d
expre ss ivity of I means that more indirect enco ding of top ological constraints i s require d than
with S thi s i s comp ensate d by I b e ing b etter suite d to automate d reasoning
Further Work
Thi s study has sought to identify advantage s and di sadvantage s of di erent p oss ible repre s entations
of spatial information and to clar ify the relationships b etween the s e di erent formali sms However
many asp ects of the s e theor ie s still remain unclear In thi s s ection I highlight a numb er of areas
which I b elieve are particularly de s erving of further re s earch Some of the s e are imp ortant b ecaus e
they concer n the foundations of spatial reasoning whilst others are areas which may lead to the
further development of the theory in new directions
CHAPTER FURTHER WORK AND CONCLUSIONS
Complete Spatial Theor ie s
In s ection I cons idere d the p oss ibility of constructing a complete and categor ical theory having
the same vo cabulary as RCC We saw that the undecidability of Grzegorczyk means that
no complete nitary st order theory of thi s kind can b e sp eci e d Neverthele ss for the sake of
providing a theoretical foundation for spatial reasoning a complete RCC like theory i s certainly
de s irable even if formulate d in a system such as nd order or in nitary st order logic for which
a complete pro of system cannot b e sp eci e d
As I was coming to the end of my work on thi s the s i s a complete region bas e d spatial theory
was indee d e stabli she d by Pratt and Scho op us ing an in nitary extens ion of st order
logic Thi s theory i s calle d and i s formulate d for a language containing a monadic pre dicate
of connecte dne ss together with Bo olean op erations cons i sts of a s et of st order axioms and an
1
in nitary inference rule and i s shown to b e complete with re sp ect to an interpretation in which
the domain of regions cons i sts of all thos e regular op en regions of the Carte s ian plane that
can b e b ounde d by some nite numb er of linear e dge s The re str iction to linear b ounde d regions
i s ine ss ential s ince every con guration of regular op en planar regions i s top ologically equivalent
to some con guration in which all the e dge s are linear
The vo cabular ie s of RCC and are inter de nable so the axiom s et could b e us e d to sp ecify
a vers ion of RCC which i s complete with re sp ect to a natural interpretation in D space However
the in nitary nature of means that it cannot b e us e d as a practical to ol for carrying out spatial
inference s The que stion also remains as to what axioms are nee de d to sp ecify a theory which i s
complete with re sp ect to a D interpretation
E ective Mo dal and Intuitioni stic Re asoning
Whilst mo dal repre s entations of spatial relations can b e shown to have a theoretical advantage over
st order repre s entations namely that deci s ion pro ce dure s are known for the mo dal language s
neverthele ss doubts may remain as to whether the mo dal repre s entations could ever b e of practical
us e After all a deci s ion pro ce dure do e s not nece ssar ily provide us with an e ective means of
computation Ideally we would like to have p olynomial algor ithms for spatial reasoning Recently
a lot of re s earch has b een directe d towards the nee d for more e cient mo dal reasoning systems
Wallen Au ray Enjalb ert and Herbrard Catach Demr i Giunchiglia and
Sebastiani Nonnengart Balbiani and Demr i Montanar i and Policr iti Hustadt
and Schmidt If the mo dal approach to qualitative reasoning i s to b e of practical us e it will
b e nece ssary to demonstrate that the mo dal repre s entations can b e e ectively manipulate d One
way to do thi s would b e to identify tractable sub language s of mo dal calculi which are capable of
repre s enting s igni cant s ets of spatial relations
1
This rule state s that if it can be shown for all n that every region which is a sum of n connecte d comp onents
has the property then one can infer x x
CHAPTER FURTHER WORK AND CONCLUSIONS
Extending Expre s s ive Power
Another imp ortant direction for further work i s to inve stigate how the expre ss ive p ower of decidable
spatial repre s entations such as I can b e extende d The main fo cus of my work has b een on
top ological relations but a fully expre ss ive language for qualitative spatial information would b e
able to de scr ib e a wide range of non top ological prop ertie s and relations For many purp os e s one
would wi sh to character i s e the relative p os ition and or ientation of regions or p oints Freksa b
It would also b e very us eful e g for the task of ob ject recognition to have a r icher vo cabulary for
di stingui shing di erent shap e s
In chapter I showe d how the expre ss ive p ower of the RCC language can b e greatly increas e d
by means of an additional conv function giving the convex hull of any region Thi s enable s many
us eful relations concer ning containment to b e de ne d Cohn has shown that within thi s
augmente d RCC theory a large class of shap e s can b e sp eci e d by de ning shap e concepts in
a hierarchical manner and the expre ss ivene ss and complexity of the language cons i sting of the
RCC relations and a convexity pre dicate i s explore d in detail in Davi s et al However
prop ertie s involving or ientation cannot b e expre ss e d within such a language A logical treatment of
or ientation convexity and relate d prop ertie s in terms of p oints has b een given by Knuth
Thi s i s bas e d on a pr imitive ter nary relation ass erting that three p oints lie in an anti clo ckwi s e
or ientation in the plane There s eems no reason why a s imilar pre dicate op erating on regions could
not b e intro duce d into RCC Appropr iate axioms determining the logical prop ertie s of the new
pr imitive would then have to b e sp eci e d
In s ection I showe d how prop ertie s of the convex hull op erator can b e capture d by means of
mo dal schemata It i s p oss ible that a s imilar technique could b e applie d to other spatial concepts
Indee d Balbiani Far inas del Cerro Tinchev and Vakarelov have shown that mo dal logics
can b e interprete d as sp ecifying con gurations in incidence geometry My metho d of sp ecifying
prop ertie s of spatially interprete d mo dalitie s in terms of axiom schemata i s somewhat ad hoc and
do e s not provide a direct interpretation of the op erator in terms of mo del structure s To do
thi s we would nee d r icher mathematical structure s as mo dels An obvious choice would b e to us e
metr ical Carte s ian space s The s e are canonical mo dels for Euclidean geometry and so provide an
interpretation for any gure or prop erty de scr ibable in thi s geometry Having metr ical space s as
mo dels for qualitative language s also f acilitate s easy integration with quantitative information as
will b e di scuss e d in s ection
Although the combination of top ological concepts and convexity provide s a very p owerful spa
tial de scr iption language the e ective reasoning pro ce dure s that I have so f ar constructe d only
cover a small f ragment of the prop ertie s and relations that can b e expre ss e d in terms of the s e con
cepts Hence the most us eful further work will p erhaps b e directe d towards expanding the range
of information that can b e handle d by e ective deci s ion pro ce dure s rather than the expre ss ive
p ower of spatial repre s entations After all highly expre ss ive but intractable mathematical nota
tions already exi st Sp eci cally it i s probable that there are e ective algor ithms for I reasoning
that can deal with much larger class e s of formulae than the re str icte d class nee de d to repre s ent
CHAPTER FURTHER WORK AND CONCLUSIONS
the RCC relations For instance it would b e us eful to sp ecify top ological constraints involving
Bo olean combinations of regions such as DC x pro d y z corre sp onding to the I mo del con
straint x x z An alter native means of increas ing the expre ss ive p ower of decidable
systems i s by means of quanti er elimination pro ce dure s such as the one given in chapter which
i s by no means as general as it could b e
The next two s ections fo cus on two asp ects of extending expre ss ive p ower that I cons ider to b e
of particular imp ortance
Re asoning with One Piece and other Simplicity Constraints
An extremely imp ortant top ological prop erty of regions i s that of b e ing self connected or in one
piece Thi s prop erty can b e quite eas ily de ne d within the RCC theory as follows
OP x y z x sum y z C y z
def
Thi s prop erty i s particularly s igni cant b ecaus e the region o ccupie d by what we think of as a
phys ical b o dy i s almost always in one piece accordingly in natural language de scr iptions of
phys ical s ituations implicit one piece constraints are ubiquitous In f act the ob jects of natural
di scours e are typically further constraine d to b e regular i e of uniform dimens ion and rmly
s elf connecte d a three dimens ional phys ical ob ject cannot b e divide d into two parts that are
connecte d only at a p oint or along a line
A s er ious de ciency with the reasoning systems de scr ib e d in thi s the s i s i s that I have not
provide d any means for reasoning under constraints sp ecifying that certain regions are in one piece
let alone more subtle s implicity constraints Handling such prop ertie s i s an imp ortant goal for
future re s earch It might b e p oss ible to apply a technique s imilar to that us e d to enforce the
convexity of regions In thi s cas e if region a i s supp os e d to b e one piece we would check all pairs
b c of regions involve d in the s ituation to s ee if a sum b c could b e prove d If so the further
condition C b c must b e adde d to the s ituation de scr iption Whatever approach i s taken it i s
likely contrary to what some might supp os e that the pre s ence of s implicity constraints will make
reasoning intr ins ically more di cult more will b e said ab out thi s at the end of the next s ection
Points and Dimens ionality
In thi s the s i s I have b een pr imar ily concer ne d with spatial relationships that can hold b etween
regions Thi s re str iction was motivate d in the intro duction by the obs ervation that most natural
forms of de scr iption make reference to ob jects which o ccupy three dimens ional volume s or le ss
commonly two dimens ional areas One can then argue that although higher dimens ional ob jects
can b e constructe d s et theoretically f rom p oints it i s much preferable f rom a computational p oint
of view to formulate theor ie s in which regions are bas ic entitie s i e constitute a domain over which
one can apply str ictly st order quanti cation rather than to employ the highly complex language
of s et theory Neverthele ss there i s also strong evidence that many natural forms of expre ss ion do
CHAPTER FURTHER WORK AND CONCLUSIONS
refer to p oint like or linear entitie s and it i s clear that we also di stingui sh b etween two dimens ional
regions on a surf ace and three dimens ional volume s
In chapter we saw how Tarski and Clarke to ok regions as the bas ic entitie s of the ir theor ie s but
then intro duce d p oints s et theoretically as corre sp onding to certain s ets of regions For Tarski thi s
was a means of render ing hi s theory categor ical by constraining it to ob ey the axioms of elementary
p oint geometry Clarke s intention in intro ducing p oints i s to show that hi s theory can encompass
class ical geometr ical and top ological concepts by the us e of s econd order de nitions Ne ither of
the s e treatments of p oints addre ss e s the i ssue of how to construct a naturali stic logical formali sm
capable of expre ss ing information ab out spatial entitie s of di erent dimens ions Preferably one
would like to have a system which allowe d thi s information to b e repre s ente d without the us e of
s econd order op erators Thi s i s particularly imp ortant for computational applications s ince higher
order formali sms are typically intractable and often not completely axiomatisable
The RCC formali sm do e s not place constraints on the dimens ionality of regions except that
b ecaus e of the exi stence of a non tangential prop er part of every region all the regions in the
domain must have the same dimens ionality For certain applications thi s will constitute a s evere
limitation in expre ss ive p ower A formali sm having much in common with RCC but capable of
expre ss ing relations b etween entitie s of di erent dimens ions has b een given in Gotts Thi s
INCH calculus i s bas e d on the pr imitive INCH x y read as x include s a chunk of y meaning
that the region x overlaps with some part of y which i s of the maximum dimens ion of any part of
y In terms of just thi s pr imitive pre dicate s identifying regions of any nite dimens ionality can b e
de ne d
Handling prop ertie s involving dimens ionality also pre s ents ma jor problems for automating spa
tial inference s The reasoning algor ithms de scr ib e d by me in chapters only enforce entailments
which hold in a very large class of top ological space s and the same i s true of reasoning us ing
comp os ition table s as de scr ib e d in chapter I mentione d at the end of s ection that if we
re str ict the domain of regions involve d in a s et RCC relational f acts to planar regions b ounde d
by Jordan curve s then te sting cons i stency of the s e f acts b ecome s NP hard Gr igni et al
Solving thi s problem also involve s enforcing the s implicity constraints mentione d in the previous
s ection
The Relation Between Logic and Algebra
The inve stigation carr ie d out in thi s the s i s was conducte d pr imar ily f rom the p oint of view of
logical analys i s That i s my pr incipal intere st was in entailment relationships and inference rule s
involving formal expre ss ions However in carrying out thi s analys i s algebraic structure s have
playe d a key role In my enco dings of spatial relationships into order logics equational algebraic
theor ie s acte d as an interme diary b etween relational formalisms and order formulae and hence
enable d me to show the correctne ss of the s e enco dings An alter native approach would b e to start
by adopting equational reasoning as a f ramework for computational inference and then lo ok at
what spatial theor ie s could b e expre ss e d equationally
CHAPTER FURTHER WORK AND CONCLUSIONS
Equational reasoning i s a large re s earch area in its elf and its metho ds cannot b e covere d here
A collection of pap ers on many asp ects of the area can b e found in A t Kaci and Nivat
in which one chapter Fear nley Sander de scr ib e s an intere sting equational repre s entation of
spatial information bas e d on vector space s thi s i s very di erent to my closure algebraic treatment
The pr imary di erence b etween equation centre d approache s and mine i s in the formali sm that i s
actually us e d for reasoning I have sugge ste d that order logics should b e us e d but there may
also b e go o d reasons why it would b e b etter to us e equational reasoning
Another i ssue concer ning the relation b etween logic and algebra i s that of notation Although
algebraic structure s often o ccur as mo dels for logical language s there do e s not s eem to b e any
standard way of stating corre sp ondence s b etween logical and algebraic prop ertie s and I found
cons iderable di culty in arr iving at a way of expre ss ing the correlation theorems nee de d to jus
tify my order enco dings The general f ramework of category theory i s well suite d to de scr ibing
relationships b etween di erent mathematical structure s and may prove us eful for thi s but for fur
ther study of the connection b etween algebraic constraints and logical entailment more sp eciali s e d
notation would b e de s irable
Comp o s itional Re asoning and Relation Algebra
In the last chapter I lo oke d at the us e of comp os ition table s for cons i stency checking of s ets of
binary relations We saw that comp os itional constraint propagation us ing such a table provide s
a cons i stency checking pro ce dure that runs in O n time for a s et of relational f acts containing
n constants However whether thi s pro ce dure i s complete dep ends on the particular relation s et
and the background theory with re sp ect to which they are interprete d Given the e ectivene ss
of comp os itional reasoning determining the conditions under which a comp os ition table can b e
complete with re sp ect to a theory in the s ens e sp eci e d in s ection i s likely to b e a f ruitful
area for further enquiry
The formali sm of Relation Algebra br ie y inve stigate d in s ection also de s erve s further
study and may prove to b e well suite d to automate d reasoning Relation Algebra provide s an
extremely expre ss ive alter native to st order logic it has almost the same expre ss ive p ower Tarski
and Givant particularly in formali s ing theor ie s where binary relations play an imp ortant
role
Spatial Re asoning in a More General Framework
In thi s the s i s I have treate d spatial relationships as an i solate d domain of information However
if one wi she s to develop a more general reasoning system capable of pro ce ss ing the divers e kinds
of information that humans routinely deal with one must nd some means by which purely spa
tial concepts and reasoning mechani sms can b e interf ace d or combine d with repre s entations and
reasoning mechani sms for non spatial concepts
CHAPTER FURTHER WORK AND CONCLUSIONS
A General Theory of the Phys ical World
A tenet of AI re s earch into Knowle dge Repre s entation i s that if an arti cial agent i s to act in
an intelligent way to accompli sh goals in the real world or in some virtual s imulation of the real
world it must have b oth f actual knowle dge ab out the actual state of the world and theoretical
knowle dge concer ning p oss ible state s of the world and p oss ible ways that one state can succee d
another Haye s Haye s b Guha and Lenat From the p oint of view of so calle d
symb olic AI the s e laws of p oss ibility and causality will constitute a formal theory of phys ical
pro ce ss e s Thi s theory might b e akin to thos e that have b een e stabli she d by phys ici sts except that
whereas the phys ici st i s pr imar ily concer ne d with the de scr iptive p ower and pre dictive accuracy of
hi s theory the computer scienti st must also cons ider computational prop ertie s of the theory such
as the kinds of inference that can b e e ectively compute d
It has b een sugge ste d that for the purp os e s of AI what i s nee de d i s not a fully scienti c theory
of the phys ical world but rather a na ve theory of thos e phys ical concepts which are relevant
to commons ens e reasoning ab out the world Haye s Haye s a Haye s b Randell
Cohn and Cui b Egenhofer and Mark But it i s clear that whatever style of theory
i s require d it must contain a sub theory of spatial concepts Just as a spatio temp oral geometry
de scr ib e s the underlying theory of co ordinate systems up on which mathematical theor ie s of phys ical
pro ce ss e s are built repre s entations of spatial and temp oral concepts must b e fundamental to any
formal de scr iption of the s e pro ce ss e s which might b e employe d in AI Formali sms for temp oral
reasoning have rece ive d a huge amount of attention f rom the AI community s ee e g Galton
and repre s entations of spatial concepts are increas ingly b e ing studie d However e stabli shing a
foundation for the sp eci cation of phys ical theor ie s will require integrate d repre s entations and
reasoning mechani sms capable of handling integrate d spatial and temp oral information and the
construction of a suitable combine d spatio temp oral theory p os e s formidable problems Some of
the more concrete prop osals can b e found in Randell and Cohn Randell Cui and Cohn
Galton and Galton I shall give some details of the s e prop osals in the next
s ection
A suitable spatio temp oral theory ought to provide a f ramework within which theor ie s of matter
kinematics and dynamics can b e develop e d in such a way that the s e theor ie s can b e us e d to
reason ab out de scr iptions of phys ical pro ce ss e s in a way which i s amenable to e ective automate d
reasoning The theory of matter whilst one of the pr incipal fo cuss e s of phys ici sts has rece ive d
comparatively little attention f rom logicians and AI re s earchers a notable exception i s Haye s
a analys i s of the ontology of liquids For instance in formali s ing problems of rob ot motion
planning it has generally b een assume d that space can b e neatly divide d into two partitions
o ccupie d space and empty space Thi s i s clearly a very coars e approximation to the real nature
and di str ibution of matter in the univers e
Quite a large b o dy of work exi sts on qualitative kinematics and dynamics for AI s ee e g Weld
and De Kleer Nearly all thi s work i s bas e d up on some kind of abstraction of the spatio
temp oral b ehaviour of a system into a s equence of trans itions within a di screte space of p oss ible
CHAPTER FURTHER WORK AND CONCLUSIONS
state s state trans itions among RCC relations will b e cons idere d in the next s ection Given
that a su ciently expre ss ive and computationally tractable repre s entation for spatial temp oral
information has not yet b een di scovere d thi s approach i s certainly well justi e d An alter native
approach has b een to develop formal theor ie s in which actions events or pro ce ss e s are prop er
entitie s constraine d by temp oral relationships e g Allen In b oth the s e approache s the
structure of space its elf s eems to all but di sapp ear once phenomena are formally analys e d Thi s
lack of expre ss ivene ss in re sp ect of spatial relationships s eems to me to b e an inherent weakne ss
of most exi sting formali sms for de scr ibing phys ical pro ce ss e s
Spatial Information and Change
One way of building a dynamical theory on top of a spatial theory i s by sp ecifying p oss ible trans
itions among relations holding b etween the regions o ccupie d by two b o die s when the b o die s un
dergo continuous di splacement and or deformation Figure taken f rom Randell Cui and
Cohn shows a graph of p oss ible trans itions among the RCC relations re sulting f rom e ither
continuous di splacements or deformations of the regions involve d Trans itions b etween qualitative
spatial state s have b een studie d in a numb er of pap ers by Antony Galton
Connecte d sub graphs of a trans ition network are known as conceptual neighbourhoods a term
that was intro duce d in Freksa s a analys i s of the Allen relations Freksa notice d that all the
entr ie s in the comp os ition table for the Allen relations corre sp ond to conceptual ne ighb ourho o ds
The relationship b etween conceptual ne ighb ourho o ds and relational comp os ition was also studie d
by me in Bennett a where I showe d that the correlation obs erve d by Freksa do e s not apply
to all s ets of spatial relations
An alter native metho d of accommo dating change into a spatial repre s entation i s to intro duce
time as an extra dimens ion dimens ional regions would then corre sp ond to the space time exten
s ions of dimens ional ob jects throughout the ir hi story Thi s approach was adopte d in Randell
and Cohn in which a theory of top ological relations b etween spatio temp oral regions was
augmente d with a relation B x y ass erting that the spatio temp oral region x wholly temp orally
prece de s region y
TPP NTPP a b
a b a a a a b =
b b b
b a a DCEC PO b
TPPi NTPPi
Figure Trans ition network for e ight top ological relations
CHAPTER FURTHER WORK AND CONCLUSIONS
Vague and Uncertain Information
In real applications of reasoning systems s ituations will often ar i s e where information i s vague or
2
has some degree of uncertainty Ideally we would like a computer system to do its b e st whatever
that means even with vague or uncertain information It i s likely that an understanding of spatial
vaguene ss will b e very imp ortant in the development of many applications Qualitative repre s enta
tions such as the RCC language have an intr ins ic advantage over numer ical repre s entations when
it come s to dealing with vague or uncertain f acts relevant qualitative di stinctions can b e made
without any commitment to the preci s e details of a s ituation For example we may not know the
exact geometry of a ro om nor the exact s ize and p os ition of a table s ituate d somewhere in the
ro om however we can b e certain that the table do e s not overlap the walls of the ro om Us ing RCC
we could s imply ass ert something like O table walls whereas in terms of numer ical co ordinate s
stating thi s f act would require a complex and clumsy s et of inequalitie s
Although certain asp ects of vaguene ss and uncertainty can b e straightforwardly capture d by
the generality encapsulate d in qualitative concepts other asp ects are not so eas ily repre s ente d
Certain typ e s of region e g a swamp or a cloud have inherently vague b oundar ie s and hence a
sharp di stinction b etween the top ological relations holding among such regions cannot b e made
An axiomatic theory which generali s e s RCC to take account of regions with vague b oundar ie s has
b een develop e d in Cohn and Gotts a Cohn and Gotts b Gotts and Cohn Cohn
and Gotts
Relating Qualitative and Metr ic Repre s entations
There has b een a tendency among some re s earchers in the eld of QSR to e schew metr ical data
in the b elief that s igni cant AI tasks can b e p erforme d us ing only qualitative information While
in certain cas e s thi s may b e p oss ible I b elieve that in the ma jor ity of practical applications one
will want to combine b oth quantitative and qualitative information and cons equently the interf ace
b etween the two typ e s of data will b e increas ingly studie d
Purely qualitative spatial reasoning systems provide an inference mechani sm for determining
whether a given qualitative f act follows f rom some s et of such f acts Such systems can b e us e d to
answer quer ie s relative to a qualitative databas e However a qualitative spatial reasoning system
nee d not b e employe d in i solation f rom co ordinate bas e d geometr ical information and other kinds of
numer ical data Indee d it i s clear that for many us eful functions numer ical information i s e ss ential
For instance we may want to p os e a query us ing qualitative concepts but requir ing a quantitative
answer e g What i s the area of the large st de s ert that lie s entirely within the b orders of one
country Moreover the combination of qualitative and quantitative repre s entations promi s e s to
b e a p owerful to ol in system de s ign and to enable novel program functionality In the re st of thi s
s ection I shall sketch a numb er of ways in which qualitative and metr ical data could b e combine d
In the intro duction to thi s the s i s I obs erve d that current computer systems repre s ent spatial
2
Although vaguene ss and uncertainty have some logical propertie s in common it is imp ortant to recognis e them
as very di erent phenomena However in the pre s ent br ief discussion the di erence s are not imp ortant
CHAPTER FURTHER WORK AND CONCLUSIONS
information almost entirely in terms of numer ical co ordinate s However a high prop ortion of te sts
made on thi s data e g in conditional statements of the form if test then command although
formulate d in numer ical terms are actually de s igne d to te st qualitative relations b etween data
ob jects For example we may wi sh to te st whether two line s egments cross Thi s i s a qualitative
relationship b etween the s egments To determine whether a qualitative relationship such as thi s
holds b etween entitie s an algor ithm i s nee de d which will op erate on numer ical data structure s so
as to extract the require d information In many cas e s including the cas e of the cross ing line
s egments thi s can b e achieve d by formulating the relationship in terms of a Bo olean combination
of equalitie s and inequalitie s involving the co ordinate s of p oints in other cas e s more complex
iterative routine s will b e require d
Whilst it may b e p oss ible on a cas e by cas e bas i s to devi s e an algor ithm to extract sp eci c
qualitative information when nee de d f rom quantitative data structure s it would b e f ar preferable
to have a general purp os e metho d of te sting all qualitative relationships which one may encounter
A qualitative repre s entation whos e interpretation i s linke d in a preci s e way to the content of quant
itative data structure s can go some way towards providing thi s capability The idea i s to asso ciate
the pr imitive s of the qualitative repre s entation with appropr iate algor ithmic op erations on quant
itative data Given thi s interpretation of the pr imitive s any complex expre ss ion in the qualitative
language would then b e evaluate d by combining the s e pr imitive op erations in accordance with the
s emantics of logical op erators in the repre s entation Constructing thi s evaluation mechani sm may
b e very di cult or even imp oss ible dep ending on the nature of the pr imitive s and the logical
op erations involve d but once achieve d it provide s a general purp os e pro ce dure for evaluating a
large probably in nite class of qualitative expre ss ions The qualitative repre s entation can thus
function directly as a query language as well as b e ing us e d inter nally for program control
A limite d vers ion of thi s approach i s already found in nearly all computer programs Whenever
one de ne s some bas ic qualitative te sts as functions retur ning Bo olean value s and then us e s
Bo olean combinations of the s e te sts in conditional statements a s imple qualitative language i s in
op eration To move f rom thi s limite d capability to the us e of a fully e dge d qualitative repre s
entation one must identify a vo cabulary of pr imitive s and logical op erators su cient to repre s ent
any qualitative f act in some particular conceptual domain The problem for the programmer then
rather than b e ing how can I co de an algor ithm to te st whether thi s relationship holds b ecome s
how can I expre ss thi s relationship in terms of my qualitative language Thi s architecture has the
advantage that the evaluation of qualitative te sts i s indep endent of the particular data structure s
us e d to store quantitative data in the system except in so f ar as op erations corre sp onding to the
pr imitive s must b e co de d
The main obstacle to achieving thi s kind of qualitative quantitative interf ace i s that as we have
s een even mo de st logical vo cabulary can give r i s e to a language which i s highly intractable In
particular a language which allows quanti cation over some p otentially in nite domain of entitie s
i e a st order language will b e undecidable unle ss by taking account of the meanings of the
sp eci c vo cabulary of the language some sp ecial purp os e deci s ion pro ce dure can b e devi s e d The
order repre s entations of spatial relations develop e d in thi s the s i s go some way towards solving
CHAPTER FURTHER WORK AND CONCLUSIONS
thi s problem by providing quanti er repre s entations capable of expre ss ing a s igni cant vo cabulary
of spatial relations
Even where a deci s ion pro ce dure can b e found it may b e that the time taken to evaluate a
qualitative te st increas e s exp onentially with the amount of information which has to b e taken
into account Thi s will make the language unsuitable for repre s enting large amounts of data
However in many cas e s it may b e safe to assume that although the databas e of quantitative spatial
information may b e very large the qualitative te sts quer ie s that the system will b e require d to
evaluate will b e comparatively conci s e The time taken to evaluate a qualitative query will b e a
function of b oth the amount of quantitative information store d and the complexity of the query
Retr ieving information f rom the quantitative databas e will typically take time which increas e s only
p olynomially in the s ize of the databas e in most cas e s retr ieval time s will increas e linearly or as
some small p ower of the databas e s ize Thus even if the query answer time increas e s exp onentially
with query complexity thi s may b e acceptable as long as all quer ie s have complexity b elow a certain
level Also on rece iving a qualitative query it would b e p oss ible for the system to e stimate the
maximum time require d to retur n an answer
A us eful generali sation of the capability of answer ing qualitative quer ie s with re sp ect to a
metr ical databas e i s the ability to generate a qualitative de scr iption f rom such a databas e A
s imple example i s that one may have a databas e cons i sting of a s et of p olygons each corre sp onding
to some geographical region and f rom thi s one might wi sh to extract a complete de scr iption of
the relationships b etween the s e regions in terms of the RCC relation s et i e generate a s et of
f acts in which each pair of regions i s relate d by one of the RCC relations Having extracte d a
qualitative de scr iption f rom a quantitative databas e one could then combine thi s with additional
purely qualitative information Bas e d on thi s idea a sophi sticate d and exible architecture can b e
envi sage d in which quantitative data can b e transparently combine d when require d with qualitative
data in order to allow quer ie s to b e addre ss e d to a hybr id information source containing b oth
quantitative and qualitative data
Yet another us eful capability would b e to generate numer ical co ordinate data sati sfying a given
s et of spatial constraints Thus for example one might wi sh to generate a p oss ible quantitative
sp eci cation for a mechanical comp onent having certain pre scr ib e d qualitative prop ertie s Perhaps
thi s could b e done by means of some mo del building automate d theorem prover An obvious
di culty i s that there i s usually no unique quantitative state sati sfying given qualitative constraints
many solutions may b e unnece ssar ily complex or deviate in subtle ways f rom what was really
wante d so it may b e hard to pick a s ens ible solution
In s ection I sugge ste d that interpreting qualitative language s in terms of metr ical mo dels
might b e a way to develop more expre ss ive language s Clearly thi s would also b e very us eful for
integrating qualitative and metr ical information The eld of QSR has tende d to e schew metr ical
mo dels on the na ve assumption that such mo dels are only appropr iate for quantitative repre s ent
ations But thi s i s to mi sunderstand the relationship b etween a logical language and its mo dels
Formal language s cannot ordinar ily fully de scr ib e the ir own mo dels the f act that a mo del sati s e s
a given formal s entence i s a matter of meta logic Nor do e s the ontological commitment of a formal
CHAPTER FURTHER WORK AND CONCLUSIONS
language dep end up on its mo dels but rather on its re source s for ass erting what exi sts and what
do e s not e g exi stential quanti cation and the concomitant exi stential imp ort of its theorems
Hence there i s no reason why qualitative language s should not have metr ical mo dels Indee d ca
nonical metr ical mo dels ar i s e naturally when the axioms of a theory enforce s eemingly qualitative
constraints which imp os e order on the domain of individuals thi s i s illustrate d by the theory of
Allen s interval relations Allen and Haye s Ladkin s ee the end of s ection
and the spatial theory of Pratt and Scho op s ee s ection
Applications
Although thi s the s i s has fo cus e d on devi s ing spatial reasoning algor ithms that can b e e ectively
implemente d concrete applications have not b een cons idere d In the intro duction I obs erve d that
spatial information was of key imp ortance to many areas of computer science including such central
elds of AI as computer vi s ion and rob otics However logical reasoning with formal language s has
not b ecome an e stabli she d technique in any of the s e areas It i s therefore incumb ent up on thos e
developing QSR algor ithms to indicate how the s e might b e exploite d to solve problems in the more
pragmatic branche s of computer science which are concer ne d with pro ce ss ing spatial data
I shall rst cons ider the p oss ibility of applying QSR to rob otics The class ical approach to rob ot
control a rob ot i s to compute preci s e movement instructions to achieve a de s ire d goal Schwartz and
Shar ir Latomb e Whilst the s e instructions are pre dominantly metr ical the goal its elf
will typically corre sp ond to a high level qualitative pre scr iption of an action e g Put the b ox into
the skip Computing the metr ical instructions to achieve thi s goal can b e s een as a generali sation
of the problem of nding a spatial region sati sfying given qualitative spatial constraints which was
mentione d at the end of the last s ection But in the cas e of a rob otic goal the constraints may not
b e purely spatial and one must generate a spatio temp oral movement path rather than s imply a
spatial region One approach to thi s problem i s to translate constraints into a numer ical form and
then us e purely numer ical constraint solving technique s Schwartz and Shar ir Ar non
Thi s can only b e done e ectively for f airly s imple motions so where more complex motions are
require d planning technique s are often us e d to nd a s equence of s impler subgoals which achieve s
the de s ire d ultimate goal Lozano P erez Schwartz Shar ir and Hop croft del Pobil and
Ser na
Computing motion plans i s p erhaps the asp ect of rob otics that i s most likely to b ene t f rom
QSR technique s Given a qualitative repre s entation of initial and goal state s and a background
theory of p oss ible state change s a qualitative movement plan can in pr inciple b e compute d by
ab ductive inference Eshghi Shanahan Denecker Mi ss iaen and Bruyno oghe
Spatial concepts will play a very s igni cant role in b oth the state de scr iptions and the background
theory However adequate sp eci cation of rob ot state s and goals will also require concepts for
de scr ibing temp oral relationships mater ial prop ertie s and p erhaps abstract entitie s such as actions
Thus the reasoning problem i s f ar f rom purely spatial One might hop e to b e able to solve the
problem in a much more general theory of phys ical s ituations and pro ce ss e s as envi sage d in
CHAPTER FURTHER WORK AND CONCLUSIONS
s ection However it i s doubtful whether a su ciently general theory which i s also tractable
can b e develop e d in the near future In order to make us e of purely spatial inference mechani sms
one would have to f actor out spatial asp ects of the reasoning problem and show how the s e can b e
handle d as a mo dular comp onent of motion planning computations In my view thi s i s a much
more reali stic approach
Applications to reasoning ab out phys ical systems f ace many of the same problems as ar i s e in
rob otics In f act a rob ot can b e s een as a rather s imple example of a phys ical system with
a limite d numb er of degree s of f ree dom As note d ab ove s ection adequate theor ie s of
phys ical pro ce ss e s will probably nee d to incorp orate a very r ich conceptual vo cabulary Hence if
qualitative spatial inference s are to b e exploite d the nee d for mo dular i sation of reasoning problems
i s even more acute
A task which i s part of rob ot motion planning but i s also us eful for many other applications
e g route nding aids for motor vehicle dr ivers i s navigation Here we are not concer ne d with
the detaile d mechanics of movement but with somewhat more abstract problems such as nding
a viable path b etween two spatial lo cations for thi s purp os e the moving ob ject can normally b e
cons idere d to b e a p oint rather than an extende d b o dy and the require d path can b e repre s ente d by
a line rather than a s equence of complex movements Navigation problems are more purely spatial
than rob otic automation and cons equently spatial reasoning technique s are eas ier to apply A
numb er of concrete prop osals have b een made for the us e of qualitative repre s entations in automate d
navigation systems Kuip ers and Levitt Schlie der
A very promi s ing application for QSR i s to GIS which are increas ingly in demand as a to ol
for bus ine ss planning and land management The nee d for qualitative spatial query language s
to interact with the s e systems i s clear Egenhofer and Franzosa Egenhofer and Herr ing
Egenhofer and Al Taha Clementini et al Egenhofer and Mark High
level quer ie s of a na ve GIS us er corre sp ond to natural language que stions and the s e typically
involve qualitative concepts In s ection b elow I shall de scr ib e a prototyp e GIS that exploits
top ological reasoning
Interpreting query language s i s a sp ecial cas e of the more general problem of interpreting
spatial expre ss ions o ccurr ing in natural language which tend to b e pre dominantly qualitative
rather than quantitative cons ider prep os itions such as in on and through Vieu But
in applying QSR to natural language one f ace s the problem that spatial expre ss ions are enme she d
in an unformali s e d and mass ively complex conceptual structure By contrast the limite d spatial
vo cabulary employe d in vi sual computer programming language s i s much more amenable to formal
de scr iption and a numb er of recent works have us e d qualitative repre s entations to sp ecify the syntax
3
and s emantics of vi sual programming language s such as Pictor ial Janus Haarslev Go o day
and Cohn Go o day and Cohn
Another branch of AI which may b e well suite d to exploit QSR technique s i s computer vi s ion
3
The s e are language s in which programs are create d by e diting picture s within a graphical environment Program
execution can also be visualis e d by means of animations of the s e graphical repre s entations This is intende d to
facilitate debugging and understanding of how a program works
CHAPTER FURTHER WORK AND CONCLUSIONS
A computer vi s ion system typically employs a f airly long s er ie s of transformation pro ce dure s
culminating in a geometr ical mo del of ob jects in the scene Within thi s kind of architecture
it i s very easy to ins ert a pro ce dure which exploits spatial reasoning Indee d the us e of s emantic
technique s b oth for image s egmentation and ob ject recognition has long b een recogni s e d Winston
Qualitative reasoning bas e d on a s et of or ientation relations has b een succe ssfully applie d
to the analys i s of tra c ow f rom video image s Fer nyhough Cohn and Hogg Fer nyhough
Cohn and Hogg
In all areas involving spatial information it i s easy to give hand waving accounts of how QSR
can b e us e d to great advantage However the obstacle s to attaining practical re sults cannot b e
overe stimate d The re sults rep orte d in thi s the s i s indicate that achieving e ective reasoning even
with a very limite d vo cabulary of spatial concepts may require complex logical apparatus and
reasoning algor ithms sp eci cally tailore d to handling that particular range of concepts How the s e
limite d repre s entations can b e put to work on real problems i s f ar f rom obvious
It i s tempting to supp os e that once a su ciently expre ss ive repre s entation has b een devi s e d
the manner in which it can b e exploite d will b ecome obvious But without a hugely radical
advance in computer hardware or software technology it s eems likely that the con ict b etween
expre ss ive p ower and tractability will always b e a strong constraint on the us e of AI technique s
in computer systems Thus to nd a practical application of QSR one will have to show how
some concrete task can b e re duce d to manipulating a small numb er of spatial concepts or at least
how the role of di erent typ e s of spatial information in carrying out thi s task can b e i solate d and
handle d in a mo dular f ashion Thi s problem i s e sp ecially acute if one attempts to work within an
architecture in which all information and reasoning i s handle d by means of a purely qualitative
repre s entation one cannot then rely on any of the well understo o d mechani sms for quantitative
data manipulation that have b een develop e d over the years In my opinion the interf ace with
quantitative information di scuss e d in the last s ection i s the key to op ening up the path towards real
applications Emb e dding qualitative reasoning mo dule s within a more conventional architecture
enable s one to explore the strengths of us ing qualitative repre s entations without exp os ing all the ir
4
weakne ss e s
As my main re sults are ab out reasoning with top ological relations and to a much le ss er extent
convexity I ought to sugge st applications for thi s limite d form of spatial reasoning Top ological
relations are fundamental and p ervas ive in all spatial information so one might exp ect the us eful
ne ss of top ological reasoning to b e equally general But what sp eci c computational tasks can b e
re duce d to top ological reasoning
I have obs erve d that in many p otential application areas adequate qualitative de scr iption of
tasks require s not only non top ological concepts but also many non spatial concepts In such cas e s
a mo dular analys i s of relevant reasoning capabilitie s will b e nece ssary in order to i solate us eful
top ological inference pro ce dure s and thi s i s a re s earch topic in its elf However I b elieve that
s igni cant s emantic constraints relevant to ob ject recognition can b e sp eci e d in terms of purely
4
Fernyhough provide s a go o d example of what can be achieve d using this kind of architecture
CHAPTER FURTHER WORK AND CONCLUSIONS
top ological conditions and thi s may well lead to practical us e s within the eld of computer vi s ion
The application for which purely top ological reasoning has the most obvious us e s i s GIS It easy
to envi sage s ituations in which a GIS us er wants to p os e a query that i s e ss entially top ological in
nature For example in s iting a f actory one might wi sh to nd an area of undevelop e d land which
i s adjacent i e exter nally connecte d to a water source such as a lake and i s part of a particular
urban di str ict What i s not so obvious i s how s igni cant the s e top ological quer ie s are to the overall
functionality of a GIS which typically provide s acce ss to a vast amount of metr ical information
Top ological Inference in a GIS Prototyp e
I shall conclude the di scuss ion of applications with a de scr iption of a prototyp e spatial AI system
b e ing develop e d as part of EPSRC pro ject GR K on Logical Theor ie s and Deci s ion Pro ce d
ure s for Reasoning ab out Phys ical Systems Thi s incorp orate s the O n top ological reasoning
algor ithm bas e d on my I enco ding which was de scr ib e d in s ection program co de i s given in
app endix C The system maintains a databas e of geographical information in the form of geo
metr ical p olygon data and also handle s qualitative data in the form of top ological relations b etween
name d regions Some of the s e name d regions are identi e d directly with p olygons in the geomet
r ical databas e whereas for others the geometry i s not preci s ely known but only constraine d by the
qualitative top ological relations The top ological relationships determine d by the the quantitative
geometr ical data can also b e rapidly compute d and acce ss e d by the top ological reasoning mech
ani sm allowing quer ie s to b e addre ss e d to the combine d qualitative and quantitative databas e
Thi s capability i s as f ar as we know not available in any other system Work i s also underway to
demonstrate the us e of top ological reasoning in the control of arti cial agents op erating in a virtual
world constitute d by geographical data
Figure shows a screen dump of the current prototyp e system Most of the co de i s wr itten
in SICStus Prolog but a Tcl Tk sub pro ce ss i s us e d to create the GUI The window at the top
left shows a s imple cartographical di splay whos e geometry i s determine d by a databas e giving the
co ordinate s and terrain typ e of a numb er of tr iangular regions Thi s data i s shown in the b ottom
left window The top r ight window pre s ents a databas e of qualitative relations b etween regions
In the middle on the r ight i s the Prolog top level query window All functions of the system can
b e acce ss e d by typing commands and quer ie s at the Prolog prompt although common op erations
are more conveniently acce ss e d via the GUI The gure shows the Prolog interpreter b e ing us e d
for querying the qualitative databas e Such quer ie s are answere d by means of the spatial reasoning
algor ithm de scr ib e d in chapter which determine s whether a relation given as a query i s cons i stent
with incons i stent with or a nece ssary cons equence of the databas e The b ottom r ight window i s
one of a numb er of information screens which can b e di splaye d via the system s help function
CHAPTER FURTHER WORK AND CONCLUSIONS
Figure A prototyp e geographical information system
Conclus ion
I shall conclude thi s the s i s by making some general remarks ab out the prosp ects for automate d
reasoning bas e d on ins ights I gaine d dur ing my re s earch
When I starte d work on spatial reasoning I was under the na ve impre ss ion that st order logic
or something like it could provide an ideal formali sm for knowle dge repre s entation and reasoning
in thi s and almost any conceptual domain Although I was aware of the theoretical undecidability
and intractability of st order reasoning I did not reali s e the s er iousne ss of the di cultie s that
the s e prop ertie s p os e for automate d reasoning I imagine d that with a p owerful enough computer
it would b e feas ible to compute entailments b etween relations as determine d by a s imple axiomatic
theory However after attempting to compute RCC inference s us ing the Otter theorem prover
McCune it so on b ecame apparent that thi s i s completely impractical Even s eemingly s imple
de ductions would very often exhaust the available computational re source s
My exp er ience of theorem proving probably has much in common with that of many others who
have entere d thi s eld It i s now widely recogni s e d that e ective automate d reasoning with logical
repre s entations cannot b e achieve d by general purp os e pro of systems but require s the construction
of sp eciali s e d reasoning algor ithms Even so it s eems to me surpr i s ing that a tractable pro of
CHAPTER FURTHER WORK AND CONCLUSIONS
pro ce dure for spatial relations should b e so f ar remove d f rom one s intuitive picture of the problem
An intere sting que stion i s whether thi s i s typical of e ective solutions to reasoning problems
That thi s may b e so was sugge ste d by Alan Robinson who having di scovere d the very
p owerful but extremely unnatural hyper resolution inference rule prop os e d that there may b e a
di erence in kind b etween the style of reasoning intelligible to humans and the typ e of reasoning
mechani sms which can b e e ciently implemente d in computer programs Thi s i s also evidence d by
the pro digious numb er crunching abilitie s but p o or conversational skills of computers Though it
do e s not give any reason for the divergence b etween style s of reasoning of humans and computers
Robinson s prop osal do e s s eem to concur with much of what has b een di scovere d in the study of
automate d reasoning
From another p oint of view the us e of mo dal and intuitioni stic logics for spatial reasoning may
not b e so p ervers e as it rst s eems It may just b e that thi s us e of the s e logics i s unf amiliar
Although mo dal logics were or iginally intende d to capture prop os itional mo di ers and intuition
i stic logic to sp ecify an ontologically pars imonious form of mathematical reasoning the structural
manipulations emb o die d in the inference rule s of the s e logics are of a very general nature Hence
it i s only to b e exp ecte d that alter native interpretations can b e given
The succe ss of the I and S enco dings of spatial relations may also she d some light on why
st order reasoning i s so intractable In st order pre dicate logic the sub structure of atomic
prop os itions has no logical content By thi s I mean that although we may analys e an atomic
prop os ition in terms of a relation b etween a numb er of functional terms the s e comp onents are ar
bitrary having no sp ecial logical prop ertie s except insof ar as they may b e constraine d by axioms
Hence the meanings of the s e symb ols are not capture d directly by rule s of inference but only
indirectly through axioms taking part in inference Moreover the s e axioms often take the form of
quite complex quanti cational formulae It i s the s e theoretical formulae that make st order reas
oning so computationally intens ive even when employe d to compute s eemly obvious cons equence s
of s imple f actual information
As an exception to thi s treatment of the meaning of pre dicate s the meaning of the equality
relation i s usually sp eci e d in terms of inference rule s rather than axioms One could treat equality
5
but it i s eas ier to capture the logical prop ertie s as a non logical symb ol constraine d by axioms
of by means of inference rule s than in axioms Axiomatic treatment of equality adds a large
numb er of formulae to the sp eci cation of a st order theory which greatly increas e s the s earch
space that an automate d theorem prover has to deal with Although adding inference rule s for
equality also increas e s the s earch space it has b een found that thi s metho d i s in most cas e s much
more conducive to automate d reasoning Wos Du y When one reasons with a theory
of equality in axiomatic form a pro of may involve a cons iderable amount of reasoning ab out the
5
The equality relation can be characteris e d e ither by the nd order axiom x y x y
or by a s et st order axioms ensur ing that is an equivalence relation and specifying all p ossible ways
that an equality justi e s substitution into the arguments of relations and function The substitution axioms
z w x y z x w z y w where is a relation symb ol of the theory and take the forms x y
x y z w x y z x w z y w where is a function symb ol z and w repre s ent p ossibly empty
s equence s of var iable s lling any additional argument place s of and
CHAPTER FURTHER WORK AND CONCLUSIONS
concept of equality its elf as well as reasoning ab out other concepts whereas if equality i s handle d
by an inference rule such as paramo dulation then the theory of equality i s encapsulate d within
thi s rule so the ramifying e ect on the s earch space i s greatly re duce d
Asso ciating inferential meaning to other pre dicate and function symb ols within a prop os ition can
obviate the nee d for auxiliary axioms and the ndings concer ning equality sugge st that thi s may b e
extremely advantageous for automate d reasoning One example of thi s i s the us e of sorte d logic s ee
e g Cohn and s ection of thi s the s i s where reasoning concer ning the sorts of pre dicate s
6
and functions i s built into inference rule s Another example i s the us e of demo dulation rule s s ee
e g Du y to rewr ite and s implify terms in accordance with known identitie s The s e
rule s must b e tailore d to the sp eci c prop ertie s of a given theory but they have prove d extremely
e ective in many domains Wos A typical us e of demo dulation i s to re duce Bo olean and
other algebraic terms to normal form to avoid proliferation of equivalent but syntactically di stinct
terms Algebraic terms are very common in mathematical theor ie s but generally do not play a
ma jor role in theor ie s of commons ens e concepts However the analys i s of RCC relations in terms
of inter ior algebraic equations s ee s ection shows that an algebraic sp eci cation of such concepts
may b e p oss ible even where it i s not imme diately obvious It i s thi s analys i s of the RCC relations
that enable s the ir meanings to b e capture d by means of inference rule s rather than axioms
Algebraic analys i s may exp os e sub structure in the meanings of relational concepts but in
its elf thi s i s probably not helpful to automate d reasoning If we s imply axiomati s e d the algebraic
op erators the re sulting theory might b e even more complex than a direct axiomatisation of the
concepts To gain computational advantage we nee d a pro of system that take s direct account
of the inferential s igni cance of the algebraic op erators and hence encapsulate s the meaning of
the concepts within its inference rule s It i s well known that class ical prop os itional logic can b e
interprete d as a Bo olean algebra and that mo dal op erators can also b e identi e d with algebraic
op erators Hence it should not b e surpr i s ing that pro of systems de s igne d to compute inference s in
the s e prop os itional language s can also b e exploite d to reason ab out algebraic equations However
the detaile d working out of how thi s can b e done i s probably the most novel asp ect of the work in
thi s the s i s
Becaus e they excee d the expre ss ive p ower of s imple Bo olean algebra but avoid the intractability
of st order logic I b elieve that decidable constraint language s bas e d on Bo olean algebras with
additional op erators are very well suite d to computational manipulation The s e encompass the
mo dal algebras which I explore d in chapter and also relation algebras di scuss e d in s ection
As well as providing a vehicle for e ective automation of spatial reasoning repre s entations bas e d
on algebraic structure s of thi s kind may b e us eful in many other areas of knowle dge repre s entation
6
Re solution bas ed inference rule s are particularly well suite d to incorp orating sortal reasoning
App endix A
Elementary Geometry
A Tarski s Axiom System
Tarski has given the following axiomatisation of elementary geometry in terms of the two
pr imitive s betweenness and equidistance Here B x y z means that p oint y i s b etween p oints x
and z Thi s relation i s taken as true if z i s equal to e ither x or z xy z w means that the di stance
b etween p oints x and y i s equal to the di stance b etween p oints y and z
B Identity Axiom for Betweenness
xy B x y x x y
B Transitivity Axiom for Betweenness
xy z u B x y u B y z u B x y z
B Connectivity Axiom for Betweenness
xy z u B x y z B x y u x y B x z u B x u z
B Reflexivity Axiom for Equidistance
xy xy y x
B Identity Axiom for Equidistance
xy z xy z z x y
B Transitivity Axiom for Equidistance
xy z uv w xy z u xy v w z u v w
B Pasch s Axiom
txy z u v B x t u B y u z B x v y B z t v
B Euclid s Axiom
txy z u v w B x u t B y u z x y
B x z v B x y w B v t w
B Five Segment Axiom
0 0 0 0 0 0 0 0 0 0 0 0
xx y y z z uu xy x y y z y z xu x u y u y u
0 0 0 0 0
B x y z B x y z x y z u z u
B Axiom of Segment Construction
xy uv z B x y z y z uv
B Lower Dimension Axiom
xy z B x y z B y z x B z x y
APPENDIX A ELEMENTARY GEOMETRY
B Upper Dimension Axiom
xy z uv xy xv y u y v z u z v u v
B x y z B y z x B z x y
B Elementary Continuity Axioms
Al l sentences of the form
v w z xy B z x y u xy B x u y
where stands for any formula in which the variables x y w but neither y nor z nor u occur free
and similarly for with x and y interchanged
B Weak Continuity Axiom
0 0 0 0 0
xy z x z u y ux ux uz uz B u x z B x y z
0 0 0 0
uy uy B x y z
A Pr imitive Geometr ical Concepts
The s equence of de nitions given b elow shows how starting f rom the fundamental ter nary relation
xy y z which i s true when two p oints x and z are equidi stant f rom a third p oint y many other
s imple geometr ical relations can b e intro duce d In the s e de nitions the juxtap os ition xy of two
var iable s x and y i s intende d to refer to the di stance b etween the s e two p oints Thus xy y z i s a
pre dicate which holds in cas e y i s clos er to x than to z The other relations are B x y z y i s
b etween x and z including the cas e where y i s identical with e ither x or z L x y z x y and
z are col linear and M x y z y i s the mid p oint b etween x and z
The relation xy y z i s of great geometr ical s igni cance as it relate s the centre p oint y of
a sphere to any pair of surf ace p oints x and z For a dimens ional gure the truth of thi s
relation for any three p oints can b e determine d by means of a compass The relations B x y z
and xy z w are taken as pr imitive s in Tarski s elementary geometry A pro of that the quater nary
relation xy z w i s de nable in terms of the ter nary xy y z i s or iginally due to Pier i The
following de nitions showing how thi s can b e done together with further di scuss ion of pr imitive
notions in geometry can b e found in Royden
xy y z w y w w z u xu uy uy y w
def
B x y z w w x xy w z z y w y
def
L x y z B x y z B y x z B x z y
def
M x y z w L w x y xy y w w x w z
def
w x y z u v M w u y M x u v v y y z def
App endix B
An Alter native Pro of of MEconv
In thi s app endix I give an alter native pro of of the theorem BEconv which was demonstrate d in
s ection The statement of BEconv i s as follows
Convexity of Di sjunctive Mo dal Algebraic Entailments MEconv
U U U U j
n m
MA
L
i
U for some i f ng U U j
i m
MA
L
The alter native pro of relie s only on the additivity of the mo dal op erator and do e s not require
that its algebraic prop ertie s b e sp eci able just in terms of equations The bas i s of the pro of i s
that given counter mo dels sati sfying the premi ss e s of the s equent and individually f als ifying each
di sjunct of its conclus ion the additive nature of the op erator allows one to construct a counter
mo del sati sfying the premi ss e s and f als ifying the conclus ion as a whole
Pro of of MEconv Let S b e the s et of s et constants o ccurr ing in a di sjunctive
entailment DE of the form given in the theorem Supp os e none of the di sjuncts on
the r h s i s entaile d by the equations on the l h s Thi s means that for each di sjunct
U there i s an ass ignment hS U m i sati sfying all the equations U
i i i i i j
but such that U We can assume without loss of generality that the univers e s
i i
U in each of the ass ignments are di sjoint From the s e ass ignments we can construct a
i
new ass ignment again sati sfying all the equations U and such that U
j i
for each
i
S S
Let hS U m i where U U i s de ne d by
i i
i i
S
for each constant S and m i s de ne d by m X m X U for every s et
i i
i
X U in each cas e the subscr ipt i range s f rom to n
We note that b ecaus e we are dealing with mo dal algebras each of the functions m
i
must b e additive m X Y m X m Y Thi s means that m i s also additive
i i i
m X Y m X Y U m X U Y U
i i i i i
i i
APPENDIX B AN ALTERNATIVE PROOF OF MECONV
m Y U m X m Y m X U m X U m Y U
i i i i i i i i
i i i
S
I now show that for any term i e the denotation of any term
i
i
under i s just the union of its denotations under the ass ignments If i s a constant
i
thi s i s ensure d directly by the sp eci cation of so we can prove it inductively for all
terms by showing that if it holds for any terms and it must also hold for the terms
and For and we have
U U U
i i i i i
i i i i
i i i
i i i
S
Whence must hold s ince
i
i
The pro of for the cas e of the mo dal op erator i s rather more involve d Since we are
S S
and s ince m we have m m assuming
i i
i i
S
m If we now replace m by its de nition in terms i s additive thi s equals
i
i
S S
of the functions m we get the expre ss ion m U where i and j b oth
i j i j
i j
range f rom to n Notice that i s always a subs et of U so b ecaus e the U s are
i i i
di sjoint U must equal if i j and otherwi s e Thi s means that the
i j i
S S
expre ss ion can b e re duce d to m which i s equivalent to
i i i
i i
S
Since for any term and the range s of the ass ignments are
i i
i
di sjoint it follows that an equation i s sati s e d by if and only if it i s sati s e d by all
of the s Thi s ensure s that sati s e s all the f rame equations of the logic L It also
i
means that must sati sfy all the equations on the l h s of the DE and none of the
equations in the di sjunction on the r h s of DE
Hence the constructe d ass ignment demonstrate s that if none of the di sjuncts on
the r h s of DE i s individually entaile d by the equations on the l h s the ir di sjunction
cannot b e entaile d So the class of entailments of mo dal algebraic equations of the form
of DE i s convex
App endix C
Prolog Co de
C Generating all Conjunctions of RCC Relations
The following program generate s all logically di stinct relations which can b e sp eci e d as a con
junction of RCC relations and the ir negations The co de include s do cumentation of how it works
Further explanation can b e found in s ection in the main the s i s and also in the s ection following
the program li sting where I pre s ent and explain the program s output
rcc cons pl
This program generates the complete set of logically distinct
relations which can be specified as conjunctions of ve and
ve literals taken from the RCC relation set
The set can be generated with or without non null constraints
on the regions involved
For the sake of generality non null constraints are represented
by adding the relations x and y to the set of RCC relations
x is true just in case the st argument of the relation is null
and y if the second argument is null
top level calls
Generate all combinations of RCC relations and null relations
There are including the impossible relation
generate rcc cons
setof X rcc con X complete X Set
showlist Set
length Set L write length L
APPENDIX C PROLOG CODE
Generate all combinations of RCC relations for which the
arguments are non null
There are including the impossible relation
generate rcc nn cons
setof X rcc nncon X complete X Set
showlist Set
length Set L write length L
Find the most specific relations specifiable between non null regions
excluding the impossible relation
This generates the RCC relations
generate nnrcc base rels
setof X rcc nncon X complete X Set
setof B member B Set member C Set proper subset B C Base
showlist Base
length Base LB write base length LB
Subsidiary Predicates
rcc con CONJ pick conjunction dc dr p pi ntpp ntppi eq x y CONJ
CONJ impossible
rcc nncon not x not y Rest
pick conjunction dc dr p pi ntpp ntppi eq Rest
rcc nncon impossible
A set of relations is complete iff it is closed under implications
complete Set member R Set implies R S
member S Set
member R Set member R Set
implies and R R S
member S Set
Implications holding between rcc relations implies dc dr
APPENDIX C PROLOG CODE
implies ntpp p
implies ntppi pi
implies eq p
implies eq pi
implies x ntpp
implies y ntppi
implies x dc
implies y dc
implies and p pi eq
implies and dr p x
implies and dr pi y
implies and ntpp ntppi x
implies and ntpp ntppi y
implies and ntpp eq x
implies and ntpp eq y
implies and ntppi eq x
implies and ntppi eq y
implies not R not S implies S R S and
implies and R not S not T implies and R T S
Additional simple predicates
proper subset X Y X Y member E X member E Y
write a list one element per line
showlist
showlist H T write H nl showlist T
pick a conjunction of ve or ve literals from a list
pick conjunction
pick conjunction T PT pick conjunction T PT
pick conjunction H T H PT pick conjunction T PT
pick conjunction H T not H PT pick conjunction T PT
minimise can be used to remove redundant rels from RCC conjunctions
this is not actually used by the predicates defined above
APPENDIX C PROLOG CODE
Minimising a set is removing all implied relations
minimise Set M extract R Set Rest
member S Rest
implies S R
minimise Rest M
minimise Set M extract R Set Rest
extract S Rest Rest
member T Rest
implies and S T R
minimise Rest M
minimise S S
extract an element from a list non deterministically
extract X List Rest append Front X End List
append Front End Rest
C Conjunctions of the RCC Relations and the ir Negations
Here i s the s et of logically di stinct conjunctions of the RCC relations and the ir negations
generate d by the program given in the last s ection The relations are given in the form of a li st of
conjuncts with negate d conjuncts given as not R Relations are denote d by the ir usual initials
but in small letters b ecaus e of the syntax of Prolog The empty li st corre sp onds to the universal
holding b etween any two regions Any conjunction containing a literal and its negation i s equivalent
to the impossible relation
The f act that one or other of the regions involve d in a relation i s null i s sp eci e d by the sp ecial
ps eudo relations x and y meaning re sp ectively that the st or nd argument i s null In the RCC
theory all regions are non null Thus only thos e conjunctions including the conjuncts not x
and not y corre sp ond to legitimate RCC relations There are such relations including the
impossible relation which implicitly include s b oth non null constraints
The conjunction s ets generate d by the program are clos e d under implication and thi s ensure s
that they are all genuinely logically di stinct It also means that there i s a lot of re dundancy in the
re sulting sp eci cation of the relations For instance every conjunction which has dc as a conjunct
also include s the weaker relation dr as a conjunct Thi s re dundancy could b e eliminate d by p ost
pro ce ss ing the s ets to remove implie d conjuncts however there i s not always a unique way to
s implify a conjunction so I have not done thi s
generate rcc cons
impossible dc dr
APPENDIX C PROLOG CODE
dc dr p ntpp x
dc dr p pi ntpp ntppi eq x y
dc dr p not pi ntpp not ntppi not eq x not y
dc dr pi ntppi y
dc dr pi ntppi not eq y
dc dr not eq
dc dr not p pi not ntpp ntppi not eq not x y
dc dr not p not ntpp not eq not x
dc dr not p not pi not ntpp not ntppi not eq not x not y
dc dr not pi not ntppi not eq not y
dr
dr not eq
dr not p not ntpp not eq not x
dr not p not pi not ntpp not ntppi not eq not x not y
dr not pi not ntppi not eq not y
p
p ntpp
p pi eq
p pi eq not x
p pi eq not x not y
p pi eq not y
p pi not ntpp eq not x
p pi not ntpp eq not x not y
p pi not ntpp not ntppi eq not x not y
p pi not ntppi eq not x not y
p pi not ntppi eq not y
p not ntpp not ntppi not x not y
p not ntpp not x
p not ntpp not x not y
p not ntppi not x not y
p not ntppi not y
p not pi ntpp not ntppi not eq not x not y
p not pi ntpp not ntppi not eq not y
p not pi not ntpp not ntppi not eq not x not y
p not pi not ntppi not eq not x not y
p not pi not ntppi not eq not y
p not x
p not x not y
p not y pi
APPENDIX C PROLOG CODE
pi ntppi
pi ntppi not eq
pi ntppi not eq not x
pi ntppi not eq not x not y
pi ntppi not eq not y
pi not eq
pi not eq not x
pi not eq not x not y
pi not eq not y
pi not ntpp ntppi not eq not x
pi not ntpp ntppi not eq not x not y
pi not ntpp not eq not x
pi not ntpp not eq not x not y
pi not ntpp not ntppi not eq not x not y
pi not ntpp not ntppi not x not y
pi not ntpp not x
pi not ntpp not x not y
pi not ntppi not eq not x not y
pi not ntppi not eq not y
pi not ntppi not x not y
pi not ntppi not y
pi not x
pi not x not y
pi not y
not dc dr not p not pi not ntpp not ntppi not eq not x not y
not dc p pi eq not x not y
not dc p pi not ntpp eq not x not y
not dc p pi not ntpp not ntppi eq not x not y
not dc p pi not ntppi eq not x not y
not dc p not ntpp not ntppi not x not y
not dc p not ntpp not x not y
not dc p not ntppi not x not y
not dc p not pi ntpp not ntppi not eq not x not y
not dc p not pi not ntpp not ntppi not eq not x not y
not dc p not pi not ntppi not eq not x not y
not dc p not x not y
not dc pi ntppi not eq not x not y
not dc pi not eq not x not y
not dc pi not ntpp ntppi not eq not x not y
not dc pi not ntpp not eq not x not y
APPENDIX C PROLOG CODE
not dc pi not ntpp not ntppi not eq not x not y
not dc pi not ntpp not ntppi not x not y
not dc pi not ntpp not x not y
not dc pi not ntppi not eq not x not y
not dc pi not ntppi not x not y
not dc pi not x not y
not dc not dr p pi eq not x not y
not dc not dr p pi not ntpp eq not x not y
not dc not dr p pi not ntpp not ntppi eq not x not y
not dc not dr p pi not ntppi eq not x not y
not dc not dr p not ntpp not ntppi not x not y
not dc not dr p not ntpp not x not y
not dc not dr p not ntppi not x not y
not dc not dr p not pi ntpp not ntppi not eq not x not y
not dc not dr p not pi not ntpp not ntppi not eq not x not y
not dc not dr p not pi not ntppi not eq not x not y
not dc not dr p not x not y
not dc not dr pi ntppi not eq not x not y
not dc not dr pi not eq not x not y
not dc not dr pi not ntpp ntppi not eq not x not y
not dc not dr pi not ntpp not eq not x not y
not dc not dr pi not ntpp not ntppi not eq not x not y
not dc not dr pi not ntpp not ntppi not x not y
not dc not dr pi not ntpp not x not y
not dc not dr pi not ntppi not eq not x not y
not dc not dr pi not ntppi not x not y
not dc not dr pi not x not y
not dc not dr not eq not x not y
not dc not dr not ntpp not eq not x not y
not dc not dr not ntpp not ntppi not eq not x not y
not dc not dr not ntpp not ntppi not x not y
not dc not dr not ntpp not x not y
not dc not dr not ntppi not eq not x not y
not dc not dr not ntppi not x not y
not dc not dr not p pi not ntpp ntppi not eq not x not y
not dc not dr not p pi not ntpp not eq not x not y
not dc not dr not p pi not ntpp not ntppi not eq not x not y
not dc not dr not p not ntpp not eq not x not y
not dc not dr not p not ntpp not ntppi not eq not x not y
not dc not dr not p not pi not ntpp not ntppi not eq not x not y
APPENDIX C PROLOG CODE
not dc not dr not pi not ntpp not ntppi not eq not x not y
not dc not dr not pi not ntppi not eq not x not y
not dc not dr not x not y
not dc not eq not x not y
not dc not ntpp not eq not x not y
not dc not ntpp not ntppi not eq not x not y
not dc not ntpp not ntppi not x not y
not dc not ntpp not x not y
not dc not ntppi not eq not x not y
not dc not ntppi not x not y
not dc not p pi not ntpp ntppi not eq not x not y
not dc not p pi not ntpp not eq not x not y
not dc not p pi not ntpp not ntppi not eq not x not y
not dc not p not ntpp not eq not x not y
not dc not p not ntpp not ntppi not eq not x not y
not dc not p not pi not ntpp not ntppi not eq not x not y
not dc not pi not ntpp not ntppi not eq not x not y
not dc not pi not ntppi not eq not x not y
not dc not x not y
not eq
not eq not x
not eq not x not y
not eq not y
not ntpp not eq not x
not ntpp not eq not x not y
not ntpp not ntppi not eq not x not y
not ntpp not ntppi not x not y
not ntpp not x
not ntpp not x not y
not ntppi not eq not x not y
not ntppi not eq not y
not ntppi not x not y
not ntppi not y
not p pi not ntpp ntppi not eq not x
not p pi not ntpp ntppi not eq not x not y
not p pi not ntpp not eq not x
not p pi not ntpp not eq not x not y
not p pi not ntpp not ntppi not eq not x not y
not p not ntpp not eq not x
not p not ntpp not eq not x not y
APPENDIX C PROLOG CODE
not p not ntpp not ntppi not eq not x not y
not p not pi not ntpp not ntppi not eq not x not y
not pi not ntpp not ntppi not eq not x not y
not pi not ntppi not eq not x not y
not pi not ntppi not eq not y
not x
not x not y
not y
length
yes
APPENDIX C PROLOG CODE
C An I Theorem Prover for Spatial Sequents
The following co de implements an intuitioni stic theorem prover bas e d on a Gentzen s equent calcu
lus The prover i s optimi s e d to p erform b etter with the class of s equents generate d by the enco ding
of RCC reasoning in I Thi s means that the prover i s not complete for arbitrary I s equents
The main s impli cation of the calculus i s that the rule for eliminating implications on the left of
the s equent i s replace d by mo dus p onens Another var iant of mo dus p onens in adde d to handle
the cas e of an implication with a conjunction as its antece dent s ee s ection
Gentzen system for propositional intuitionistic logic
Restricted to give better performance on sets of spatial
constraint formulae as given in KR
set prooftrace to on to see trace or use prooftr
command below to toggle mode
dynamic prooftrace
prooftrace off
Output current goal if in tracing mode
entail Prems Conc prooftrace on
format trying to prove p p n Prems Conc
fail
SEQUENT RULES FOR I
Terminating conditions
entail Prems Conc member Conc Prems
entailtrace Proven conc is prem n
entail Prems member absurd Prems
entailtrace Proven absurd prem n
Simple sequent re writes
equiv
entail Prems equiv P Q
setadd P Prems P Prems
entail P Prems Q
APPENDIX C PROLOG CODE
setadd Q Prems Q Prems
entail Q Prems P
equiv
entail Prems Conc
extract equiv P Q Prems Rest
setadd if P Q if Q P Rest NewPrems
entail NewPrems Conc
and
entail Prems Conc
extract and P Q Prems Rest
setadd P Q Rest P Q Prems
entail P Q Prems Conc
if
entail Prems if P Q
setadd P Prems P Prems
entail P Prems Q
not
entail Prems not P
setadd P Prems P Prems
entail P Prems absurd
Conjunctive Splitting Rules deterministic
and
entail Prems and P Q
entail Prems P
entail Prems Q
or
entail Prems Conc
extract or P Q Prems Rest
setadd P Rest P Prems
entail P Prems Conc
setadd Q Rest Q Prems
entail Q Prems Conc
APPENDIX C PROLOG CODE
Pruning Rules
Not necessary for completeness but save a lot of search
Implications are redundant if you have the conclusion
entail Prems Conc
extract if Q Prems Rest
member Q Rest
entail Rest Conc
More such rules could be added for greater efficiency
Non deterministic Rules
Application of these rules reduces sequent to a logically
stronger form so must backtrack for completeness
rules xentail are not used for the spatial reasoner
but would be needed for complete intuitionistic reasoning
if
The if rule is not used for the spatial constraints
It is replaced by modus ponens and another similar rule
see below
disabled entail Prems Conc
extract if P Q Prems Rest
entail Rest P
setadd Q Rest Q Prems
entail Q Prems Conc
not
re write not X to if X absurd
entail Prems Conc extract not P Prems Rest
setadd if P absurd Rest NewPrems
entail NewPrems Conc
Using modus ponens for if is not complete for I in
general but it is complete for the topological constraints
if used together with the similar rule following
entail Prems Conc
APPENDIX C PROLOG CODE
extract if P Q Prems Rest
member P Rest
setadd Q Rest Q Prems
entail Q Prems Conc
Rule for constraint not and X Y on left
a Modus Ponens variant
entail Prems Conc
extract if and X Y Q Prems Rest
member X Rest
member Y Rest
setadd Q Rest Q Prems
entail Q Prems Conc
We still have non determinism for disjunctive conclsions
This could also be eliminated by adding more prunig rules
or
entail Prems or P Q
entail Prems P
entail Prems Q
Conclusion may also be given as singleton list
for compatibility with other sequent progs
entail Prems Conc entail Prems Conc
FAIL
If no rule applicable fail current entail goal
entail entailtrace Failed n
fail
alternative top level call for single premiss sequents
entails P Q entail P Q
APPENDIX C PROLOG CODE
Simple predicates used above
extract X List Rest X occurs in List remainder is Rest
the definition is now kept in prolog lib mylib pl
extract X List Rest append A X B List
append A B Rest
add an element to a set
setadd Elt Set Set member Elt Set
setadd Elt Set Elt Set
add two elts to a set
setadd Elt Elt SetIn SetOut
setadd Elt SetIn SetOut
setadd Elt SetOut SetOut
Tracing the prover
output with format if prooftrace is on
entailtrace Str Args prooftrace on
format Str Args
true
Toggle proof tracing
prooftr prooftrace on
retractall prooftrace
assert prooftrace off
write prooftrace off
retractall prooftrace
assert prooftrace on
write prooftrace on
Some example test problems
APPENDIX C PROLOG CODE
check emi entail or p not p
check dn entails p not not p
check dn entails not not p p
check test
entail not and c con a not and b con c
if c con b or not c not b
not and a con c if c con a
or not c not a
not and c con b
hard test Not so hard with restriced rules
entail not and a b not and b a
or not b not a not and b c
if c con b or not c not b
not and a c not and c a
or not c not a if b con b
if a con a if c con c
not and con b con a
APPENDIX C PROLOG CODE
C A Sp ecial Purp o s e O n Algor ithm for Spatial Sequents
The following co de implements the I reasoning algor ithm bas e d on the optimi s e d s equent calculus
rule s given in s ection As with the program given in app endix C thi s means that the pro of
system i s only complete for s equents ar i s ing f rom the I enco ding of RCC cons i stency problems
and not for arbitrary s equents of I For thi s re str icte d class of s equents it can b e shown that the
worst cas e run time i s of O n in the numb er formulae on the l h s of the s equent Thi s numb er
i s of the order of the numb er of RCC relations which are to b e te ste d for cons i stency however
checking cons i stency of n relations require s O n s eparate I s equents to b e te ste d Thus checking
cons i stency of n RCC relations require s O n time Thi s re sult applie s more generally to any
s et of relations which can b e repre s ente d as a conjunction of RCC relations and the ir negations
n top pl
A decision procedure for spatial entailments encoded into sequents
of the binary fragment of intuitionistic propositional logic
Declare dynamic predicates to store model and entailment constraints
The last argument is a status flag used to keep track of formulae
which are asserted temporarily in the course of testing an entailment
There are four kinds of model constraint
dynamic mcon or
dynamic mcon if
dynamic mcon nand
Atoms are stored as mcon AtomName Status
dynamic mcon atom
Entailment constraints are stored as econ Formula Status
dynamic econ
Three status flags are used
db formula is part of a consistent database encoding spatial facts
test formula is associated with a putative spatial fact whose
consistency is to be tested
pr formula is asserted temporarily while testing particular sequent
The flags ought to be further parameterised by some database id
ie we would have db id test id and pr id
Then we could use multiple databases
APPENDIX C PROLOG CODE
Predicates for adding to the database
add mcon Formula Status
Add a model constraint formula to the database
also add extra implications entailed by disjunctions
and add closure of all implications
All added formulae have status S
add mcon if X Y S
add imp and close if X Y S
add mcon or if X f Y S
add the entailed implication if X Y
add imp and close if X Y S
assert if new mcon or if X f Y S
add mcon or if X f if Y f S
add imp and close if X if Y f S
assert if new mcon or if X f if Y f S
add mcon A S atom A
assert if new mcon atom A S
add mcon list
add mcon list H T S add mcon H S
add mcon list T S
add econ list
add econ list H T S assert econ H S
add econ list T S
add imp and close Implication Status
Add an implication to the database together with all its consequences
Status flag S also added which allows temporary formulae to be removed
if already there do nothing
add imp and close if X Y mcon if X Y
if subsumed do nothing
add imp and close if X if Y f
APPENDIX C PROLOG CODE
mcon nand X Y
mcon nand Y X
mcon if X f
mcon if Y f
add if X if Y f and consequences
add imp and close if X if Y f S
sweep mcon if A X
assert if new mcon nand A Y S
sweep mcon if B Y
assert if new mcon nand X B S
assert mcon nand X Y S
add simple if X Y and consequences
add imp and close if X Y S
sweep mcon if Y Z
assert if new mcon if X Z S
sweep mcon if Z X
assert if new mcon if Z Y S
assert mcon if X Y S
prove Formula
This is true if Formula is a consequence of the model constraints
stored in the database
prove if X f
prune ors wrt if X f
assert if new mcon atom X pr
derive by modus ponens f
APPENDIX C PROLOG CODE
prove if X Y assert mcon atom X pr
derive by modus ponens Y
prove if X if Y f assert mcon atom X pr
assert mcon atom Y pr
derive by modus ponens f
prove or if X f if Y f
prune ors wrt if X f
prune ors wrt if Y f
assert mcon atom X pr
derive by modus ponens f
clean pr
assert if new mcon atom Y pr
derive by modus ponens f
prove or if X f Y
prune ors wrt if X f
prune ors wrt Y
assert mcon atom X pr
derive by modus ponens f
clean pr
derive by modus ponens Y
Specification of the PROOF RULES
Add all consequences of the pruning rule for disjunctions
prune ors wrt F
sweep mcon or X F add mcon X pr
sweep mcon or F X add mcon X pr
APPENDIX C PROLOG CODE
derive by modus ponens Conc
probably wont terminate as soon as Conc found
First sweep over all MP applications
sweep mcon atom A
mcon atom f stop if inconsistent
mcon atom Conc stop if proved
mcon if A B
assert if new mcon atom B pr
mcon nand A B
assert if new mcon if B f pr
mcon nand C A
assert if new mcon if C f pr
Could also subsume if X A clauses
But must replace them if using an incremental DB
Then test whether Conc or f has been derived
mcon atom Conc mcon atom f
Top level predicate for testing intuitionistic sequents
test sequent Prems Conc
clean
add mcon list Prems test
prove Conc
clean test
query database
Use check new cons wrt db to check consistency
Necessary if all query Mcons also entailed by db Mcons
and all query Econs entailed by db Econs
This version only checks consistency
query db Rel Ans
rcc i Rel Mcons Econs
APPENDIX C PROLOG CODE
check new cons wrt db Mcons Econs
Ans consistent
Ans inconsistent
clean test clean pr
check new cons wrt db Mcons Econs
add mcon list Mcons test
add econ list Econs test
all econs consistent
all econs consistent
econ F
clean pr
prove F
Time random queries wrt a fixed database
time random queries No Regs T AvT
statistics runtime
do n random queries Regs
statistics runtime T
AvT is T No
do n random queries
do n random queries N RegNo
random rel RegNo RR
gen out testing rel RR
query db RR
NextN is N
do n random queries NextN RegNo
generate random db
generate random db RegNo RelNo RelsTried
random rel RegNo RR
APPENDIX C PROLOG CODE
gen out testing rel RR
rcc i RR Mcons Econs
add if consistent Mcons Econs
MoreRels is RelNo
gen out Consistent more rels to add MoreRels
generate random db RegNo MoreRels MoreTries
RelsTried is MoreTries
gen out Inconsistent wrt DB
generate random db RegNo RelNo MoreTries
RelsTried is MoreTries
add if consistent Mcons Econs
check new cons wrt db Mcons Econs
clean pr
change status test db
add if consistent
clean pr clean test fail
time random db Regs Rels Tried T
clean
statistics runtime
generate random db Regs Rels Tried
statistics runtime T
nl write regions Regs
nl write relations Rels
nl write tried Tried
nl write time T nl ttyflush
Predicates for adding removing and changing status of formulae
in the database
Add mcon unless already present
Note that the existing fact need not have same status
APPENDIX C PROLOG CODE
assert if new mcon if X Y S
mcon if X Y assert mcon if X Y S
assert if new mcon nand X Y S
mcon nand X Y assert mcon nand X Y S
assert if new mcon or X Y S
mcon or X Y assert mcon or X Y S
assert if new mcon atom X S
mcon atom X assert mcon atom X S
clean S remove from the database all dynamic facts with status S
clean S retractall mcon or S
retractall mcon if S
retractall mcon nand S
retractall mcon atom S
retractall econ S
clean
remove all dynamic facts from the database
clean clean
Change status of all mcons with status S to S
change status S S
sweep retract mcon or X Y S
assert mcon or X Y S
retract mcon if X Y S
assert mcon if X Y S
retract mcon nand X Y S
assert mcon nand X Y S
retract mcon atom X S
assert mcon atom X Y S
rcc i
This predicate specifies the mapping from RCC relations
to intuitionistic model and entailment constraint formulae
RCC rel Model Constraints Entailment Constraints
APPENDIX C PROLOG CODE
rcc i dc X Y or if X f if Y f if X f if Y f
rcc i ec X Y if X if Y f or if X f if Y f
if X f if Y f
rcc i po X Y if X if Y f if X Y
if Y X if X f if Y f
rcc i tpp X Y if X Y or if X f Y
if X f if Y f
rcc i tppi X Y if Y X or if Y f X
if X f if Y f
rcc i ntpp X Y or if X f Y if X f if Y f
rcc i ntppi X Y or if Y f X if X f if Y f
rcc i eq X Y if X Y if Y X if X f if Y f
Auxilliary Minor Predicates
gen out flag on
gen out gen out flag off
gen out O write O nl ttyflush
use module library random
random rel RegNo Rel
random elt dc ec po tpp tppi ntpp ntppi eq R
random RegNo R
random RegNo R
Rel R r R r R
random elt L E length L Len
Lim is Len
random Lim R nth R L E
App endix D
Re dundancy in Comp o s ition Table s
Thi s app endix summar i s e s the main re sults that were publi she d in Bennett a concer ning the
re dundancy of information in comp os ition table s
If a bas i s s et contains n relations then there will b e n table entr ie s and if computing each
entry require s making n cons i stency checks then the total numb er of cons i stency checks require d
to construct the table will b e n However a cons ideration of the structure of a comp os ition table
will reveal that it contains a large amount of re dundant information Hence much of the work
done in cons i stency checking to compute such a table i s also re dundant One sort of re dundancy
o ccurs b ecaus e if we compute each cell of a comp os ition table s eparately we end up checking the
cons i stency of identical s ets of relations s everal time s Further re dundancy i s intro duce d by the
f act that any relation can b e wr itten in two ways by inverting the relation and swapping the order
of the arguments
Clearly a comp os ition table can b e constructe d very eas ily once we know the s et of cons i stent
tr iangular con gurations of relations drawn f rom the bas i s s et under cons ideration Furthermore
once we have determine d whether a tr iangle i s cons i stent we have already determine d the con
s i stency of the e ss entially equivalent tr iangle s obtaine d by rotating the or iginal or inverting each
of its relations The exact numb er of tr iangle s equivalent to a given tr iangle dep ends up on the
di str ibution of symmetr ic and asymmetr ic relations and whether it contains duplicate relations
The que stion I now addre ss i s how many e ss entially di stinct tr iangle s can b e forme d f rom
s symmetr ic and a asymmetr ic relations Cons ider an arbitrary s et of relations cons i sting of s
symmetr ic relations a asymmetr ic relations and a further asymmetr ic relations which are the ir
convers e s Figure D shows all p oss ible con gurations of symmetr ic and asymmetr ic relations
in a tr iangle mo dulo rotation and ipping The capital letters S and A stand for symmetr ic
and asymmetr ic and indicate the numb ers of each typ e of relation pre s ent in the tr iangle The
small letters c d and f stand for converging diverging and following which de scr ib e the
di erent ways in which two asymmetr ic relations can b e arrange d r and n denote rotating and
not rotating con gurations of three asymmetr ic relations
To calculate the total numb er of e ss entially di erent tr iangle s the numb ers of p oss ible instan
tiations of each of the s e con gurations were calculate d cas e by cas e After some manipulation the
APPENDIX D REDUNDANCY IN COMPOSITION TABLES
SSS SSA
SAAc SAAd SAAf
AAAr AAAn
Figure D Poss ible con gurations of symmetr ic and asymmetr ic relations
following p olynomial giving the total numb er T of e ss entially di stinct tr iangle s in terms of s and
a was arr ive d at
s s s s a s a a a a T
We also know that the total numb er n of relations in a theory i s equal to s a so s n a By
substituting n a for s in the p olynomial we end up with a s impler equation pr imar ily involving
n
T n n n na
As the numb er of relations increas e s the n terms of the s econd equation will dominate Thus
for large n the numb er of di stinct tr iangle s will approach n
The following table shows value s of s a n and T for a numb er of theor ie s for which com
p os ition table s have b een constructe d RCC i s the bas i s of e ight top ological relations de ne d in
Randell Cui and Cohn RCC i s a bas i s of spatial relations involving containment whos e
de nition i s di scuss e d in Cohn et al the complete comp os ition table i s given in Bennett
b IC i s Allen s temp oral interval calculus and LOS i s Galton s Line
of Sight calculus The nal column give s T as a p ercentage of n
Bas i s Set s a n T
RCC
RCC
IC
LOS
Table D Comp os ition table re dundancy gure s for four relation s ets
Hence by lo oking at relational comp os itions as b e ing charater i s e d by a s et of cons i stent tr i
angle s rather than by a table and by taking advantage of rotational and mirror symmetry exhibite d
APPENDIX D REDUNDANCY IN COMPOSITION TABLES
by the s e tr iangle s the computational work nee de d to determine the comp os itions of a s et of re
lations can b e re duce d to approximately one s ixth of what would b e require d us ing the na ve
table bas e d approach Moreover rather than stor ing a comp os ition table it i s su cient to store
just the cons i stent tr iangle s or the incons i stent one s if there are le ss of thos e It i s easy to s ee
that f rom thi s information comp os ition table entr ie s can b e compute d by a constant time b ounde d
algor ithm
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