Terminological knowledge representation
systems in a pro cess engineering application
Von der Mathematisch Naturwissenschaftlichen Fakult at der
Rheinisch Westf alischen Technischen Ho chschule Aachen
zur Erlangung des akademischen Grades einer
Doktorin der Naturwissenschaften
genehmigte Dissertation
vorgelegt von
Diplom Informatikerin Ulrike Sattler
aus M unchen
Referent Universit atsprofessor Dr Ing F Baader
Korreferent Universit atsprofessor Dr Ing W Marquardt
Tag der m undlichen Pr ufung Mai
Contents
Introduction
Pro cess Systems Engineering
Terminological Knowledge Representation Systems
Structure of the Thesis
Pro cess Mo delling
Mo deling chemical pro cesses
VeDa a chemical engineering data mo del
Part whole relations
A well de ned taxonomy of part whole relations
Mereological collections
Integral part whole relations
Comp osed part whole relations
Comparison with other part whole relations
Description Logics
A brief history of Description Logics
The basic Description Logic ALC
Knowledge representation based on Description Logics
A tableau based algorithm for ALC
ii CONTENTS
Extensions of ALC
ALC extended by Complex Roles
ALC extended by Number Restrictions
Expressive p ower of the basic Description Logics
Prototype Implementation
The mo deling to ol ModKit
The terminological knowledge representation system Crack
The integration of Crack into ModKit
Exp eriences
Expressive Number Restrictions
Number Restrictions on Complex Roles
Undecidability results
A decidable extension
Symbolic Number Restrictions
An undecidability result
A decidability result
Related work
Transitive relations in Description Logics
ALC extended by the Transitive Closure of Roles
ALC extended by Transitive Roles
ALC extended by Transitive Orbits
Representation of part whole relations
Conclusion
Consequences for the application
A App endix
CONTENTS iii
B Bibliography
C Index
D List of Figures
I would like to thank all those who help ed me start set up and complete
this thesis by providing scienti c and technical supp ort and guidance co ee
laughter friendship dart games well being optimism fun fuzz and love
Chapter
Introduction
This thesis is concerned with the question of how far terminological knowledge
representation sytems can supp ort the development of mathematical mo dels
of chemical pro cesses In this chapter after a very brief description of the
goals of pro cess systems engineering it will b e argued that the structuring
of the application domain is crucial for the development of pro cess mo dels
Then the main ideas and features of terminological knowledge representation
sytems are sketched and it is argued that the system services provided by a
terminological knowledge representation system can supp ort the structuring
of an application domain Furthermore the reasons why the pro cess systems
engineering application asks for the extension of the expressive p ower of the
formalism underlying terminological knowledge respresentation systems so
called Description Logics will b e discussed
Pro cess Systems Engineering
Pro cess systems engineering is concerned with the construction of mo dels of
chemical pro cesses We can think of these pro cesses as the ones taking place
in p ossibly huge chemical plants even though pro cess systems engineering is
not restricted to those With this image in mind it should b e clear that trial
and error metho ds for the design and optimisation of these pro cesses are not
only to o costly and time consuming but also to o dangerous Furthermore
as the comp etition in the market of chemical pro ducts is very strong the
developers of chemical pro cesses aim at a high level of optimisation This
do es not only concern the quality and quantity of the pro cess pro duct that
is the substance the pro cess is intended to pro duce but also the resources
Chapter Introduction
H2O TC EO+H2O
cooling water 1
TI CSTR H2O, EO
EO evaporator H2O
EG H2O DIEG EG TREG DIEG
TREG
Figure Flowsheet of the ethylene glycol pro cess
the pro cess needs for its pro duction the amount and comp osition of other
substances pro duced by the pro cess the level of safety etc Hence instead of
trial and error metho ds pro cess engineers construct models whose purp ose it is
to enable the numerical simulation analysis and understanding of chemical
pro cesses
Mo dels of chemical pro cesses come in di erent shap es according to the sp eci c
requirements of the mo del and the state of its development They can b e
natural language descriptions of the pro cess miniature plants in a lab oratory
ow sheets graphics using some standard notations for reactors pip es valves
thermometers etc and other graphical mo dels or sets of equations describing
the b ehaviour of the pro cess For example a owsheet representation of an
ethylene glycol pro cess can b e found in Figure It consists mainly of a
reactor which is equipp ed with a stirring unit and a co oling jacket and an
evaporator and these two parts are connected via a pip e equipp ed with a valve
In Figure the same pro cess can b e found in the ModKit representation
which is very close to the one used within VeDa A graphical representation
in VeDa notation of the reactor in the ethylene glycol pro cess can b e found
in Figure In this representation the connection b etween the reactor or
more precisely the co oling jacket and the environment is emphasised Finally
the decomp osition of the gas phase material balance equation describing the
b ehaviour of the evaporator can b e found in Figure It is denoted using a
so called and or graph in order to represent b oth the parts an equation consists
of using and no des and alternatives one has for the description of a single
term in the equation using or no des
Pro cess Systems Engineering
Figure The ethylene glycol pro cess in ModKit representation
Mo dels can also b e classi ed according to the goal they are designed to achieve
For example there are mo dels designed for the precise simulation of the b e
haviour of the pro cess mo dels designed in order to get a b etter understanding
of the way in which the pro cess functions and in which the parts of this pro cess
interact and there are decision supp ort mo dels whose goal it is to supp ort
decisions that are to b e taken while working on an appropriate solution for the
design problem
Mathematical mo dels namely sets of di erential integral or algebraic equa
tions and mixtures of these equations are of a sp ecial interest since they are
the prerequisites for the analysis and simulation of the b ehaviour of a pro
cess A mathematical mo del consists in general of a huge number of equa
tions hundreds or thousands of equations dep ending on the pro cess and the
degree of detailing of the pro cess mo del and it can thus hardly b e solved
exactly neither by hand nor by a symbolic solver but only approximately
by a numerical solver or simulation to ol To use such a to ol it is necessary to
transform the mathematical mo del into the input format of this to ol Then
the engineer can simulate the in uence which particular parameters like the
inlet temp erature to the reactor have on the outcome of the pro cess Other
to ols allow the optimisation of particular pro cess parameters like the time the
pro cess needs for the pro duction of the target substance or the ratio b etween
Chapter Introduction
Reactor with Environment Cooling Jacket
Reactor Cooling Jacket
Thermo- Controller
unit
Figure Decomp osition of the reactor of the ethylene glycol pro cess into
building blo cks
the target and the input substances
There are mainly three factors contributing to the complexity of the construc
tion of a precise mathematical mo del
The p ossibly very high complexity of a pro cess together with the great
variety of phenomena that can take place within a pro cess
The fact that the more precisely a mathematical mo del describ es the
b ehaviour of a pro cess the more equations are contained in it
The di culties human b eings have in the p erception of facts and inter
relationships that are denoted in mathematical formulae
Thus in general di erent graphical mo dels serve as outlines for the construc
tion of a mathematical mo del
Summing up we are confronted with various transitions from one representa
tion formalism to another From one graphical formalism to another one from
a graphical formalism to a mathematical one and from a mathematical one
into a language that is understo o d by a numerical solver or simulation to ol
Pro cess Systems Engineering
GAS PHASE MATERIAL BALANCE
dm
i
m x R j A
out s
i i i
dt
and
MASS TRANSFER RATE j
i
or
j ak g c
i i i i
and
EQUILIBRIUM
MASS TRANSFER
CONCENTRATION g
COEFFICIENT k
i
SPECIFIC SURFACE
AREA a
or
or
or
k
m n
i
S c Re
i
c
i
P
g
i
and
a c
j i
j
DENSITY REYNOLDS NUMBER Re
SCHMIDT NUMBER S c
Figure Decomp osition of the gas phase material balance equation
Unfortunately there is no software to ol available that supp orts these tran
sitions b etween representation formalisms Furthermore there is neither an
agreed metho dology for the development of pro cess mo dels nor a description
of the ob jects these mo dels can b e viewed to consist of Marquardt With
out going into detail here these ob jects will b e called basic building blo cks
and can b e thought of as devices connections or descriptions of the b ehaviour
of devices or connections Using these building blo cks a mo del could b e char
acterised by a description of the building blo cks it consists of together with
a description of how these buildings blo cks are interrelated Until now these
building blo cks and their prop erties strongly dep ended on the representation
formalism used for a particular kind of mo del As a consequence during the
life cycle of a pro cess mo del one has to build several mo dels using several for
malisms First one starts with natural language mo dels then moves to various
graphical formalisms then designs a mathematical mo del and or writes the
source co de for a numerical solver or simulation to ol It is not p ossible to use
one single mo del while increasing its degree of detailing or p ossibly changing
the graphical representation of its building blo cks according to the goal and
state of the mo del This fact not only increases the cost of mo deling chem
ical pro cesses but it also makes the reuse of already developed mo dels more
Chapter Introduction
di cult and is an additional source of errors
To overcome this shortage W Marquardt and co workers at the Lehrstuhl
f ur Proze technik of RWTH Aachen Aachen University of Technology are
developing such a description of the standard building blo cks the so called
VeDa Verfahrenstechnisches Datenmo dell german for chemical engineer
ing data mo del It consists of three parts a frame based formalism for
the representation of standard building blo cks the description of these
standard building blo cks within this formalism and the description of
mo deling steps i e the steps an engineer undertakes while building a pro cess
mo del VeDa is a step towards a mo del development environment supp orting
the ab ove mentioned transitions from one representation formalism to another
one The standard building blo cks are indep endent from the graphical repre
sentation they are generic in that they comprise all ob jects a pro cess mo del
might comprise and they are exible in that they can b e changed according
to the sp eci c requirements of a mo del
Since the number of standard building blo cks is relatively large they must b e
stored in a structured way otherwise the reuse of already de ned building
blo cks b ecomes di cult b ecause it is hard to nd the ones one is lo oking for In
VeDa ob jects are group ed into classes and classes are ordered with resp ect
to the inheritance relation which should according to the intended semantics
of VeDa coincide with the is a sp ecialisation of relation That is to say that
if a class B inherited some of its prop erties from a class A then B should also
b e more sp eci c than A Now due to the strong expressive p ower of VeDa
this cannot always b e guaranteed and the consequence of this fact is shown
in the following example
Example A user of VeDa could de ne a class reactor using inheritance
from a class tank and sp ecify its prop erties by mistake in such a way that
an instance of reactor is not necessarily also an instance of tank As a
consequence the user and p ossibly other users as well might get confused
since the structure he or she thinks the classes have is di erent from the one
they actually have One can think of a situation where an engineer is searching
the instances of tank exp ecting to nd a certain t in it Now it is p ossible
that this t b elongs to the sub class reactor of tank and not to tank and
that the engineer will not nd t b ecause he or she was lo oking for t in the
wrong place
In order to avoid misunderstandings and to really pro t from the structured
storage it should always b e kept consistent with the way the structure is
de ned More precisely if classes are structured with resp ect to the is a
sp ecialisation of relation then the system should guarantee that a class A
Terminological Knowledge Representation Systems
which is said to b e more sp eci c than a class B is really more sp eci c than
B that is each ob ject b elonging to A also b elongs to B
The number of building blo cks is continuously growing hence the structure
should also b e exible and allow for changes according to new building blo cks
This means that it should b e p ossible to add a new appropriate class for
ob jects that do not t well into already existing ones or to divide a class A
into sub classes if A has b ecome to o unsp eci c b ecause of the growing number
of ob jects in it
To guarantee the consistency of this structure and to help the engineers in
using and extending it automatic reasoning services could b e used Whether
these automatic reasoning services are realizable or not dep ends only on the
formalism which is used to describ e these building blo cks This formalism
has to allow for the computation of these reasoning services For example
these reasoning services and the ab ove mentioned exibility are not provided
by standard data base systems which have other features but they are pro
vided by some knowledge representation systems In fact the family of knowl
edge representation systems based on Description Logics are equipp ed with a
p owerful language to describ e interesting prop erties of the ob jects in an appli
cation domain and they provide the reasoning services necessary to guarantee
the consistency of the structured storage of these ob jects Furthermore for
a variety of these knowledge representation systems questions concerning the
expressive p ower and the computational complexity of the according reasoning
services are already formulated and investigated
In a nutshell the motivation b ehind the co op eration b etween the developers of
VeDa at the Lehrstuhl f ur Proze technik and the Lehr und Forschungsgebiet
Theoretische Informatik was the question Which representation formalism
has enough expressive p ower for describing the relevant prop erties of the build
ing blo cks of pro cess mo dels and allows for the automatic computation of the
reasoning services needed to guarantee consistency of their structured stor
age The rst results obtained within this co op eration are presented in this
thesis
Terminological Knowledge Representation
Systems and Description Logics
Terminological Knowledge Representation Systems are based up on Descrip
tion Logics a formalism also known as Concept Languages KL One based
knowledge representation languages or Terminological Logics
Chapter Introduction
After a brief introduction of Description Logics it is argued that terminological
knowledge representation systems DL systems are go o d candidates for to ols
supp orting the structured storage of building blo cks
Description Logics are a family of logic based representation formalisms whose
origin can b e found in semantic networks Beside the pure formalism there are
various implementations of knowledge representation systems that are based on
Description Logics Patel Schneider et al MacGregor Baader et al
Bresciani et al Horro cks Rector Starting in the late eight
ies a large variety of di erent Description Logics has b een de ned and inves
tigated almost all of whom have the following prop erties in common
They are equipp ed with precise well de ned semantics This makes
the b ehaviour of the knowledge representation system indep endent of a
particular implementation and enables the precise de nition of inference
problems and reasoning services
In most but not all cases they allow only the use of unary predicates
such as Device or Connection and binary predicates such as connec
ted to or has part
System services of DL systems deduce implicit knowledge from the one
explicitly given by the user Most Description Logics are built such that
they have decidable inference problems see for example Hop croft
Ullman for an introduction to the theory of complexity and com
putability Hence the system services of a DL system are e ectively
computable even if most of them are of a relatively high computational
complexity For an extensive overview on reasoning problems and tech
niques see Donini et al
These prop erties the last one in particular lead to the assumption that De
scription Logics are a go o d starting p oint for searching a representation for
malism as describ ed at the end of Section
A Description Logic is mainly characterised by a set of constructors that can
b e used to build complex concepts and roles from atomic concepts unary pred
icates and roles binary predicates Throughout this thesis to distinguish
more easily b etween concepts and roles the latter start with lower case letters
whereas concepts start with upp er case letters For example devices which
are only connected to pip es and have a co oling jacket can b e describ ed by the
Terminological Knowledge Representation Systems
following concept
Device u connected to Pipe u has part Cooling jacket
Devices having no co oling jacket as parts and at least parts can b e describ ed
by
Device u has part Cooling jacket u has part
Using these constructors the terminology of an application domain can then
b e de ned in a so called TBox by a set of concept and role de nitions as for
example the following two concept de nitions describing co oled reactors and
co oled reactors equipp ed with a stirring unit
CoolReactor Reactor u has part Cooling Jacket
StCoolReactor CoolReactor u has part Stirring Unit
where we assume that the concepts o ccurring in these de nitions are already
de ned Referring to these concepts concrete ob jects of a concrete world
are describ ed in a so called ABox An example of which b eing
REACTOR Reactor
REACTOR has part COOL
COOL Cooling Jacket
which states that the ob ject REACTOR is an instance of Reactor and related
via the role has part to the ob ject COOL which is a Cooling Jacket Given
the ab ove concept de nitions a knowledge representation system should then
b e able to infer that REACTOR is a CoolReactor This inference classifying
ob jects with resp ect to the concepts de ned in a TBox is one of the system
services that are provided by DL systems Other system services comprise for
example
the computation of the implicit subsumption relation b etween two con
cepts or more generally the computation of the taxonomy i e the
hierarchy of concepts de ned in a TBox with resp ect to the subsumption
relation
deciding whether a given concept is satis able i e whether it can ever
b e instantiated
Precise de nitions of syntax and semantics of Description Logics are given in Section
an overview can b e found in the App endix
This relation is de ned in De nition Roughly sp oken a concept D subsumes a
concept C if each instance of C is always also an instance of D
Chapter Introduction
classi cation of ob jects i e enumerating the most sp eci c concepts a
given ob ject is an instance of or
retrieving all ob jects o ccurring in an ABox that are instances of a given
concept
Given that ob jects are group ed into concepts that corresp ond to classes the
subsumption relation on concepts is equivalent to the is a sp ecialisation of re
lation on classes mentioned in Section Thus the automatic computation
of the subsumption relation and the automatic classi cation of ob jects are the
reasoning services available in DL systems which can b e used to guarantee the
consistency of the structured storage of ob jects
Since the late eighties numerous investigations concerning the expressive p ower
and the computational prop erties of various Description Logics have b een car
ried out Donini et al a b These investigation were often moti
vated by the use of certain constructors in implemented systems Neb el
Borgida Patel Schneider or the need for these constructors in sp eci c
applications Baader Hanschke Franconi Sattler and the re
sults in uenced the design of new systems Horro cks
An imp ortant constructor for concepts in Description Logics are the so called
number restrictions These allow to describ e ob jects by restricting the num
b er of ob jects which are related to them by some role For example
has part describ es all those ob jects having at least parts i e those
ob jects having at least ob jects related to them via the role has part
There are two main reasons why number restrictions are present in almost all
implemented systems Firstly human b eings tend to describ e ob jects using
number restrictions Secondly number restrictions can express information
concerning the way in which ob jects are interrelated by roles This information
on the relation b etween ob jects is of great imp ortance for a pro cess systems
engineering application b ecause pro cess mo dels strongly dep end on the way in
which their buildings blo cks are interrelated
Unfortunately most Description Logics only allow the use of atomic roles in
side number restrictions whereas complex roles are excluded from b eing used
in number restrictions Complex roles are built using role forming construc
tors such as union intersection inverse and comp osition For example these
constructors can b e used to describ e the role connected to as the union of
input connected to and output connected to and the role is part of
as the inverse of has part
Terminological Knowledge Representation Systems
This restriction to atomic roles is unsatisfactory since we must distinguish
roles that are also allowed inside number restrictions from those that can b e
used in other places of concepts but not inside number restrictions This makes
the syntax of Description Logics more complicated and more di cult to learn
for the user and the expressive p ower of number restrictions increases
if complex roles built by the ab ove mentioned role constructors can b e used
inside them Using comp osition of roles for example one could describ e a
device such that all devices connected to it are controlled by the same control
unit by
Device u connected to controlled by
The number restriction in states that each of these devices is related to
at most one ob ject via the role connected to followed by controlled by To
ensure that the device itself is also controlled by the unit that controls the
devices connected to it we additionally need union in the number restriction
Device u controlled by t connected to controlled by
Inversion of roles comes in if we need the role controls as well An imp ortant
part of this thesis is concerned with the in uence complex roles in number re
strictions have on the computational complexity of the corresp onding inference
problems
Another weakness of traditional number restrictions comes up if we try to
describ e devices having fewer inputs than outputs referred to by Output devs
in the following or devices whose parts have the same number of inputs and
outputs referred to by Equal parts devs If an upp er b ound n on the
number of inputs can b e xed Output devs can b e describ ed by the following
disjunction
Device u input u output t
input u output t
input u output t
n input u n output
If we cannot x this upp er b ound it is not obvious how Output devs can b e
describ ed since in nite disjunction is not allowed in Description Logics
To overcome this lack of expressiveness we introduce numerical variables
to b e used in number restrictions Thus Output devs can b e de
scrib ed by
Device u input u output
Chapter Introduction
where stands for some nonnegative integer Similarly Equal parts devs
can b e describ ed by
Device u has part input u output
This concept reveals a certain ambiguity It describ es devices where each part
has the same number of inputs and outputs However it dep ends on the
reading whether di erent parts can have di erent numbers of inputs or not
To overcome this ambiguity we introduced explicit existential quanti cation
of numerical variables denoted by to distinguish b etween a device
where for each of its parts the number of its inputs equals the number of its
outputs
Device u has part input u output
and a device where for some number n all parts have n inputs and n
outputs
Device u has part input u output
The investigation of these so called symbolic number restrictions forms another
imp ortant part of this thesis Finally this application asks for the adequate
representation of comp osite ob jects The high complexity of most pro cess mo d
els asks for the p ossibility to mo del a pro cess with di erent degrees of detailing
by decomp osing devices into their parts or aggregating parts of a mo del to a
device or to a connection Hence the relation b etween parts and the whole they
b elong to the part whole relation plays an imp ortant r ole for the adequate
representation of pro cess mo dels Beside its other prop erties the part whole
relation is transitive Thus the adequate representation of complex ob jects
asks for a Description Logics that allows for transitive roles A Description
Logic has b een presented in Baader that allows the transitive closure
of roles inside concepts Using this role forming constructor it is p ossible to
distinguish b etween ob jects having a direct part that is a valve connection
Device u has part Valve Connection
and ob jects having at some arbitrary but nite level of decomp osition a valve
connection
Device u has part Valve Connection
where has part denotes the transitive closure of the role has part The rea
soning services of DL systems allowing for the transitive closure constructor are
In this thesis the sp elling r ole is used to refer to roles in general in order to distinguish
them from roles in the context of Description Logics where they denote binary relations
Structure of the Thesis
still decidable and there exist decision pro cedures for these reasoning services
However this constructor strongly increases the computational complexity of
these services Lo oking for a p ossibility to evade this high complexity other
ways of extending Description Logics by transitive roles were investigated The
results of this investigation form another imp ortant part of this thesis
The ab ove mentioned part whole relation has several subrelations such as
the one b etween a whole and its ingredients or the one b etween a whole and
its comp onents Intuitively these subrelations have di erent prop erties and
interact in sp eci c ways Hence they are also of imp ortance for the appropriate
representation of complex ob jects In the literature several characterisations
of these subrelations can b e found Simons Winston et al Gerstl
Pribb enow Pribb enow but as none of these characterisations was
precise enough to decide which are the relevant part whole relations for the
pro cess systems engineering application an investigation of these sub part
whole relations is also describ ed within this thesis
Besides these theoretical investigations the assumption that DL systems are
appropriate for supp orting the structured storage of pro cess mo del building
blo cks was tested empirically The DL system Crack Bresciani et al
was integrated into the pro cess mo deling to ol ModKit Bogusch et al
ModKit is developed under the real time exp ert system shell G Gen
which is a p owerful to ol that lacks however means for the structured stor
age of ob jects As a consequence in ModKit classes of standard building
blo cks were only p o orly structured and Crack was integrated to overcome
this shortcoming A description of this integration as well as a r esum e of the
exp erience we had with this integration can b e found in Chapter
Structure of the Thesis
This thesis was written while I was a member of Postgraduate College In
formatics and Technology Graduiertenkolleg Informatik und Technik at
the RWTH Aachen The aim of this Graduiertenkolleg is to establish a close
co op eration b etween computer science and technological applications in order
to b oth identify applications for new computer science techniques and to work
on and solve complex problems in engineering applications Hence this thesis
addresses b oth engineers and computer scientists I tried to write most parts
of all chapters except Chapters and in a way that they can also b e read
by someone who is not a computer scientist The computer science readers
are asked to ap ologise for the remarks which might b e in their eyes unneces
Chapter Introduction
sary and redundant and which were made for readers who are not computer
scientists
The rest of this thesis is organised as follows
Chapter discusses the problems one encounters while constructing mo d
els of chemical pro cesses and sketches the pro cess mo deling metho dology
developed within VeDa
In Chapter a framework for the classi cation of part whole relations
is presented These relations play an imp ortant r ole for the representa
tion of aggregated ob ject and had to b e understo o d b efore designing a
representation formalism for pro cess systems engineering
Chapter contains an introduction to Description Logics in order to
make the thesis self contained Besides the de nition of syntax and se
mantics of the Description Logic the future investigations and extensions
are based on it includes a sketch of how Description Logics are used
within a knowledge representation system and the idea of the basic in
ference algorithm Finally it presents already existing extensions of this
basic Description Logic together with a discussion of their expressive
p ower
In Chapter the integration of a DL system into a pro cess mo deling
to ol is describ ed This integration was realised during my time at the
Postgraduate College and motivated by the wish to learn more ab out the
usefulness of DL systems from concrete scenarios esp ecially more ab out
the way in which the p owerful inference services of these systems can b e
used within such a pro cess mo deling to ol
Chapter describ es the extensions of the basic Description Logics by
so called expressive number restrictions These extensions increase the
expressive p ower of Description Logics in a way that is useful for the
pro cess systems engineering application The rst extension allows the
user to restrict the number of ob jects an ob ject is related to via complex
relations The second extension allows the user to state for example that
an ob ject has the same number of inputs and output without b eing
more precise on this number Both extensions are motivated by the
pro cess systems engineering application and investigated with resp ect to
the computational complexity of the relevant inference problems
In Chapter three extensions of the basic Description Logic by transitive
relations are investigated These extension are mainly motivated by the
Structure of the Thesis
need for an appropriate representation of aggregated ob jects All three
extensions have decidable inference problems and it is proved that one
is of a much lower computational complexity than the others Then the
consequences of these results for the representation of aggregated ob jects
are describ ed
Finally Chapter contains the conclusion and summary of the thesis
After this chapter the app endix contains a short summary of the syntax
and semantics of the Description Logics used within this thesis
Chapter
Pro cess Mo delling
In this chapter we describ e the pro cess systems engineering application and the
problems one encounters when mo deling chemical pro cesses The development
of these mo dels is a rather creative and complex task which is describ ed in
the rst part of this chapter Metho dologies of how to develop such a mo del
evolved only recently One of these metho dologies was developed within VeDa
Marquardt and its main ideas are brie y sketched in the second part of
this chapter
Mo deling chemical pro cesses
Pro cess systems engineering is concerned with setting up mo dels of chemical
pro cesses Mostly these pro cesses take place in chemical plants that are run
by chemical industries for the pro duction of chemical substances
More precisely a process is an amalgamation of physical chemical biological
and informational events which are undertaken in order to change substances
with resp ect to their nature prop erties and comp osition A model is a goal
driven simpli cation of the reality obtained by abstraction Hence a process
model is a mo del of a pro cess and its goal is to enable the simulation analysis
and understanding of a given pro cess This understanding is the prerequisite
for improving already developed pro cesses and for designing new high quality
pro cesses
As stated in Section setting up a pro cess mo del of some precision is a
complex task This is mainly due to the following p eculiarities of pro cess
systems engineering
Mo deling chemical pro cesses
There is a great variety of di erent chemical devices and connections
such as reactors pip es valves distillation columns etc which have to
b e considered on an abstract level
There is a great variety of di erent physical phenomena such as heat
transfer reaction evaporation disp ersion mixing etc which also have
to b e considered on an abstract level
Abstracting from a physico chemical phenomenon is nondeterministic
Many phenomena can b e viewed from di erent angles and thus ab
stracted by di erent means and their b ehaviour can b e approximated
by various di erent heuristics
The requirements with resp ect to the granularity and the precision of
the mo dels is increasing with the state of the art the comp etition in
the market and the standards concerning environmental and safety de
mands
The overall number of di erent mo dels is in nite b ecause the size of a mo del
i e the number of the devices and connections it contains is not b ounded
This large number of mo dels would b e of purely theoretical interest but the
number of di erent pro cesses that are describ ed by these mo dels is also very
large Moreover even one particular pro cess taking place in a sp eci c plant
can b e mo deled in many di erent ways which di er not only in their degrees
of detailing but also in the way phenomena are abstracted This freedom of
choice contributes tremendously to the complexity of building a pro cess mo del
Various software to ols which supp ort the development of pro cess mo dels help
the engineers to accomplish this task increase the e ciency of the mo deled
pro cesses and contribute to an improvement of the security of chemical plants
Typically the nal pro cess mo del that is to b e built is a mathematical mo del
i e a set of di erential integral and algebraic equations as well as mixtures
of these types of equations Dep ending on the degree of detailing and the com
plexity of the pro cess such a set can contain several thousands of equations
The route that is taken to reach this mo del can b e viewed as taking place in
a three dimensional space Pohl Jarke Marquardt
The speci cation dimension relates to the degree of understanding the
engineers have of the pro cess and thus to the degree of detailing of the
mo del In general the engineers start the mo deling at the plant level
The mo del is then re ned p ossibly down to the level of physico chemical
phenomena at the molecular level
Chapter Pro cess Mo delling
The representation dimension relates to the representation formalism
that is used to describ e the current mo del In the early stages the
pro cess can b e describ ed in a natural language sp eci cation Then semi
formal graphical representation formalisms can b e used for a more precise
description of the pro cess Next the informal graphical formalism can b e
replaced by a more formal graphical representation formalism Finally
the b ehaviour of the pro cess is describ ed by a mathematical mo del
Following the agreement dimension the degree of agreement among the
di erent engineers involved in the development of a pro cess mo del should
increase while setting up this mo del
In general pro cess mo dels can b e viewed as b eing built from so called building
blocks which are the entities pro cess mo dels consist of In the VeDa metho d
ology the central building blo cks are devices connections entities describing
the b ehaviour of devices and connections and entities describing the inter
faces b etween devices and connections By standard building blo cks we mean
generic building blo cks commonly used The following is a p ossible scenario of
the development of a pro cess mo del
The engineer chooses standard building blo cks such as devices connec
tions etc from a catalogue This is a well structured collection of build
ing blo cks together with descriptions of their prop erties and p ossibly
equipp ed with p owerful search facilities
If necessary these building blo cks are mo di ed according to the require
ments of the current mo del These mo di cations can include b oth a more
detailed sp eci cation of prop erties that are unsp eci ed so far as well as
complete changes of the prop erties of a building blo ck
The building blo cks are connected to each other p ossibly using their
already sp eci ed interfaces
The building blo cks are broken down into their comp onents until an
appropriate degree of detailing is reached
The b ehaviour of the building blo cks is sp eci ed First for each building
blo ck the phenomena that are asso ciated to it are xed Then pro cess
quantities i e variables standing for relevant pro cess quantities are
introduced for the basic phenomena Equations relate these variables to
each other and might re ne some of them Finally if necessary initial
values are xed for pro cess variables
Mo deling chemical pro cesses
The information sp eci ed so far is used to automatically generate the
source co de for a numerical solver which describ es the b ehaviour of this
pro cess
If it is likely that parts of this mo del can b e reused in other mo dels they
are stored as new building blo cks in the catalogue
To assist the engineer each of these steps should b e controlled by a mechanism
that guarantees the consistency of the mo del and the mo di cations that are
carried out by these steps This means for example that adaptations of the
standard building blo cks to the requirements of the current mo del see Step
are compared with the actual descriptions of the building blo cks in order to
guarantee consistency of the mo del If for example the engineer chooses
from the catalogue a building blo ck that represents a stirred reactor and then
removes the stirring unit the system should notice that this building blo ck
do es no longer represent a stirred reactor but a reactor
Unfortunately no mo deling to ol supp orting the ab ove scenario is available In
fact current mo deling to ols for chemical pro cesses can b e divided into two
groups Marquardt
Mo deling using an equation oriented to ol e g Speedup or Diva con
sists of either the implementation of a mathematical mo del using a
declarative programming language together with a mo del library or its
implementation using a pro cedural programming language and a set of
subroutine templates These to ols are very exible in that they allow
for a large variety of di erent mo dels The price one has to pay for this
exibility is the high e ort required for mo deling
Mo deling using a block oriented to ol e g Aspen Plus takes place on
the ow sheet level where the engineer selects sp eci es and connects
standard building blo cks so called pro cess units from a mo del library
using either a mo deling language or a graphical editor A severe restric
tion of these to ols is due to the fact that standard building blo cks can
only b e mo di ed in a restricted way to match the sp eci c needs of a
mo deling context Hence the degree of detailing of the mo dels one can
construct using these to ols is b ounded which restricts the applicability
of these to ols As a recomp ense they are rather comfortable to use
Even if these di erent to ols could b e integrated into one general mo deling to ol
p oints and of the ab ove mentioned scenario would not automatically
Chapter Pro cess Mo delling
b e supp orted The blo ck oriented to ols are restricted to the prede ned building
blo cks which cannot b e broken down arbitrarily Furthermore a metho dology
is still lacking which describ es the b ehaviour of devices or connections in such
a precise way that the according set of mathematical equations and the com
plying source co de for numerical solvers can b e generated automatically First
attempts at the automatic generation of source co de from mo dels developed
using an ob ject oriented mo deling to ol were made within ModKit a mo deling
to ol partially describ ed in Chapter
In order to realise the ab ove scenario the new building blo cks have to b e stored
in a way that enables the user to nd a building blo ck he or she is lo oking
for This means that they must b e arranged with resp ect to some meaningful
order otherwise it would b e hard to nd the building blo cks one wants to
reuse In the following we will refer to a storage of building blo cks with resp ect
to such a meaningful order by structured storage
The more di cult it is to nd the building blo cks one is lo oking for the less
one will try to reuse already sp eci ed blo cks Hence the structured storage of
building blo cks will lead to a higher degree of e ciency in the development of
pro cess mo dels
As a consequence of the ab ove observations the need for a representation
formalism which can b e used for b oth the description of the structural and b e
havioural asp ects of building blo cks arises Additionally this formalism should
b e able to supp ort the structured storage of building blo cks A promising can
didate for the corresp onding meaningful order is the is a sp ecialisation of
relation More precisely one unites similar building blo cks into classes and
asso ciates a description of the prop erties of its members to each class Then
the resulting set of classes and thus their instances can b e arranged with
resp ect to the is a sp ecialisation of relation In order to avoid inconsisten
cies this sp ecialisation relation should not b e hand co ded by the engineers
but it should b e computed automatically using the descriptions of the classes
Furthermore class de nitions should b e tested automatically for inconsisten
cies or contradictions In order to enable the automatic computation of the
sp ecialisation relation and the consistency test the prop erties of a class i e
the prop erties of the ob jects b elonging to this class should b e describ ed in a
declarative way using a formalism with well de ned semantics Furthermore
the problem of whether one class is a sp ecialisation of another class or whether
a class is inconsistent should b e e ectively decidable In Chapter a family
of knowledge representation formalisms having these prop erties the so called
Description Logics is introduced The integration of a knowledge represen
tation system based on Description Logics into a mo deling to ol for chemical
VeDa a chemical engineering data mo del
pro cesses is then describ ed in Chapter
VeDa a chemical engineering data mo del
VeDa is short for Verfahrenstechnisches Datenmodel chemical engineering
data mo del It was rst developed by W Marquardt Marquardt VeDa
is a frame based formalism see Baumeister for a description of the un
derlying formalism designed for the formal representation of the mo deling
metho dology developed at the Lehrstuhl f ur Proze technik of RWTH Aachen
Both the formalism and the metho dology are describ ed in Marquardt
The main characteristics of this metho dology are the following
VeDa distinguishes two asp ects of a pro cess mo del Its structural and its
behavioural asp ects This distinction is useful b ecause of the following freedom
of choice Firstly the structure of a mo del is not uniquely determined by the
sp eci c pro cess it mo dels and the degree of detailing of the mo del Secondly
having xed the structure of a mo del its b ehaviour can b e mostly mo deled
in various di erent ways This is due to the fact that b oth the mo deling
of the b ehaviour and of the structure strongly dep end on the goal and the
precision of the mo del For example the b ehaviour of a stirring reactor can
b e mo deled in one mo del by describing the kinetics of the reaction taking
place in this reactor In this case the equations describing these kinetics are
still not determined In most cases one can choose b etween several heuristics
found in literature or carry out exp eriments to determine them In another
mo del of the same pro cess one might neglect these phenomena and describ e
only the generalised uxes in the reactor
Furthermore VeDa also includes a formalism for representing the steps the
engineer executes while building a mo del
Structural asp ects Souza VeDa divides the set of ob jects repre
senting structural asp ects of a pro cess mo del into devices such as a reactor
or a distillation column and connections such as signal lines or valve connec
tions These classes can b e further re ned into more sp eci c sub classes such
as heterogeneous phase or directed permeable valve co nne ctio n It is
not the case that each sub class is necessarily related to a sp eci c function such
as mixing reacting etc the instances of this class have or a concrete gadget
that can b e found in chemicals plants In most cases sub classes are related to
certain abstraction mechanisms that are employed in pro cess mo deling such
as a heterogeneous well mixed or homogeneous phases
Chapter Pro cess Mo delling
Ob jects representing structural asp ects can b e b oth atomic or aggregated
In order to appropriately supp ort the top down or b ottom up mo deling ag
gregated ob jects have to b e describ ed in such a way that for example the
prop erties of the aggregated ob ject can b e deduced from the prop erties of its
parts and from the way in which the interfaces of the parts match with the
interfaces of the whole
Behavioural asp ects Bogusch The b ehaviour of an ob ject is in a
rst place describ ed by the phenomena that are are asso ciated to this ob ject
These phenomena lead to the introduction of asso ciated pro cess quantities
which are related to each other by appropriate equations and which can b e
re ned by constitutive equations Hence the b ehavioural part of the VeDa
class hierarchy contains classes for di erent types of phenomena equations
and variables
Mo deling steps Lohmann Krobb The development of a high
quality pro cess mo del is a strongly creative pro cedure hence it cannot b e
completely automated However it is p ossible to guide the engineer by for
example prop osing p ossible next steps which can b e executed in the current
state of the mo del These steps as well as the description of mo deling states
are also formalised within VeDa A step is characterised by
a goal such as the introduction of a new device
preconditions which have to hold for the step to b e activated
p ostconditions which are guaranteed to hold after the execution of a step
and
further prop erties
An imp ortant prop erty of this formalisation is that the syntax for the descrip
tion of the pre and p ostconditions was de ned in such a way that all
realistic conditions can b e expressed and the reasoning services such as
checking whether a comp osite step is correctly decomp osed into sub steps or
checking whether one step can b e substituted by another one are e ectively computable
VeDa a chemical engineering data mo del
Implementation of VeDa
In order to evaluate the VeDa approach the mo deling to ol ModKit was de
veloped and implemented at the Lehrstuhl f ur Proze technik of RWTH Aachen
Bogusch et al It builds on the exp ert system shell G and is describ ed
in more detail in Section
Chapter
Part whole relations
As already mentioned the pro cess engineering application asks for a Descrip
tion Logic with enough expressive p ower to represent the relevant prop erties of
ob jects in the application domain namely building blo cks of pro cess mo dels
For the adequate representation of aggregated ob jects the part whole relation
as well as those sub part whole relations relevant for the application have to
b e represented appropriately This appropriateness can only b e achieved if the
prop erties of the part whole relation and its sub relations are known In this
chapter we investigate in detail the prop erties of the part whole relation and
its subrelations and present a general scheme for the classi cation of di erent
kinds of part whole relations This scheme was partially developed while I was
visiting A Artale N Guarino and E Franconi at Ladseb CNR in Padova
Italy
The imp ortance of part whole relations in general led to the development of a
philosophy based on the notion of parts and wholes namely the classical ex
tensional mereology as introduced in Le sniewski for an introduction and
exhaustive overview see Simons Classical extensional mereology is a
formalism quite similar to set theory with the di erence that instead of b eing
based on the membership relation it is based on the part whole relation As
for set theory various logical formalisms are based on mereology and di erent
ways of axiomatising the world based on these formalisms were presented
di ering for example in whether the world is supp osed to b e comp osed from
atoms or not where atoms are ob jects having no parts Furthermore mereo
logical logics were also extended to deal with mo dalities actions and pro cesses
Finally Simons presents an exhaustive list of prop erties which can ad
ditionally hold b etween parts and wholes such as a part b elonging exclusively
to a whole
Later when the development in computer science gave rise to the eld of knowl
edge representation researchers in this eld realised that part whole relations
have particular prop erties which have to b e taken into account for the adequate
representation of complex ob jects This observation led to a great variety of in
vestigations of these prop erties of part whole relations Winston et al Iris
et al Gerstl Pribb enow Franconi Padgham Lambrix
Artale et al Pribb enow all motivated by the goal of developing a
formalism for the adequate representation of complex ob jects
A p oint of view supp orted by most of them is that there are di erent part whole
relations such as comp onent aggregate and ingredient ob ject with di erent
prop erties and that there is a general transitive part whole relation contain
ing all of these part whole relations Because of the particular prop erties of
the di erent part whole relations the transitivity of the general part whole
relation and the interaction b etween di erent sp eci c part whole relations
the part whole relations cannot b e represented adequately by simple binary
relations For example consider a device d that has a part d which in turn
has a carcinogenic part z A system answering no when asked whether d
has a carcinogenic part can hardly b e called adequate but a system in which
the part whole relation is not mo deled by a transitive relation can only answer
with no to this question
However despite the fact that many authors b elieve in the relevance of di er
ent kinds of part whole relations and underline the necessity to have a go o d
understanding of the interaction b etween them to our knowledge there is no
taxonomy of part whole relations Winston et al present di erent
part whole relations but the authors only present examples to explain their
meaning and no exact de nitions are given From counterexamples which
strongly dep end on the intuition of the reader they infer that these di erent
part whole relations do not interact with each other This is to say that from
the fact that x is a part of y with resp ect to one sub part whole relation and
y is a part of z with resp ect to another sub part whole relation one can only
conclude that x is a part of z with resp ect to the general part whole relation
It seems reasonable that this rather weak conclusion could b e strengthened for
some sub part whole relations
In the following a taxonomy of part whole relations together with de nitions
of part whole relations will b e presented From this framework the way in
which the part whole relations interact follows by de nition
Chapter Part whole relations
A well de ned taxonomy of part whole re
lations
Given the need for a b etter understanding of the di erent part whole relations
we developed a scheme for the classi cation of part whole relations The re
sulting taxonomy of part whole relations is given in Figure It shows classes
of part whole relations in a sub class sup erclass hierarchy At the top of this
hierarchy the general part whole relation is found It is even more general
than mereological collections in that the principles imp osed on mereological
collections are stronger than the ones imp osed on which is the union of all
sub part whole relations Hence strongly dep ends on these sub relations
but the only prop erty we want to imp ose on is that it is a strict partial or
der and thus transitive and irre exive These two prop erties are axiomatised
by A and A
x y y z x z A
x x A
Mereological collections are presented in Section whereas the other two
sub relations of are brie y presented in this section and in more detail in
Section and
An integral part whole relation is a subrelation of the general part whole rela
tion which involves some integrity condition on the parts For x to b e a part
of y with resp ect to an integral part whole relation x has to b e a part of y
and x has to satisfy the integrity condition asso ciated with this relation For
example for the ingredient mass relation to hold b etween x and y x has to
b e integral with resp ect to the substantial asp ect This means that y contains
some material such as our or oxygen and that x is exactly all of this material
in y Another example to mention is the segment entity relation which holds
b etween z and d if z is a geometrically integral part of d This means that z
forms some geometric b o dy within d It is imp ortant to note that all these
integrity conditions are imp osed on the parts only indep endent of the whole
This indep endence is lacking for part whole relations b elonging to the second
sp ecialisation of the general part whole relation A composed part whole re
lation is characterised by an additional relation which has to hold b etween a
part and its whole For example the component aggregate relation holds b e
tween x and y if x is a part of y x is an integral ob ject with resp ect to some
integrity condition and x is functional for y i e the functioning of y dep ends
on x This additional relation such as b eing functional for in the previous
example is the reason why the comp osed part whole relations are in general
A well de ned taxonomy of part whole relations
not transitive If the additional relation is not transitive then the comp osed
part whole relation cannot b e transitive For example let a pro cedure b e a
comp onent of thus functional for a small program which is a comp onent of
a large program If the pro cedure is never called by the large program b ecause
the large program uses the small one in a restricted way then this pro cedure
cannot b e said to b e a comp onent of the large program
All these relations can b e sp ecialised by re ning the integrity conditions or the
additional relations which have to hold b etween a part and its whole As a
consequence of the prop erties of integral and comp osed part whole relations
integral part whole relations are transitive and interact in a transitive way
with other part whole relations whereas comp osed part whole relations have
in general none of these prop erties In the sequel these relations are de ned
in detail
Mereological collections
From a mathematical p oint of view a mereology is a partial order of a very
particular kind In this section only the basic prop erties of this order are
sketched Most of the axioms given in the following that enforce these prop
erties are taken from Simons Let S b e some bag of individuals Then
classical mereology is built around a primitive part whole relation which
is axiomatised in the following using rst order logic with equality The rst
two axioms enforce that is a strict partial order and SA de nes as the
re exive closure of
x y y x SA
x y y z x z SA
x y x y x y SA
Still this is axiomatisation is not su cient On the one hand the supple
mentation principle says that if a whole has a strict part then it must have
another part di erent from the rst one see SF However SF do es not
exclude ascending chains from b eing mo dels Axiom SF enforces that each
comp osed ob ject has at least two parts which are not part of each other
x y z z y z x SF
x x z SF x y z z y z
We use the notion bag instead of set to underline the fact that in the following
notions like subset or element are not used in the set theoretic sense
Chapter Part whole relations
General
p w relation
Mereological
collections
Integral Comp osed
p w relation p w relation
Substantial Perceptive Spatial
Integral Integral Integral
p w relation p w relation p w relation
Member collection Comp onent aggregate
relation relation
Integral additiv Integral functional
Geometric
Topologic
Integral Integral
p w relation p w relation
Di erent integrity conditions on parts Additional relations b etween parts and wholes
not transitive
Figure A taxonomy of part whole relations
Now dep ending on the intuition one has of the relation these axioms are
still not su cient Thus other axioms are added according to the ontological
principles assumed These principles determine the structures one accepts as
mo dels and those one wants to exclude from b eing mo dels For example the
weak supplementation principle says that if a whole has a part then it has to
have another part disjoint from the rst one This principle is axiomatised in
SA which in contrast to SF asks for an additional part z which has no
part in common with x
x y z z y w w z w x SA
Furthermore one can restrict the mo dels to those where if two ob jects x y
have a part in common then there exists another ob ject which has x y as
prop er parts Another principle atomicity excludes structures with in nite
A well de ned taxonomy of part whole relations
ascending chains from b eing mo dels It assumes the existence of indivisible
ob jects so called atoms of which all other ob jects are comp osed A last
principle to b e mentioned here is the proper part principle which is similar to
the extensionality principle of set theory It assumes that no two comp osed
ob jects have exactly the same set of strict parts In this spirit many other
principles can b e axiomatised which lead to di erent mereologies and whose
enumeration go es b eyond the scop e of this work for details see Simons
However the strong restrictions made on the order in all these mereologies
have as a consequence that and its sp ecialisations are sp ecialisations of the
general part whole relation
Integral part whole relations
An entity x is an integral part of an entity y with resp ect to the integrity
condition Int if and only if x is a part of y with resp ect to the general part
whole relation and if the integrity condition Int holds on x
def
x y x y Int x
Int
The integrity condition Int is the most general integrity condition and simply
imp oses that x is integral with resp ect to some criteria as for example its
spatial extension the rectangle in the picture its substantial comp osition
the our in the cake the p erception of an observer the nice part of
the house etc Whether an ob ject x satis es an integrity condition or not is
indep endent of its context hence from the transitivity of follows immediately
that the integral part whole relation is also transitive Moreover it interacts
in a transitivity like manner with that is
x y y z x z Int x x z
Int Int
Re ning the integrity conditions
The integral part whole relation can b e re ned in a natural way by re ning the
integrity condition Int For example three ways of re nement are presented in
Figure namely according to whether a part x is integral with resp ect to
its spatial extension which means that x forms a connected b o dy This
condition can b e further re ned by sp ecifying whether a part is supp osed
Chapter Part whole relations
to b e integral with resp ect to top ological conditions a convex segment
of a cake to geometrical conditions a cubic segment of a cake
whether it is allowed to contain holes or not etc The segment entity
relation discussed in Gerstl Pribb enow b elongs to this class of
integral part whole relations
its substantial composition Again we can re ne this integrity condition
to whether the part is integral with resp ect to its chemical comp osition
the protein in a cake its physical state the ice cub es in a glass of
water its molecular structure the diamonds in a heap of carb on
etc
the perception of an observer This condition can b e further re ned by
sp ecifying for example the sense organs involved in the p erception the
dark part of the ro om the loud part of the piece of music or whether
a part is supp osed to b e integral with resp ect to emotional criteria the
most b eautiful moment in the holidays etc
Let Sp ecInt b e a re ned integrity condition i e Sp ecInt x Int x Then
the corresp onding re ned integral part whole relation is de ned similar
Sp ezInt
to namely
Int
def
x y x y Sp ecInt x
Sp ezInt
Again as for the general integral part whole relation re ned integral part
whole relations interact in a transitivity like manner with other part whole
relations namely
x y y z x y Sp ecInt x y z x z
Sp ezInt Sp ezInt
0
y Furthermore if Sp ecInt implies Sp ecInt then it is obvious that x
Sp ecInt
implies x y hence the re nement of the integrity conditions yields a
Sp ezInt
re nement of the integral part whole relations in a straightforward way
Comp osed part whole relations
The third class of sub relations of the general part whole relation composed
part whole relations are characterised by the fact that they imp ose additional
conditions which have to hold b etween a part and its whole These additional
conditions can no longer b e viewed as integrity conditions on the part b ecause
they dep end on b oth the part and its whole One example for such an addi
tional condition is functionality A part engine can b e b oth functional for a
A well de ned taxonomy of part whole relations
whole car which works prop erly only if engine is present and intact and non
functional for another whole garage which has engine as its part but works
also without it In general these part whole relations are of the form
def
x y x y x CompRel y
Comp
where CompRel is a binary relation characterising For example CompRel
Comp
could b e re ned to Funct with the intended meaning that x Funct y holds if and
only if x is functional for y There are no restrictions on the relation CompRel
i e it has not to b e transitive As a consequence comp osed part whole rela
tions are no longer transitive nor do they interact in a transitivity like manner
with other part whole relations
Another example for a comp osed part whole relation is the so called member
collection relation as describ ed in Winston et al This relation holds
for example b etween a tree and a forest or b etween a grain of salt and a
sp o onful of salt According to our understanding an ob ject x is a member of
a collection y if and only if the following conditions are satis ed see Figure
for a b etter intuition
x is in fx x g a set of parts of y where each x satis es a predicate Q such
i
as b eing a grain which implies some kind of integrity condition Furthermore
y is the sum of the ob jects x This is to say that each part x of y di erent
i
from all x is either a part of some x or some x is a part of x see of the
i i i
formal de nition b elow Furthermore these x are supp osed to b e disjoint
i
that is they do not have any parts in common
The formalisation of these conditions is rather complicated and involves a
second order variable Horizon which holds on the x that sum up to y Fur
i
thermore Overlap z w is an abbreviation for v v z v w and the
member collection relation dep ends on the predicate Q
def
Q
x y x y Coll x y where
Q
Memb er
def
Coll x y
Q
Horizon Horizon x
z Horizon z
Q z z y
w w y w z z w Horizon w
w w y Horizon w Overlap z w z w
z z y w Horizon w z w w z w z
Chapter Part whole relations
collection y
members x i x
Horizon
Figure Example of a collection y and members x
i
Comparison with other part whole rela
tions
In this section the taxonomy of part whole relations presented in Winston et
al is compared with the taxonomy presented in this section
Comp onent Integral Ob ject This relation is a comp osed part whole re
lation with the additional condition that the part is functional for the whole
Formally according to the de nition in Winston et al it can b e writ
ten as x y x Functional y where x Functional y holds only if x is integral
with resp ect to some integrity conditions which is not sp eci ed in Winston
et al Furthermore the placement of the part within a whole might
b e relevant for the functionality but this condition is not explained in more
detail The authors assume that this condition leads to a subrelation of the
Comp onent Integral Ob ject relation
Member Collection The Member Collection relation de ned in Winston
et al di ers from the the one de ned in Section in that it do es
Comparison with other part whole relations
neither ask for any similarity b etween the members nor do es it ensure dis
jointness of the members instead it only asks for spatial proximity or so cial
connection Hence the Member Collection relation is also a comp osed part
whole relation in that it holds whenever a part as well as all other parts can
b e found in the same place or is so cially connected to the whole However
this do es not really match the intuition given by the examples in Winston et
al such as tree forest ship eet or juror jury These examples t b etter
to the member collection relation de ned in Section b ecause the forest
resp jury eet is the sum of trees resp jurors ships and the forest resp
jury eet can b e decomp osed into trees resp jurors ships which are disjoint
from each other and similar to each other
Portion Mass The example given in Winston et al for this relation
is a slice of pie and it is characterised by the fact that the part called
piece here is not functional for the whole whereas it has to b e similar to
other pieces as well as to the whole in b eing pie The authors say that if the
Portion Mass relation holds b etween two ob jects x y then one can say x is
some of y as in Could you please pass me some of the pie where it would b e
incorrect to pass only the crust of the pie Since this relation strongly dep ends
on the similarity b etween the p ortion and the whole it is parametrised by a
predicate Q which in the example is b eing pie The Portion Mass relation can
b e de ned as follows and thus b elongs to the comp osed part whole relations
Q
x y x y Piece x y where
Q
Portion
Piece x y Q x Q y z z y Q z Overlap x z
Q
w w y Overlap x w Overlap z w
Stu Ob ject According to Winston et al if this relation holds b e
tween x and y one can say that x is made of y and y cannot b e re
moved without altering y s identity Unfortunately the authors do not explain
what a thing s identity is Examples given are alcohol martini steel bike and
hydrogen water In the taxonomy presented in this chapter Stu Ob ject is
clearly a substantial integral part whole relation
Feature Activity The only di erence b etween comp onent integral ob ject
and feature activity is that b oth the part and the whole of the latter have to
b e actions It app ears that this di erence were made only b ecause the inverse
of the former translates in English to has whereas this translation do es not
work for the latter
Chapter Part whole relations
Place Area This relation b etween sp ecial places and areas is said to b e
similar to the p ortion mass relation b esides the fact that places cannot b e
separated from the area they o ccupy in the sense a slice of pie can b e taken away
from a pie The authors distinguish b etween Place Area and the top ological
inclusion relation The rst one also demands that the place is part of the
area it can b e found in whereas for b eing top ologically included by something
it su ces to b e surrounded by it
Consequences for knowledge representation issues
As we have seen in this section there is a great variety of di erent part whole
relations with di erent prop erties and some of them are rather complicated
such as the member collection relation or the comp onent aggregate relation
The rst one is complicated since it describ es a whole as the sum of disjoint
parts all of them satisfying some additional integrity condition The second
one is at a rst glance less complicated but the description or axiomatisation
of functionality is very complex It requires the de nition of correctly func
tioning The expressive p ower of decidable formalisms such as Description
Logics is surely to o weak to de ne the member collection or the comp onent
aggregate relation
The in uences of the ontological principles on the required expressive p ower
are only sketched for two examples namely the prop er part and the atomicity
principle If a Description Logic has the tree mo del prop erty see Section
for a de nition of this prop erty then a concept C is satis able i C is satis
able in a mo del ob eying the prop er part principle Each satis able concept
has a tree mo del which obviously ob eys the prop er part principle Other in
ference problems such as subsumption are either likely to b e reducible to
satis ability in this case they are also invariant under the assumption of the
prop er part principle or the corresp onding inference algorithms can b e easily
extended to ob ey the prop er part principle this is the case for ABox consis
tency Similarly the atomicity principle comes for free in Description Logics
having the nite tree mo del prop erty Satis able concepts of such a Descrip
tion Logic are always satis able in a mo del that has no in nite ascending
chain If a Description Logic do es not have the nite tree mo del prop erty
then one can enforce in nite ascending chains and sp ecial techniques have to
b e applied for testing satis ability in a mo del ob eying the atomicity principle
Such a technique is describ ed in Calvanese
The investigations of the requirements made by all these principles for the ex
pressive p ower of Description Logics go es b eyond the scop e of this work We
Comparison with other part whole relations
concentrated on di erent p ossibilities to extend Description Logics by transi
tive relations and to relate transitive relations to other p ossibly non transitive
relations The results of these investigations together with the consequences
of these results for the representation of part whole relations can b e found in
Chapter
Chapter
Description Logics
In this section the basic Description Logic ALC is formally introduced Start
ing with a short history of Description Logics syntax and semantics of ALC are
introduced as well as a formal de nition of the corresp onding inference prob
lems For a b etter understanding of the inference algorithms presented later
the reasoning techniques used within this work are rst given for ALC Then
it is explained how these reasoning techniques can b e used within a knowl
edge representation system based on Description Logics The main ideas of
knowledge representation systems based on Description Logics are p ointed out
including the description of the main reasoning services of such systems Next
the extensions of ALC by number restrictions and by the transitive clo
sure of roles are introduced Finally the expressive p ower and some mo del
theoretic prop erties of these basic languages are discussed
A brief history of Description Logics
Since the late eighties Description Logics play an imp ortant r ole in the eld
of knowledge representation The origin of Description Logics can b e said
to lie in the necessity to formalise the semantics of semantic networks Sowa
This is a vivid graphical representation formalism where concepts and
individuals are related to each other via lab elled edges For example the
semantic network given in Figure states that elephants are a sp ecialisa
tion of mammals and that Clyde is an elephant Despite the vividness of
semantic networks their lack in formal semantics b ecame irritating esp ecially
when trying to understand answers given by knowledge representation sys
tems based on semantic networks Brachman To overcome this problem
A brief history of Description Logics
Mammal
isa
isa Clyde Elephant
trunk leg color 4 Cylinder
Gray
Figure A semantic network
a new graphical formalism structured inheritance networks was introduced
and implemented Brachman et al Kaczmarek et al Furthermore
structured inheritance networks were provided with a well de ned semantics
which xed the precise meaning of its graphical constructs which led to the
de nition of the rst Description Logics Brachman Schmolze
Unfortunately it turned out that the main inference problems of these rst
Description Logics were undecidable Patel Schneider Schmidt Schauss
This insight and the fact that sound and complete inference algorithms
seem to b e crucial for the tasks knowledge representation systems based on
Description Logics are applied to led to the restriction of the expressive p ower
of subsequent Description Logics and to numerous investigations concerning
their computational complexity Hollunder et al Baader Baader
Hanschke Donini et al a b Hollunder Baader Schmidt
Schau Smolka Hanschke Baader et al Baader Hanschke
Baader Sattler a Sattler Baader Sattler b De Giacomo
et al
Chapter Description Logics
In the close relationship b etween Mo dal Logics and Description Log
ics was discovered and published Schild De Giacomo Lenzerini
a which led to new complexity results esp ecially in the eld of Description
Logics De Giacomo Lenzerini b De Giacomo
In a nutshell these investigations revealed that a tractable Description Logic is
of rather low expressive p ower and that extensions which add some more ex
pressiveness lead at least to PSpace completeness However all these results
only concern worst case complexity Several knowledge representation systems
based on intractable Description Logics are used in applications Rychtyckyj
Kirk et al and tests were undertaken which have shown that De
scription Logics with a high worst case complexity b ehave quite well on real
world or random knowledge bases Bresciani et al Baader et al
Horro cks
The relationship b etween Description Logics and other class based represen
tation formalisms such as ob ject oriented databases entity relationship dia
grams frame based systems has b een investigated since On the one hand
Description Logics can b e viewed as a uni ed framework for these formalism
with the pro t that the precise semantics of Description Logics is inherited to
these formalisms and that reasoning techniques developed for Description Log
ics can b e used for reasoning in these formalisms Bergamaschi Sartori
Buchheit et al Calvanese et al Go ni et al Lenzerini Schaerf
Levy et al Beeri et al On the other hand several exten
sions of Description Logics were motivated by the necessity to capture some
asp ects of the expressiveness of other formalisms Calvanese Lenzerini
Calvanese et al De Giacomo Lenzerini
The basic Description Logic ALC
The Description Logic underlying all investigations in this work is ALC a well
known Description Logic introduced and investigated in Schmidt Schau
Smolka Further investigations concerning extensions and restrictions of
ALC can b e found for example in Hollunder et al Donini et al a
In ALC concepts can b e built using
prop ositional op erators i e and u or t and not
value restrictions on those individuals asso ciated to an individual via a
certain role This includes existential restrictions as in has child Girl
as well as universal restrictions such as has child Human
The basic Description Logic ALC
De nition Let N b e a set of concept names and let N b e a set of role
C R
names The set of ALC concepts is the smallest set such that
every concept name is a concept and
if C and D are concepts and R is a role name then C u D C t D
C R C R C are concepts
In order to x the exact meaning of these concepts their semantics is given in
a mo del theoretic way
I I I
De nition An interpretation I consists of a set called the
I I
domain of I and a function which maps every concept to a subset of
I I
and every role to a subset of such that
I I I
C u D C D
I I I
C t D C D
I I I
C n C
I I I I I
R C fd j There exists an e with d e R and e C g
I I I I I
R C fd j For all e if d e R then e C g
A concept C is called satis able i there is some interpretation I such that
I
C Such an interpretation is called a model of C A concept D subsumes
I I
a concept C written C v D i C D holds for each interpretation I Two
concepts are said to b e equivalent written C D if they mutually subsume
each other
I
For an interpretation I an individual x is called an instance of a concept
I I
C i x C Finally x is said to b e an R successor of an individual w
I
i w x R
Additional prop ositional op erators such as implication will b e used as abbre
viations for example A B stands for A t B In complex concepts such
as those used in the reductions later these abbreviations increase the read
ability Furthermore is used as an abbreviation for the universal concept
i e A t A and is used as an abbreviation for the empty concept i e
A u A
Remark An alternative for the presentation of the semantics of ALC is
to translate concepts into rst order formulae with one free variable This
translation can b e de ned in such a way that the resulting formulae involve
Chapter Description Logics
only variables x y The translation is given by two translation mappings
t t from ALC concepts into the set of rst order formulae over two variables
x y
For each concept name A a unary predicate is introduced and for each
A
role name R a binary relation is introduced For C D p ossibly complex
R
concepts the translation is de ned as follows
t A x t A y
x A y A
t C u D t C t D t C u D t C t D
x x x y y y
t C t D t C t D t C t D t C t D
x x x y y y
t R C y x y t C t R C x y x t C
x R y y R x
t R C y x y t C t R C x y x t C
x R y y R x
Obviously a concept C is satis able i its translation t C is satis able and
x
a concept C is subsumed by a concept D i t C D is valid
x
Remark If a Description Logic allows for negation and conjunction of
concepts such as ALC subsumption and un satis ability can b e reduced to
each other
C v D i C u D is unsatis able
C is unsatis able i C v A u A for some concept name A
Since most Description Logics considered here are in fact prop ositionally closed
this connection b etween satis ability and subsumption will b e heavily ex
ploited We may restrict our attention to one of these problems b oth in the
decidability and in the undecidability pro ofs
Knowledge representation based on De
scription Logics
Until now Description Logics were introduced without mentioning how these
Logics can b e used for knowledge representation In this section the main
ideas concerning knowledge representation based on Description Logics are
introduced as well as an example for more details see Neb el Roughly
sp oken knowledge representation systems based on Description Logics DL
systems for short consist of parts
a reasoner often based on a tableau algorithm
Knowledge representation based on Description Logics
knowledge bases in which the user mo dels the explicit knowledge of an
application domain and
interfaces to the functionalities of the system
In most DL systems there are two di erent kinds of knowledge bases
The terminological knowledge of an application domain is stored in the so
called TBox To this purp ose the TBox contains a set of concept de nitions
which are in their simple version of the form A C for a concept name
A N and a p ossibly complex concept C Concepts o ccurring on the left
C
hand side of a concept de nition are called de ned concept and it dep ends on
the underlying Description Logic which constructors can b e used to build the
concepts on the right hand side
In addition to the terminological knowledge describ ed in a TBox concrete
situations can b e describ ed in the so called ABox p ossibly using the concepts
de ned in the TBox In order to describ e concrete situations individual names
a b are used to refer explicitly to elements of an interpretation domain
Then ABox statements can express that an individual a is an instance of a
concept C written a C or that two individuals a b are related via a role
R written aR b The similarity b etween ABox statements and constraints of
the completion algorithm is not incidental Constraint systems can b e viewed
as sp ecial cases of ABoxes where the sp ecialisation is due to the fact that
constraint systems for ALC and several extension do not contain cycles whereas
ABoxes are not required to b e acyclic
It is straightforward to extend the notion of a model to a TBox T and an ABox
A An interpretation I asso ciates additionally each individual a o ccurring in
I
A to an element a Then I satisfy A and T i they satisfy each of
their statements This is to say that I have the following prop erties
I I
for all A C in T A C
I
for all a C in A a C
I
for all aR b in A a b R
Such an interpretation is then called a mo del of A and T Furthermore a
concept C is said to b e T subsumed by a concept D written C v D i
T
I I
C D holds for all mo dels I of T An individual a is said to b e an instance
I
of a concept C with resp ect to A and T i a C for all mo dels of A and T
Finally an ABox A is called consistent with a TBox T i there exists a mo del
for b oth A and T
A cyclic ABox contains statements a R a a R a a R a
n n
Chapter Description Logics
DL systems vary in whether they allow for cyclic concept de nitions in a TBox
or not A concept A is said to use a concept C if C o ccurs on the right hand
side of A s de nition If the transitive closure of uses contains a concept
that uses itself this TBox is called cyclic For acyclic TBoxes reasoning
with resp ect to a TBox T can b e reduced to reasoning with resp ect to the
empty TBox by unfolding see Neb el Unfolding means substituting all
concepts A which are de ned in the TBox by A C by their de nition C
i i i i
Applying these substitutions to the concepts o ccurring in an ABox consistency
with resp ect to a TBox can thus b e reduced to consistency with resp ect to the
empty TBox Similar subsumption and satis ability of concepts and the
problem whether an individual is an instance of a given concept with resp ect
to a TBox can b e reduced to the corresp onding problems with resp ect to the
empty TBox
In general DL systems employ the unique name assumption which means that
a mo del has to interpret di erent individual names of A as di erent elements
of the interpretation domain Furthermore the open world assumption allows
that statements which are not implied by an ABox and a TBox may hold or
may not hold This will b ecome more clear in the next example
Example The TBox given in Figure contains some basic concept def
initions of the pro cess engineering application There the b ehavioural asp ects
of a device or a connection are separated from their structural asp ects by stor
ing the rst one in its implementation and the second asp ect in its interfaces
This example is incomplete in that concepts such as device implementation
or atomic are not de ned hence there are mo dels of this knowledge base which
are not conform to the engineer s intuition It is easy to see that Device
subsumes b oth Atomic device and Comp device A closer investigation re
veals that Atomic device and Comp device are disjoint which means that an
ABox containing a Atomic device and a Comp device is inconsistent with
this TBox This is due to the fact that a device must b e implemented by
an implementation which must have a description of its b ehaviour and that
device and not this b ehaviour description is atomic for an instance of Atomic
atomic for an instance of Comp device
The ABox in Figure describ es some ob jects of a pro cess mo del Since the
concepts on the right hand side of a are not restricted to de ned concepts
they can b e complex concepts as in the rst statement
With resp ect to the ab ove given knowledge base the question Is valve an
instance of Comp device must b e answered with no One can construct a
mo del of this ABox and TBox where valve has a third part which is neither
Knowledge representation based on Description Logics
Device implemented by Device implementation u
interfaces Device interface
Connection implemented by Connection implementation u
interfaces Connection interfaces u Device
device Device u has part Device t Connection u Comp
implemented by Comp device implementation
Atomic device Device u has part Anything u
implemented by Atomic device implementation
Device implementation
Implementation u behaviour Behav descr
device implementation Atomic
Device implementation u behaviour Atomic
Comp device implementation
Device implementation u behaviour Atomic
Figure Example TBox
valve Device u
implemented by Valve implementation
float Device
chamber Device
valve implementat Comp device implementat
valve has part float
valve has part chamber
valve implemented by valve implementation
Figure Example ABox
a connection nor a device This is due to the ab ove mentioned op en world
assumption Everything implied by the statements in a knowledge base has to
hold in each of its mo dels but prop ositions which are neither implied by these
statements nor contradicting them might hold or might not hold in its mo dels
The basic system services and inference problems of a DL system are the
following
Classi cation Computes the ordering of the concepts de ned in a TBox
with resp ect to the subsumption relation
Chapter Description Logics
Subsumption Tests whether a given concept subsumes another given con
cept p ossibly with resp ect to a TBox
Satis ability Checks whether a given concept can ever b e interpreted as a
non empty set p ossibly with resp ect to a TBox
Consistency Tests whether a given ABox is consistent with resp ect to a
given TBox
Retrieval Finds all those individuals of a given ABox that are instances of
a given concept
Most sp eci c concepts Given an individual it computes the most sp eci c
classes this individual is an instance of
In the following this work will fo cus on satis ability problems for the follow
ing reasons Given that the Description Logics to b e investigated are mostly
prop ositionally closed
undecidability of satis ability of concepts implies that none of the ab ove
mentioned system services are computable
subsumption can b e reduced to satis ability hence algorithms deciding
satis ability can b e used for deciding subsumption see Remark
Furthermore this pro cess engineering application do es not ask for cyclic def
initions if transitive roles are present Hence subsumption with resp ect to a
TBox can b e reduced to subsumption which can b e reduced to satis ability
for Description Logic that are prop ositionally closed
Given that the TBox is acyclic the only inference problems that can not
b e reduced to concept satis ability are the ones involving ABoxes namely
consistency retrieval and most sp eci c concepts Ignoring these problems was
on the one hand the only way to investigate several extensions relevant for
the pro cess engineering application including ABox reasoning to the set of
problems to b e investigated would have gone b eyond the scop e of this work
On the other hand tableau based algorithms deciding concept satis ability
can b e used as a basis for the development of algorithms for ABox inference
problems
For many Description Logics tableau based satis ability algorithms can b e
easily mo di ed such that they are able to decide ABox consistency This is
true for several extensions of ALC namely for those by simple and qualifying
A tableau based algorithm for ALC
number restrictions Hollunder admissible concrete domains Baader
Hanschke admissible concrete domains and generalised existential and
universal value restrictions Hanschke by inverse and b o olean op erators
on roles De Giacomo and many more
A tableau based algorithm for ALC
In general to show decidability of satis ability of concepts of a Description
Logic D one has two p ossibilities
One gives a translation from D concepts to D concepts where D is a
Description Logic whose satis ability problem is known to b e decidable
This translation has to b e invariant for satis ability i e a D concept C
is satis able if and only if its translation C is a satis able D concept
This technique is applied for example in De Giacomo
One presents an algorithm and shows that it decides satis ability of D
concepts That is started with some D concept C it always terminates
and it answers with C is satis able if and only if C is satis able Such
an algorithm is called sound and complete where soundness refers to
the fact that all concepts said to b e satis able are indeed satis able and
completeness refers to the fact that all satis able concepts are found to
b e satis able
Throughout this work the second p ossibility has always b een chosen and
several decision algorithms are presented All of them are tableau based and
the techniques used to prove their termination soundness and completeness
are quite similar In order to understand these techniques b etter and to make
their presentation easier they are demonstrated rst for ALC
Please note that these algorithms are not thought to b e implemented di
rectly but to serve as an implementation base Several optimisation tech
niques have already b een developed and tested and their impact on the p er
formance of these algorithms has b een evaluated In a nutshell it turned out
that these techniques have an enormous in uence on the e ciency of these
algorithms Esp ecially techniques for the e cient treatment of the prop o
sitional part of Description Logics Giunchiglia Sebastiani and several
techniques for early clash detection Baader et al Bresciani et al
Horro cks turned out to b e extremely p owerful The algorithms presented
Chapter Description Logics
here are designed in such a way that interesting prop erties such as soundness
completeness and termination can b e proved in a rather elegant way
The tableau based algorithm presented in the following decides satis ability
of ALC concepts Given an ALC concept C it tries to build a mo del for C
It breaks C down to its sub concepts and tries to satisfy C by satisfying step
by step the constraints induced by sub concepts of C If all these attempts
fail C is unsatis able otherwise one can easily build a mo del for C from
the structures generated while breaking down C
For simplicity all concepts are supp osed to b e in negation normal form NNF
for short This means that negation is applied to concept names only An
ALC concept can b e transformed into an equivalent one in NNF by using the
following equivalences to push negation into concepts
C t D C u D
C u D C t D
R C R C
R C R C
The basic data structure our algorithm works on are constraints For ALC we
use two di erent kinds of constraints For example the constraint x connected
to y ensures that the two individuals x and y stand for are related via the role
connected to and the constraint x Device ensures that the individual x
stands for is an instance of the concept Device
De nition Let fx y z g b e a countably in nite set of individual
variables A constraint is either of the form
xR y where R is a role name in N and x y
R
x D for some ALC concept D and some x
A constraint system is a set of constraints For a constraint system S let
denote the individual variables o ccurring in S
S
An interpretation I is a model of a constraint system S i there is a mapping
I
such that I satisfy each constraint in S i e
S
I
x y R for all xR y S
I
x D for all x D S
A tableau based algorithm for ALC
Conjunction If x C u C S and x C S or x C S then
S S fx C x C g
Disjunction If x C t C S and x C S and x C S then
S S S fx C g
S S S fx C g
Value restriction If x R C S y is an R successor
of x in S and y C S then
S S fy C g
Existential restriction If x R C S and there is no
R successor y of x in S with y C S then
S S fxR z z C g for a new variable z n
S
Figure The completion rules for ALC
For a constraint system S individual variables x y and role names R we say
i
that y is an R R successor of x in S i there are y y such
m m
that x y y y and fy R y j i m g S S contains a clash
m i i i
i fx A x Ag S for some concept name A and some variable x A
S
constraint system S is called complete i none of the completion rules given
in Figure can b e applied to S
Figure introduces the completion rules that are used to test ALC concepts
for satis ability The completion algorithm works on a tree where each no de is
lab elled with a constraint system It starts with the tree consisting of a single
leaf the ro ot lab elled with S fx C g where C is the ALC concept in NNF
to b e tested for satis ability A rule can only b e applied to a leaf lab elled with
a clash free constraint system Applying a rule S S for i n to such
i
a leaf leads to the creation of n new successors of this no de each lab elled with
one of the constraint systems S The algorithm terminates if none of the rules
i
can b e applied to any of the leaves In this situation it answers with C is
satis able i one of the leaves is lab elled with a clash free constraint system
Example If this algorithm tests
C A u R C t D u R C
Chapter Description Logics
for satis ability it starts with the constraint system fx C g and after two
applications of Rule we have
S fx C x R C t D u R C x A x R C t D x R C g
Next only Rule can b e applied which adds xR y and y C to S Then
Rule can b e applied which adds y C t D and yields S Finally Rule
yields
S S fy C g
S S fy D g
The constraint system S contains a clash b ecause fy C y C g S In
contrast S do es not contain a clash nor can any rule b e applied to it Hence it
is a complete clash free constraint system and C is satis able Furthermore
I
S can b e used to build a mo del I of C The interpretation domain is
simply the set of variables o ccurring in S and concepts as well as roles are
interpreted according to the constraints in S i e
I
fx y g
S
I I
A fx j x A S g fxg
I I
D fx j x D S g fy g
I I
C fx j x C S g
I I I
R f x y j xR y S g f x y g
In order to show that this completion algorithm really yields a decision pro ce
dure for ALC it su ces to pro of the following lemma
Lemma Let C b e an ALC concept in NNF and let S b e a constraint
system obtained by applying the completion rules to fx C g Then
For each completion rule R that can b e applied to S and for each in
terpretation I we have I is a mo del of S i I is a mo del of one of the
systems S obtained by applying R
i
If S is a complete and clash free constraint system then S has a mo del
If S contains a clash then S do es not have a mo del
The completion algorithm terminates when applied to fx C g
A tableau based algorithm for ALC
Remark Indeed Lemma immediately implies decidability of satis
ability of ALC concepts When the completion algorithm terminates it stops
with a tree whose leaves are all lab elled with complete constraint systems If
C is satis able then fx C g is also satis able and thus one of the complete
constraint systems is satis able by By this system must b e clash free
Conversely if one of the complete constraint systems is clash free then it is
satis able by and b ecause of this implies that fx C g is satis able
Hence the algorithm is a decision pro cedure for satis ability of ALC concepts
Lemma can b e proved as follows
is proved by investigating all rules separately Since all constraint systems
S generated by a rule are sup ersets of their ancestor S it follows immediately
i
that each mo del of S is also a mo del of S The other direction dep ends on
i
the resp ective rule but is straightforward in most cases
can b e proved by constructing a so called canonical model for S from a
clash free complete constraint system S In Example this was realized as
follows Variables in the constraint system b ecame individuals of the canonical
mo del and concepts and roles were interpreted according to the constraint
system For other Description Logics the construction of the canonical mo del
can b e more complicated
is straightforward It su ces to show that no interpretation can satisfy a
constraint system containing fx A x Ag
can b e proved by giving a termination order This is a set A ordered
by some well founded order Well foundedness ensures that there is no
in nite chain a a a Each constraint system S is asso ciated
to size S A in such a way that if S is obtained from S by application of
i
one of the completion rules then size S size S This prop erty together
i
with well foundedness and the fact that each rule generates only nitely many
constraint systems implies termination For extensions of ALC this technique
is applied in Baader Hanschke and Section
Intuitively the following prop erties of the completion algorithm are resp onsible
for its termination
All concepts of constraints added by the completion rules are subconcepts
of the concept C with which the algorithm started It is not di cult to
see that the number of sub concepts of a concept C is linear in the length
of C
Chapter Description Logics
If a rule R can b e applied to a constraint system S then this is b ecause
of the presence of a particular constraint a C in S and the application
of R adds constraints whose concepts are strictly shorter than C
The introduction of role successors is always triggered by a constraint
of the form x R C hence each variable has only nitely many role
successors
Once a variable is introduced it is present in all subsequent constraint
systems and constraints are never removed these two prop erties prevent
the algorithm from lo oping
Although the completion algorithm is sound and complete it is far from b eing
optimal For example when started with the following concept C whose
n
length is quadratic in n it generates a constraint system whose size is ex
p onential in the length of C i e the variables introduced by the algorithm
n
together with the role successorship induced by constraints xR y build a binary
tree of depth n
C R A u R A u
R R A u R A u
R R R A u R A u
R R R R A u R A u
R R R R R A u R A u
R A u R A R R
z
n times
When started with C the completion algorithm needs space exp onential in
n
the size of its input C which is more than necessary In Schmidt Schau
n
Smolka Donini et al a a di erent completion algorithm is presented
which needs space that is only p olynomial in the length of the input concept
and which works also for extensions of ALC Indeed it was shown that this
result is optimal since satis ability of ALC concepts is PSpace complete
Similar to the completion algorithm presented here these PSpace algorithms
exploit the fact that for ALC concepts it is su cient to try to build a tree
model for the input concept see Section for details As a consequence
of this fact the variables of a constraint system together with the role succes
sorship induced by constraints of the form xR y build a tree In contrast to
the completion algorithm presented here the PSpace algorithms handle the
branches of this tree indep endently Once a branch has b een investigated one
A tableau based algorithm for ALC
a R R A a R R A a R R A a R R A a R R A a R R A
a R A a R A a R A a R A a R A a R A
a R a R a R a R a R a R
a A a A
x A x R A x R A x R A
x A
x A x A x A
x A
Figure A non terminating application of the completion rules
can forget ab out it and reuse the space where it was stored provided one keeps
track of the results of these partial investigations
As already mentioned at the end of Section tableau based algorithms like
the completion algorithm presented ab ove for ALC that decide satis ability of
concepts can b e often mo di ed such that they decide ABox consistency To
underline this fact we have de ned ABoxes and constraint systems in such a
way that the latter are a sp ecial case of the former In most cases it su ces
to de ne a strategy for the application of the completion rules in order to
guarantee termination of the completion algorithm A strategy is mainly an
ordering in which the completion rules applied For example the following
ALC ABox can lead to an in nite application of completion rules if the rules
are applied in the same ordering as in Figure More precisely the non
termination of this example is due to the fact that generating and identifying
rules are applied in an arbitrary order Termination for ALC ABoxes can b e
easily guaranteed by a simple strategy which prefers rules on old individuals
to rules on individual variables introduced by the completion algorithms
A f a R R A
a R A
a R
aR ag
Chapter Description Logics
Extensions of ALC
In this section several known extensions of ALC are introduced They will
serve as a basis for further extensions which are describ ed in the Chapters
and The expressive p ower of ALC and these extensions are sketched in
Section
ALC extended by Complex Roles
Transitivity in Description Logics came until recently Sattler always
in the shap e of the transitive closure of roles Baader Schild De
Giacomo Lenzerini Calvanese et al The rst extension of ALC
by the transitive closure of roles was presented in Baader The author
presents the Description Logic ALC which extends ALC by allowing for regu
reg
lar expressions over role names in the place of role names In Baader a a
sound and complete tableau based algorithm is given that decides satis ability
of ALC concepts In the following ALC as well as a restricted version of
reg reg
this logic ALC is presented
De nition Starting with role names in N regular roles are built using
R
the role constructors comp osition R S union R t S and transitive closure
R
ALC is obtained from ALC by allowing additionally for regular roles inside
reg
concepts
ALC is the extension of ALC obtained by allowing additionally for each role
R N the use of its transitive closure R inside concepts
R
The extension of the semantics to these complex roles is straightforward
I I
De nition An interpretation I of an ALC concept resp
reg
ALC concept is an interpretation of ALC concepts that satis es additionally
the following equations
I I
I
R t R R R
I I I I I I
R R R R f d f j There exists e with
I I
d e R e f R g
I I I
I I i I n
R R where R R R R
i
z
n times
Extensions of ALC
For a regular role R an individual x said to b e an R successor of an individual
I
w in I i w x R
Extending ALC by comp osition or disjunction of roles do es not change the
expressive p ower of ALC b ecause of the following equivalences
R S C R S C R t S C R C t S C
R S C R S C and R t S C R C u S C
In contrast extending ALC by the transitive closure of roles really increases
its expressive p ower see Baader b for a formal de nition of expressive
p ower On one hand ALC and ALC concepts cannot b e translated to
reg
equivalent rst order formulae whereas this is p ossible for ALC concepts see
Remark This is due to the fact that the transitive closure of a relation
cannot b e expressed in rst order logic in contrast to transitivity On the
other hand ALC loses the nite tree model property when extended by the
transitive closure of roles
As a consequence of this second fact algorithms such as tableau based algo
rithms that try to construct a mo del of a concept containing the transitive
closure of roles need sp ecial cycle detection mechanisms Baader that
prevent them from lo oping For example a naive mo di cation of the comple
tion algorithm for ALC presented in Section would introduce an in nite
number of R successors when started with x C for
C A u R A u R R A
A terminating mo di cation of this algorithm must notice that all these R
successors involve the same constraints and then stop the generation of new
successors Moreover this mo di cation must recognise cases where the sat
isfaction of constraints is p ostp oned in each step This could happ en for
example while trying to construct a mo del of
A u R A C A u R
A naive mo di cation of the completion algorithm for ALC would start with
fx C g and would run into an in nite lo op while trying to generate an R
successor of x in A This is due to the fact that the constraint x R A
i
can b e satis ed by any R successor of x and that the length i of this R chain
A Description Logic has the nite tree mo del prop erty if each satis able concept has
also a mo del that has the shap e of a nite tree see Section for details
Chapter Description Logics
is not xed or b ounded by C in an obvious way Summing up a sound and
complete mo di cation of the completion algorithm for ALC has to distinguish
b etween cases where constraints on individuals propagated along some p ossi
bly in nite role chain are simply re generated but satis ed as in the example
with C and cases where the satisfaction of constraints is always p ostp oned
as in the example with C
Unfortunately these cycle detection mechanisms need to store a large amount
of information and this cannot b e accomplished using p olynomial space As
stated in Fischer Ladner Pratt satis ability of Prop ositional Dy
namic Logic PDL formulae is ExpTime complete A translation of this
result from PDL to the vocabulary of Description Logics namely to ALC
reg
can b e found in Schild This translation is b eside others satis ability
preserving That is a PDL formula is satis able if and only if its translation
into an ALC concept C is satis able Since ALC is prop ositionally closed
reg reg
subsumption can b e reduced to satis ability and vice versa and since these re
duction are linear subsumption of ALC concepts is also ExpTime complete
reg
Finally the hardness pro of in Fischer Ladner do es not make any use
of comp osition or disjunction of roles hence ALC is also ExpTime complete
ALC extended by Number Restrictions
Since ob jects are often characterised by the number of ob jects they are related
to via some relation number restrictions are present in almost all implemented
systems Brachman Schmolze Patel Schneider et al Peltason
MacGregor Brill Baader et al They o ccur in their most sim
ple form in ALCN an extension of ALC introduced in Hollunder et al
Donini et al a ALCN is obtained from ALC by allowing additionally for
constructs which describ e for example that an individual has at least parts
or at most interfaces
De nition ALCN resp ALC N is obtained from ALC resp ALC by
allowing additionally for concepts of the form n R and n R where
are nonnegative integers and R is a role name n n
I I
An interpretation I of an ALCN concept resp ALC N concept
is an interpretation of ALC concepts that additionally satis es the following
equations
I I I I
n R fd j fe j d e R g ng
I I I I
n R fd j fe j d e R g ng
Extensions of ALC
where M denotes the cardinality of a set M
Similar ALC N and ALC N denote the extensions of ALC and ALC by
reg reg
number restrictions on role names This is to say that in ALC N and ALC N
reg
the complex roles available in ALC and ALC are not allowed inside number
reg
restrictions here only role names can b e used
Since ALC is prop ositionally closed we can express all relations in f g
inside number restrictions for example n R n R and n R
n R u n R
As shown in Donini et al a adding number restrictions to ALC do es not
really increase its computational complexity Satis ability and subsumption
of concepts are PSpace complete
In Hollunder Baader a more expressive variant of number restrictions
namely so called qualifying number restrictions are presented and investigated
Qualifying number restrictions are of the form n R C or n R C and
they are interpreted as the set of those individuals having at least resp at
most n R successors that are instances of C i e
I I I I I
n R C fd j fe j d e R g C ng
I I I I I
n R C fd j fe j d e R g C ng
As a consequence of the tableau based algorithm presented in Hollunder
Baader adding qualifying number restrictions to ALC preserves the nite
tree mo del prop erty
Expressive p ower of the basic Description Logics
In general logics can b e classi ed with resp ect to the form the mo dels of
satis able formulae have Considering the form that mo dels of satis able con
cepts of a certain Description Logic have is useful b ecause this form is closely
related to its expressive p ower For example if a Description Logic has the
nite mo del prop erty then no concept can b e built within this logic which
enforces an in nite mo del
More precisely each satis able concept of a logic having the nite model prop
erty has a mo del with nite interpretation domain Given that mo del checking
is decidable the nite mo del prop erty implies that the set of all satis able
concepts is recursively enumerable Furthermore most Description Logics can
b e viewed as fragments of rst order logic see Remark for a translation
Chapter Description Logics
from ALC to rst order logic for which unsatis able formulae are known to
b e recursively enumerable Together this yields decidability of satis ability
of these logics Deciding whether a concept C is satis able can b e done by
rst translating C into a corresp onding rst order formula x and then
C
testing in parallel whether x is unsatis able and whether interpretations
C
I are mo dels of x Given that these interpretations are completely enu
j C
merated from small ones to larger ones this yields a decision pro cedure for
Description logics having the nite mo del prop erty However in general this
pro cedure is far from b eing optimal and do es not provide any upp er b ounds for
the computational complexity of the satis ability problem under consideration
A language not having the nite mo del prop erty is not necessarily undecidable
see Gr adel et al for example but a lack of this prop erty is a go o d
hint that reasoning might b e very di cult In fact all Description Logics
investigated in this thesis which lack the nite mo del prop erty turn out to
have undecidable satis ability problems
Satis able concepts of a Description Logic having the tree model property have
some mo del in the form of a tree If C is satis able then it has a mo del I
such that
I I
there is some x C that has no R predecessor in for all roles R
I
each element of can b e reached from x via a role chain and
I I
each y x in is an R successor of exactly one y for one role
name R
Finally if a Description Logic has the nite tree model property then each
satis able concept has a nite tree mo del
In contrast to ALC ALCN has the nite tree model property which will not
b e lost even when extended by allowing for quali ed number restrictions For
ALC the p ossibility to use the transitive closure of a role inside value restric
tions is the reason for the lack of the nite tree mo del prop erty For exam
ple the following concept describ es instances of A having some R successor
in A and where each individual reachable over some R path has itself some
R successor in A
A u R A u R R A
Obviously this concepts is satis able but each of its mo dels has either an
in nite R chain or it contains some R cycle Nevertheless ALC still has the
Role chains are roles of the form R R
n
Extensions of ALC
tree model property which means that each satis able ALC concept has a not
necessarily nite tree mo del
The expressive p ower added by the transitive closure to ALC b ecomes also
apparent in the following example
part is made of Carcinogenic Device u has
part one can refer to parts at some level Using the transitive closure of has
of decomp osition not known in advance without restricting this level a priori
If this maximum level could b e restricted to say n those dangerous devices
can b e describ ed by
has part is made of Carcinogenic t
has part has part is made of Carcinogenic t
has part has part has part is made of Carcinogenic t
part has part made of Carcinogenic has is
z
n times
Unfortunately b eside the necessity of xing an upp er b ound n a large number
of disjunctions over complex concepts is involved which leads in general to
p erformance degradation of the inference algorithm One can think of opti
mising techniques which exploit the fact that these disjuncts are of a similar
structure but it is probably more e cient to directly use the transitive closure
of roles together with sp eci c optimisation techniques
When constructing an algorithm for testing satis ability of concepts tree
mo del prop erties turn out to b e very useful Tableau based algorithms in
general test satis ability by trying to build a canonical mo del For logics
having the nite tree mo del prop erty one has only to consider mo dels having
a nite tree structure which makes the design of these algorithms easier For
most of these logics the attempt can b e stopp ed when a tree is constructed
whose depth is p olynomially or exp onentially b ounded by the length of the
input concept Furthermore it is easy to avoid the algorithm from lo oping and
thus to ensure termination while investigating tree structures given that the
outdegree can b e b ounded which is usually the case In arbitrary structures
sp ecial cycle detection techniques have to b e applied to yield termination
Additionally to pro of the algorithm s soundness is mostly easy if the canonical
mo del constructed for a satis able concept C has the form of a nite tree By
induction on the depth of this tree it can b e shown that the canonical mo del
satis es all constraint on variables at this depth In the presence of cycles or
in nite chains this is no longer p ossible and the pro of might b ecome more di cult
Chapter Description Logics
Summing up these mo del prop erties strongly in uence the reasoning tech
niques one has to apply As a rule of thumb reasoning in tree mo dels is
mostly easier than in arbitrary ones and reasoning in nite mo dels is mostly
easier than in p ossibly in nite ones On the other hand these mo del prop er
ties are closely related to the expressive p ower of a logic For example using
a Description Logic with the tree mo del prop erty it is imp ossible to describ e
devices where all those devices it is connected to are controlled by the same
control unit see Chapter concept and that are connected to at least
two devices A concept describing devices with these prop erties has clearly
no tree mo del since such a device is always related to a control unit via at
least two di erent paths For an application where it is crucial to describ e and
reason ab out similar concepts the expressive p ower of a Description Logic
having the tree mo del prop erty is not adequate Similarly if the application
asks for concepts describing in nite structures only a Description Logic having
the nite mo del prop erty is not expressive enough
Chapter
Prototype Implementation
In this chapter the integration of the DL system Crack Bresciani et al
into the pro cess mo deling to ol ModKit Bogusch et al is motivated and
describ ed This integration is a part of the co op eration b etween the Lehrstuhl
f ur Proze technik and the Lehr und Forschungsgebiet Theoretische Informatik
of RWTH Aachen and was realised with the supp ort and involvement of R
Bogusch and W Ge ers
The mo deling to ol ModKit
ModKit was developed at the Lehrstuhl f ur Proze technik at the University
of Technology in Aachen in order to empirically evaluate the ideas formalised
within VeDa Baumeister et al It is implemented in G which is a
to ol for developing and executing real time exp ert systems Gen G was
originally applied to pro cess monitoring and control tasks within the chemical
and manufacturing industries Its exible development environment has led to
many other application domains not all of which are centred around real time
pro cessing
G allows for the de nition of classes rules and pro cedures activated by the
user or triggered by particular events However despite the fact that G allows
for multiple inheritance the p ossibilities to describ e classes resp ectively the
ob jects b elonging to these classes are rather restricted For example it is not
p ossible to re ne the range of an attribute whose range is already sp eci ed in a
This can b e compared with the p ossibility to de ne in Description Logics one concept
as a sub concept of more than one concept de ned in the TBox
Chapter Prototype Implementation
sup erclass As a consequence the class hierarchy in ModKit is p o orly struc
tured in that the description of a class contains mostly only the name of the
according sup erclasses and no further information ab out what distinguishes
the instances of this class from the instances of its sup erclasses
Apart from this weakness pro cess mo deling using ModKit can b e said to
b e rather comfortable In general setting up a mo del in ModKit works
as follows First the structure of the mo del is sp eci ed by choosing devices
and connections from a catalogue see Figure This is done by dragging
and dropping items representing devices or connections from the catalogue
into the current workspace Next these items are connected using prede ned
connection p osts and their prop erties and b ehaviour is sp eci ed using the
dialogues built for these purp oses see Figure If comp osed items were
chosen their comp onents are sp eci ed in a way similar to the one that is
tro dden to sp ecify a whole mo del This pro cedure is strongly supp orted by on
line help and the do cumentation of the decisions taken and the choices made
during this pro cedure is supp orted by prede ned forms attached to the ob jects
representing devices and connections Once the mo del is completely sp eci ed
the simulation of its b ehaviour can easily b e realised due to the integrated
simulation to ols and source co de generators The latter automatically generate
the source co de for the former from the sp eci ed mo del Finally the whole
pro cess of setting up a mo del is supp orted by the formalisation of the actions
undertaken on the way towards a pro cess mo del Lohmann Krobb
However the design of a mo del is a highly creative pro cess and can thus only
partially b e guided
The terminological knowledge representa
tion system Crack
Crack implemented at the Istituto p er la Ricerca Scienti ca e Tecnologica
IRST in Trento Italy Bresciani et al is a DL system whose develop
ment was driven by the need for a DL system
that implements a Description Logic with high expressive p ower namely
an extension of ALC by features i e functional roles which are a simple
form of number restrictions inverse roles the intersection of roles and
individuals which allow the user to enumerate concrete individuals and
state for example that ob jects are related to one of these via a sp eci c
role by connected to one of fvalve valve valve g
The terminological knowledge representation system Crack
Figure A ModKit screenshot showing the catalogue and the structure of
the ethylene glycol pro cess
whose algorithms are based on sound and complete tableau based algo
rithms and
that is mo dular which means that it can easily b e extended to more
expressive Description Logics namely those encompassing new concept
or role forming constructors
The implementation of Crack had to cop e with two challenges On the one
hand the goal of b eing mo dular can only b e achieved by a rather direct imple
mentation of the completion rules of the tableau algorithm which were not
designed for implementation purp oses but for pro of purp oses On the other
hand the implementation has to b e highly optimised otherwise the high
computational complexity of the underlying Description Logic would prevent
any realistic application Fortunately the implementation reached b oth goals
Chapter Prototype Implementation
Figure A ModKit screenshot showing the dialogue for the prop erty sp ec
i cation of a reactor
The optimisation techniques applied proved to b e so e cient that even large
knowledge bases can b e classi ed within an acceptable amount of time
The authors of Crack and Fact Horro cks Gough are working on a
descendant of b oth Crack and Fact Fact is an implementation of an exten
introduced in Chapter which was motivated by the promising sion of ALC
R
results concerning the computational complexity describ ed there We hop e
to b e able to replace Crack by this new DL system b ecause it will b e more
appropriate for the representation of aggregated ob jects see the discussion at
the end of Section
The integration of Crack into ModKit
The ideas concerning the integration of Crack into ModKit are results of
a fruitful co op eration with R Bogusch In order to use the system services
The integration of Crack into ModKit
provided by Crack for a b etter structuring of the classes and ob jects in Mod
Kit these classes must b e translated into Crack concepts Given that the
engineer should b e supp orted when de ning new classes this translation has to
b e generated automatically In Figure the architecture of the integration
is presented The translator is a ModKit program which realises the ab ove
mentioned translation of ModKit class descriptions into concepts in Crack
syntax For the translation of a ModKit class C b esides the information
sp eci ed in the de nition of C explicitly the translator takes also into account
information concerning instances of C that is available elsewhere For example
if a ModKit relation R is de ned in such a way that its domain is C and its
range is another class D then the translator explicitly states in the translation
of C that all ob jects related to instances of C via the relation R are instances
of D While realising the integration the ModKit class de nitions were mo d
i ed in such a way that the ModKit class hierarchy and the taxonomy of the
corresp onding concepts agree This could b e realised rather easily b ecause the
number of classes was still small In the following we assume that for already
de ned classes its place in the ModKit hierarchy is identical to the place of
its corresp onding Crack concept in the taxonomy
The communication b etween ModKit and Crack is realised by a bridge
process which was implemented in C by A Beb er of IRST in Trento Italy
when he was a guest at the Lehrstuhl f ur Proze technik of RWTH Aachen
The bridge pro cess is invoked via the GSI interface of ModKit using Unix
so ckets The bridge pro cess and ModKit communicate via the GSI interface
of ModKit This communication as well as the tailoring of the bridge pro cess
to the sp eci c requirements of the overall integration was accomplished by W
Ge ers of the Lehrstuhl f ur Proze technik
For mo dularity reasons Crack is not launched automatically together with
ModKit Crack is launched and connected to ModKit by simply clicking
on a ModKit action button and then the basic system services are called
across the bridge pro cess These system services are called automatically from
built in routines in order to supp ort the engineer in structuring b oth ModKit
classes as well as ModKit individuals
ModKit classes have an attribute crack description whose value a string
is generated automatically and set by the translator The way actions trig
ger the invocation of the translator guarantees that the translation of each
class is up to date b efore Crack system services are invoked The attribute
crack description contains in Crack syntax all the information concern
ing the corresp onding class that are available from ModKit As a matter of
fact this description cannot contain any knowledge concerning triggers rules
Chapter Prototype Implementation
G2 Unix Lisp
knowledge- base TBox ABox
C- Inference translator bridge System
user Crack interface
ModKit
GSI- API
interface
Figure The architecture of the integration of Crack into ModKit
or pro cedures asso ciated to a class Hence the information concerning a class
that is visible to Crack is incomplete As a consequence no ModKit class
will b e migrated in the class hierarchy automatically b ecause Crack inferred
that the corresp onding concept has a place in the taxonomy that is di erent
from the place of the corresp onding ModKit class
The answers given by Crack are used as follows The simplest way in which
Crack can b e used is the visualisation of the taxonomy of the concepts that
are translations of ModKit classes More sophisticated uses of Crack s sys
tem services are describ ed in the following
Mo difying and reclassifying prototypes
When creating mo dels using ModKit a common technique is to copy an ob
ject o a so called prototype representing a device or a connection from the
catalogue or from another mo del and to mo dify it according to the sp eci c
needs of the current mo del If o was instance of class C when it was copied it
continues to b e an instance of C even if its prop erties are completely changed
In ModKit like in many ob ject oriented systems an ob ject continues b e
ing an instance of the class it was instantiated from until this relationship is
changed by hand For example when lo oking more closely at mo dels that were
created using ModKit we found an atomic device that was mo di ed in such
The integration of Crack into ModKit
a way that it had parts For a b etter reuse of already de ned ob jects and for
a b etter co op eration among several engineers an ob ject o should always b e an
instance of one of the classes it ts b est The meaning of tting b est is
explained in the following We assume that o s prop erties are p ossibly partially
given that is o has all the prop erties that are stated explicitly plus p ossibly
other prop erties Then a class C is said to t if o could b e an instance of C
without causing inconsistencies whatever additional prop erties o might have
A class C is said to t b est an ob ject o if no class that is more sp eci c than C
also ts o For one ob ject o there might b e several classes that t b est o Like
other ob ject oriented systems ModKit allows only for single instantiation
that is each ob ject is an instance of exactly one class As stated ab ove an
ob ject o is not necessarily an instance of some class it ts b est and it would
already b e helpful if this could b e ensured
However if the migration of ob jects to instances of classes they t b est is
realised by the engineer it b ecomes costly p ossibly incorrect and can cause
inconsistencies It can b e supp orted by applying the system services of Crack
namely the retrieval of the most sp eci c concepts for an ob ject The integration
of Crack into ModKit supp orts this migration as follows Each time the
prop erties of an ob ject are changed Crack is automatically asked to compute
the most sp eci c concepts the translation of this ob ject is an instance of The
corresp onding classes are recommended to the user to migrate this ob ject to
He or she can then choose one of these classes as the new class the ob ject under
consideration is an instance of
De ning a new class
For the de nition of a new class O Crack is used as follows Roughly sp oken
the user sketches the class de nition in a dialogue designed for this purp ose
Then this class de nition is automatically translated into a Crack concept
and classi ed into the already existing taxonomy The results of this classi
cation are shown to the user which can then investigate the subsumers and
subsumees of O s translation and change the de nition of the class if its place
in the taxonomy do es not match his or her intuition Only when O s place
in the taxonomy is accepted by the user O is added as a new class having
the subsumers and subsumees prop osed by Crackas sup er and sub classes
In this way ModKit s class hierarchy and the corresp onding taxonomy are
always kept identical
Chapter Prototype Implementation
Exp eriences
Following the argumentation of this thesis the expressive p ower of Crack is
not high enough for this application However Crack was the most expressive
DL system with sound and complete inference algorithms available at the time
the integration started Fortunately all automatically deducible information
concerning ModKit classes and ob jects can b e expressed in Crack Hence
even a more sophisticated translator would only output concepts that can b e
handled by Crack On the other hand there are prop erties of classes and
ob jects that cannot b e deduced by any translator for example those implied
by triggers or rules As a consequence no automatic changes of the class
hierarchy or ob jects are triggered by answers of Crack
The rst p ositive asp ect of this integration showed up while realising the in
tegration The ModKit class hierarchy which has grown constantly over the
last years was examined in detail Inconsistencies were detected and improve
ments towards a b etter deep er structure of this hierarchy were made This
included the mo di cation of ModKit class descriptions When the integration
was started there were many classes for which the only information present in
ModKit was the set of their sup er and sub classes The motivation for the
introduction and the meaning of these classes could only b e guessed from their
names Translating these classes and arranging the corresp onding concepts
with resp ect to the subsumption relation revealed this lack of description led
to discussions among the pro cess engineers and to a b etter declarative de
scription of these classes An obvious advantage of these mo di cations is that
new engineers working with ModKit can understand more easily the class
hierarchy
The second p ositive e ect is due to the supp ort the integration of Crack
into ModKit provides for the migration of ob jects and classes The system
services provided by Crack guide the user in the migration of ob jects whose
prop erties were mo di ed into classes that t b est and in the de nition of new
classes for new types of ob jects This cleaning up has still to b e done by
hand by the engineers but they are now supp orted by the system services
of Crack This supp orts b oth increases the motivation of the engineers to
do this cleaning up as well as decreases the risk of senseless migrations or
inconsistencies
However the system services available in current DL system can still b e ex
tended to provide an even b etter supp ort For example given a set of ob jects
a system service that automatically generates a class description for a p ossibly
new class that ts b est this set would b e very useful Furthermore a system
Exp eriences
service that given a set of ob jects partitions this set into subsets of ob jects
which are relatively similar to each other would b e useful It could b e used to
determine sets of ob jects for which it would b e useful to introduce new classes
These system services are not yet available in current DL systems To provide
these services a precise de nition of their functionality is required and ade
quate algorithms have to b e developed These investigations will b e part of
future work carried out in co op eration b etween the Lehr und Forschungsgebiet
Theoretische Informatik and the Lehrstuhl f ur Proze technik at the University
of Technology in Aachen
Chapter
Extending Description Logics by
Expressive Number Restrictions
There are two ways in which we increase the expressive p ower of traditional
number restrictions in Description Logics First we allow the use of complex
roles inside number restrictions These constructors enable us for example
to express statements such as for this ob ject we mo del at least phenom
ena that are also mo deled for its neighbours by using the number restriction
connected to has phenomena u has phenomena In this example
comp osition and conjunction u of roles are used inside a number restric
tion Other role forming constructors investigated in this chapter are union
t and inversion Second in symbolic number restrictions numerical
variables can b e used in the place of nonnegative integers Thus one has
no longer to x a number for the lower or upp er b ound of role successors
but one can express statements such as having the same number of input s
and output s or having less output s than the ob jects connected to it have
input s
Before investigating each of these Description Logics in detail these extensions
of ALCN resp ALC N and ALC N are introduced and their expressive
reg
p ower is highlighted by some examples In the following sections the compu
tational complexity of the corresp onding inference problems is investigated
ALC extended by Number Restrictions on Complex Roles
Several extensions of ALCN and ALC N obtained by allowing di erent kinds
of complex roles inside number restrictions will b e investigated and we start
by introducing a scheme for building these extensions The name of such a
scheme consists of the name of the basic language followed by the set of role
constructors that are allowed inside number restrictions
De nition Starting with atomic roles from a set N of role names com
R
plex roles are built using the role constructors comp osition R S union
R t S intersection R u S and inversion R
For a set M ft u g of role constructors we call a complex role R an
M role i R is built using only constructors from M
The set of ALCN M concepts resp ALC N M concepts and ALC N M
reg
concepts is obtained from ALCN concepts resp ALC N and ALC concepts
reg
by additionally allowing for M roles inside number restrictions
I I
An interpretation I of an ALCN M concept resp ALC N M
concept is an interpretation of ALC N concepts that satis es additionally
the following equations
I I
I
R t R R R
I I
I
R u R R R
I I I I I I
R R f d f j e d e R e f R g
I I I I
R f e d j d e R g
Comp osition is present in all extensions investigated in this thesis On the
one hand comp osition strongly increases the expressive p ower in that it allows
to restrict the number of role chain successors In simple cases this can also
b e realized using pure ALC For example using three disjoint concepts A A
and A the concept R S can b e rewritten as
R S A u R S A u R S A
However this do es not always work ALCN has the tree mo del prop erty even
when extended by qualifying number restrictions As a consequence ALCN
is not able to express upp er b ounds for the number of role chain successors
thus enforcing intermediate successors to share successors which can easily b e
expressed using ALCN For example the concept
D R u R S u R S
is obviously satis able but each of its instances x has an R S successor which
is connected to x via at least di erent role paths This example shows that
ALCN do es not have the tree mo del prop erty Furthermore ALCN can
Chapter Expressive Number Restrictions
b e said to b e more expressive than ALCN b ecause the fact that ALCN has the
tree mo del prop erty implies that no ALCN concept is equivalent to D
This kind of expressive p ower added by comp osition in number restrictions is of
strong interest for the pro cess engineering application b ecause the devices and
connections of a pro cess mo del are in general not interrelated in a tree like
way but in more complex structures Hence the p ossibility to describ e non
tree structures is imp ortant for this application Since comp osition in number
restrictions provides this p ossibility this constructor will b e investigated in
detail while lo oking for an Description Logic adequate for the representation
of the standard building blo cks of pro cess mo dels
On the other hand decidability results for ALCN M without comp osition
follows immediately from results in Gr adel et al They are discussed in
Section
Intersection on roles inside number restrictions is needed if one wants to
express for example that a device is p owered by something it is connected to
i e
Device u powered by u connected to
Intersection in combination with comp osition comes in for example if we
want to describ e devices that share at least p ower supply with devices they
are connected to i e
Device u powered by u connected to powered by
Union of roles is needed to restrict the total number of individuals that are
related via several not necessarily disjoint roles For example connected to
and has part are not necessarily disjoint roles and union can b e used to
describ e complex devices which are connected to or have as a part at least
say ob jects i e
Device u connected to t has part
The ab ove concept could b e rewritten by a disjunction of concepts involving
intersection of roles inside number restrictions but only in a very complicated
concept i e
Device u has part t
connected to u has part u has part u connected to t
connected to u has part u has part u connected to t
connected to u has part u has part u connected to t
connected to u has part u has part u connected to t
connected to u has part u has part u connected to t
connected to
Apart from the unreadability of the ab ove concept there is another argument
against the use of intersection instead of union inside number restrictions As
we will see later intersection leads to an even higher complexity than union
Union in combination with comp osition comes in for example if we want to
describ e a lo osely connected part of a mo del The following concept describ es
a device that b elongs to such a lo osely connected part
Device u connected to t
connected to connected to t
connected to connected to connected to
Inversion of roles can b e used to express for example that a p ower supply
supplies p ower for more than one device i e
Power supply u powered by
Inversion in combination with comp osition is needed for example if we
want to express that all parts of a device b elong exclusively to this device
Device u has part has part
In Section the complexity of satis ability and subsumption of these exten
sions will b e investigated
ALC extended by Symbolic Number Restrictions
De nition Let N f g b e a set of numerical variables Then
V
S
ALCN is obtained from ALCN by additionally allowing for
Chapter Expressive Number Restrictions
symbolic number restrictions R and R for a role name R
and a numerical variable and
the existential quanti cation C of numerical variables where C is
S
an ALCN concept
S
Since ALCN allows for full negation of concepts universal quanti cation of
numerical variables can b e expressed In the following we us C as a
shorthand for C
S
Before giving the semantics of ALCN concepts we have to de ne what it
means that a numerical variable o ccurs free in a concept
De nition The o ccurrence of a variable N is said to b e bound in C
V
i o ccurs in the scop e C of a quanti ed subterm C of C Otherwise the
o ccurrence is said to b e free The set free C N denotes the set of variables
V
that o ccur free in C A concept C is closed i free C The concept
C n is obtained from a concept C by substituting all free o ccurrences of
by n
Note that as usual a variable can o ccur b oth free and b ound in a concept
as for example in R u R R
S
Using this notation we can de ne the semantics of ALCN concepts
S
De nition An interpretation of a closed ALCN concept is an interpre
tation of an ALCN concept which satis es additionally
I I
C n C
n IN
If C is not closed and free C f g for n then
n
I I
C C
n
This de nition reduces symbolic number restrictions to traditional ones hence
their semantics is well de ned Since C is an abbreviation for C
we can give its semantics directly by
I I
C n C
n IN
S
Similar to ALCN it can b e shown that ALCN still has the tree mo del prop erty
This can b e proved by unravelling an arbitrary mo del I of C to a tree mo del I
of C This construction is similar to the one presented in Thomas Let
I
x C b e an instance of C in I Then I is obtained by unwinding cyclic
paths from x to in nite paths that is each path x R x R R x in
n n
I corresp onds to an individual in I and vice versa More precisely
0
I I
f x R x R R x j x x R for i n g
n n i i
i
0 0
I I I
j x A g for concept names A f x R x R R x A
n n n
0 0
I I I
g R f x R x R R x x R x R R x R y
n n n n
By induction on the structure of concept it can easily b e shown that I is
a tree mo del of C However the resulting tree can b e of in nite outdegree
S
In contrast to ALCN ALCN do es not have the nite mo del prop erty For
example the concept
R
is satis able but each instance of has in nitely many R successors On
the one hand the interpretation I where
I
fx y y y g
I
R f x y j i IN g
i
is clearly a mo del of On the other hand each mo del of satis es
I I
n R hence in there are in nitely many R successors of some
n IN
x
S
This example shows that ALCN can enforce in nite mo dels which intu
itively makes reasoning more complex However the attempt to construct a
satis able concept that has only in nite mo dels and that involves only exis
tential quanti cation of numerical variables will fail such a concept can only
b e constructed using universal quanti cation of numerical variables Hence
S S
we introduce ALUEN a sublanguage of ALCN which is obtained by allowing
only for existential quanti cation of numerical variables
S S
De nition ALUEN concepts are those ALCN concepts where negation
o ccurs only in front of concept names and number restrictions
S
Since in ALCN universal quanti cation of numerical variables came only in as
an abbreviation of negated existential quanti cation all numerical variables in
Chapter Expressive Number Restrictions
S
ALUEN are therefore quanti ed existentially Nevertheless it is still a sup er
language of ALCN Furthermore all examples given in Section to motivate
S
the introduction of symbolic number restrictions are ALUEN concepts The
S
reason for introducing rst ALCN is that it is prop ositionally closed which
enables the reduction of subsumption to satis ability and makes the syntax of
S S S
ALCN easier to use The reason for restricting it to ALUEN is that ALCN
can b e shown to have undecidable inference problems whereas satis ability
S
of ALUEN concepts will b e shown to b e satis able
Number Restrictions on Complex Roles
The examples given in the b eginning of this chapter already highlighted the
impact complex roles inside number restrictions have on the expressive p ower
of ALC For a deep er insight into the expressiveness of these extensions we
rst give the undecidability results The concepts used in the pro ofs reveal
the amount of expressive p ower added to ALCN by complex roles in number
restrictions
Undecidability results
In this section undecidability of ALC N t ALCN t ALCN u
and ALC N will b e shown by a reduction of the domino problem This
well known undecidable problem Wang Berger Knuth is used
for the pro of of all undecidability results in this thesis and this section starts
with its introduction
The problem problem asks for the tiling of the in nite plane using a nite
set of domino types see Figure Intuitively each domino is a square with
coloured edges The tiling may not have any holes edges of neighbouring
dominos must have the same colour on the touching edges and dominos may
not b e turned or ipp ed
De nition A tiling system D D H V is given by a non empty set
D fD D g of domino types and by horizontal and vertical matching
pairs H D D V D D The domino problem asks for a compatible
tiling of the rst quadrant IN IN of the plane i e a mapping t IN IN D
such that for all m n IN
t m n t m n H and t m n t m n V
Number Restrictions on Complex Roles
Figure A set of domino types and a rst part of a tiling
The standard domino problem asks for a compatible tiling of the whole plane
However a compatible tiling of the rst quadrant yields compatible tilings of
arbitrarily large nite rectangles which in turn yield a compatible tiling of the
plane Knuth Thus the undecidability result for the standard problem
Berger carries over to this variant
In order to reduce the domino problem to satis ability of concepts we must
show how a given tiling system D can b e translated into a concept E of
D
the language under consideration such that E is satis able i D allows for
D
a compatible tiling This task can b e split into three subtasks which will b e
rst explained on an intuitive level b efore showing how they can b e achieved
for the four Description Logics under consideration
Task It must b e p ossible to represent a single square of IN IN which
consists of p oints n m n m n m and n m The idea
is to introduce roles X Y where X go es one step into the horizontal i e
x direction and Y go es one step into the vertical i e y direction
The concept language must b e expressive enough to describ e that an
individual a p oint n m has exactly one X successor the p oint n
Chapter Expressive Number Restrictions
m exactly one Y successor the p oint n m and that the X Y
successor coincides with the Y X successor the p oint n m
Task It must b e p ossible to express that a tiling is lo cally compatible i e
that the X and Y successors of a p oint have an admissible domino type
The idea is to asso ciate each domino type D with an atomic concept D
i i
and to express the horizontal and vertical matching conditions via value
restrictions on the roles X Y
Task It must b e p ossible to imp ose the ab ove local conditions on all p oints
in IN IN This can b e achieved by constructing a universal role U and
a start individual such that every p oint is a U successor of this start
individual The lo cal conditions can then b e imp osed on all p oints via
value restrictions on U for the start individual
Task is rather easy and can b e realized using the ALC concept C given
D
in Figure The rst conjunct expresses that every p oint has exactly one
domino type and the value restrictions in the second conjunct express the
horizontal and vertical matching conditions Disjunction o ccurs in b oth con
juncts In the rst disjunction is used to express that each p oint is asso ciated
to a domino type D or a domino type D etc In the second conjunct disjunc
tion is used to express the compatibility condition which is of the form if a
p oint is asso ciated to a domino D then its X successor resp its Y successor
i
must b e asso ciated to a domino D such that D D H resp V In
j i j
contrast to this Task and Task can b e achieved without using disjunction
of concepts
Task can b e achieved in any extension of ALCN with either union or
intersection of roles in number restrictions see the concepts C and C in
u t
Figure
Task is easy for languages that extend ALC and more di cult for lan
reg
guages without at least the transitive closure The general idea is that the
start individual s is an instance of the concept E to b e constructed From
D
this individual one can reach via U the origin of IN IN and each p oint
that is connected with the origin via some arbitrary X and Y path
With the image of these tasks in mind the reduction concepts are now ex
plained in detail for each undecidable extension of ALCN ALC N and ALC N
reg
by complex number restrictions
Number Restrictions on Complex Roles
C t D u u D u
D i j
i m
j m j
i
u D X t D u Y t D
i j j
i m
D D H D D V
i j i j
C X u Y u X Y u Y X u
t
Y X t X Y
C X u Y u X Y u Y X u
u
Y X u X Y
0
E R u R C u C u R u R t X t Y
t D
D
E U u U C u C u X U u Y U u
t D
D
U t Y U t X U
E R u R T R u
D
R T R C u C u T u
u D
Y T u X T u
T u X T u Y T u
X u X T R u Y u Y T R
where A B is an abbreviation for A t B and
n R is an abbreviation for n R u n R
Figure Concepts used in the pro of of Theorem
We start with ALC N since here it is rather easy to reach all individuals
reg
representing p oints in the plane from the start individual In extensions of
ALC N we can use the complex role X t Y to reach every p oint Thus
reg
for each tiling system D the ALC N concept
reg
E C u C u X t Y C u C
t D t D
D
can b e constructed which is obviously satis able if and only if D admits a
compatible tiling
The complex role in the value restriction can even b e restricted to a simple
transitive closure of an atomic role Intuitively a starting p oint outside the
plane is used which is connected to each p oint in the plane via some R path
0
To achieve this the concept E in Figure makes sure that the X and
D
the Y successors of each p oint in the plane are also R successors of this p oint
Hence R can b e used in place of X t Y as universal role and thus the
0
concept E is in ALC N t
D
Chapter Expressive Number Restrictions
In ALCN t a role name U for the universal role is explicitly
introduced and number restrictions which involve comp osition union and
inversion of roles are used to make sure that the start individual is directly
connected via U with every p oint see the concept E in Figure The
D
number restrictions inside the value restriction make sure that every p oint
p that is reached via U from the start individual satis es the following Its
X successor and its Y successor each have exactly one U predecessor which
coincides with the unique U predecessor of p i e the start individual Thus
the X successor and the Y successor of p are also U successors of the start
individual
For ALCN u a similar construction is p ossible Since inversion of roles
is not allowed in ALCN u two role names R and T are needed for the
construction of the universal role The intuition is that T plays the r ole of the
inverse of R except for one individual and the universal role corresp onds
to the comp osition R T R The start individual s which is an instance of E
D
has at least one R successor p which coincides with its R T R successor
The individual p corresp onds to the origin of IN IN Let s b e the R T
successor of s The number restrictions of E make sure that p satis es
D
the following It has exactly one T successor namely s which coincides with
the unique T successors of its X and Y successors In addition the unique
X successor of p is also an X T R successor of p which makes sure that
the X successor of p is an R successor of s and thus an R T R successor
of s The same holds for the Y successor One can now continue the argument
with the X successor resp Y successor of p in place of p
With the intuition given ab ove it is not hard to show for all i i that
i
a tiling system D has a compatible tiling i E is satis able
D
Theorem Satis ability and thus also subsumption of concepts is unde
cidable for ALC N t ALCN t and ALCN u
Pro of Since the other two cases are similar and easier it will only b e shown
that the ALCN u concept E is satis able if and only if D admits a
D
compatible tiling
The if direction Let D b e a tiling system E the corresp onding concept D
Number Restrictions on Complex Roles
and t a compatible tiling Then we de ne a mo del I of E as follows
D
I
fsg fp j n m IN g
n m
I
X f p p j n m INg
n m n m
I
Y f p p j n m INg
n m n m
I
R f s p j n m INg
n m
I
T f p s j n m INg
n m
I
D fp j t n m D g for each D D
i i
n m
i
Since each p is b oth an R and an R T R successor of s s obviously is
n m
an instance of the rst conjunct of E Furthermore the value restriction
D
R T R applies to each p Now since t is compatible each individual
n m
I
p is an element of exactly one D and b oth the vertical and horizontal
n m
i
matching conditions are satis ed Hence each p is an instance of C
D
n m
Furthermore X Y are de ned in such a way that each p has exactly
n m
one X successor p one Y successor p and one X Y successor
n m n m
p which coincides with its Y X successor Hence each p is an
n m n m
I
element of C
u
Now since s is the only T successor each p is clearly an instance of
n m
T u Y T u X T
Furthermore s is the common T successor of all p hence each p is
n m n m
an instance of T u X T u Y T Finally since each p has exactly
n m
one X successor p and one Y successor p and since p and
n m n m n m
p are their own T R successors each p is clearly an instance of
n m n m
X u X T R u Y u Y T R
The only if direction Let I b e some mo del of E with s E let p
D D
b e some R successor of r which is also an R T R successor s b ecause of the
rst conjunct of E such a p exists and let s b e the R T successor of
D
s which p ossibly coincides with s Since each R T R successor of s is an
instance of C p is an instance of exactly one D D De ne a compatible
D j
tiling t by
t D
j
n m I
t n m D if p is an X Y successor of p and p D
i
i
It remains to b e shown that t is well de ned and the horizontal and
the vertical matching conditions are satis ed
Chapter Expressive Number Restrictions
t is well de ned b ecause a p has for each n m IN exactly one
n m
X Y successor which b is furthermore instance of exactly one D a holds
i
b ecause the value restriction R T R applies to p hence p is an
instance of C which means that it has exactly one X successor one Y
u
successor and one X Y successor which coincides with its unique Y X
successor Furthermore p is an instance of
T u Y T u X T u
T u X T u Y T u X u X T R u Y u Y T R
Hence s is also a T successor of the X and Y successors p p of p
and p p are b oth R successors of s Therefore p p are b oth
R T R successors of s and thus the value restriction R T R applies
also to p p The same arguments as for p can then b e used on
n m
p p hence for each n m there is exactly one X Y successor of p
n m
b is due to fact that p as well as all its X Y successors are an instance
of C
D
Obviously C ensures that the horizontal and the vertical matching con
D
ditions are satis ed
In the overview given in Figure decidable extensions are hatched horizon
tally whereas undecidable ones are hatched vertically Given
the ab ove results
the fact that as a consequence of a result in Gr adel et al
ALCN u t is decidable and
the fact that ALCN is decidable which will b e proved in the next
section
the only problems which remain op en for the extensions of ALCN concern
ALCN and ALCN t Unfortunately these problems must remain
op en for the time b eing neither a decision pro cedure for one of these extensions
nor a pro of of their undecidability could b e found
Extensions of ALC N
So far un decidability of two extensions of ALC N by complex roles was
proved In De Giacomo decidability of ALC N is shown and Theo
rem is concerned with the undecidability of ALC N t It will now b e
Number Restrictions on Complex Roles
t u
et al Gr adel
t u t u t u
t t u t u u
t u ALC N
Figure Un decidability results for extension of ALCN
shown that in ALC N it su ces to allow for comp osition in number restric
tions in order to lose decidability In Figure an overview of these results is
given again with decidable extensions hatched horizontally and undecidable
ones vertically
Again a reduction of the domino problem to concept satis ability is used to
show undecidability of ALC N Since this reduction is rather di erent from
the ones ab ove and more complicated it is treated separately The redundant
reduction for ALC N t was given since it served to give the intuition for
ALCN t and ALCN u The concepts used for the reduction of the
domino problem to ALC N concept satis ability are given in Figure
As neither C nor C is an ALC N concept Task must b e accomplished
u t
di erently Instead of using a horizontal role X and a vertical role Y
only a single role X can b e used otherwise one could not express that the
horizontal vertical successor coincides with the vertical horizontal successor of
a p oint
The three tasks introduced in the b eginning of Section are then accom
plished as follows
Chapter Expressive Number Restrictions
t u
t u t u t u
t u t t u u
Giacomo De
t u
N ALC
Figure Un decidability results for extension of ALC N
Task The concept C describ es a square by using a single role X Each
2
instance of C has two X successors that in turn each have two X successors
2
The conjunct X X makes sure that the X successors of an instance of
C have one common X successor
2
Task is easy b ecause ALC N allows for the transitive closure of roles If
I
s E then s has exactly one X successor say p which is an instance
D
n
of A Each p oint in the grid is an X successor of s Thus the lo cal conditions
on all p oints in the grid are imp osed by X C u C u C u C
2 prim diag D
Task is di cult b ecause we must distinguish b etween the horizontal and
the vertical X successor of a p oint For this purp ose the concepts A B
and C are used in the following way see Figure for a b etter intuition
The concept C enforces that each p oint is an instance of either A or B or
prim
C and that exactly one domino type D is asso ciated with each p oint The
i
concept C makes sure that each instance of A has one X successor in B
diag
and one in C and similar for instances of B and C Without loss of generality
when visualising the grid in Figure we have drawn the X successor of p
which is in C to its right and called it p The other X successor of p
Number Restrictions on Complex Roles
A C B A C B C B A C B A B A C B A C A C B A C B
Figure Visualisation of the grid as enforced by the ALC N reduction
concept
which is in B is called p and is drawn ab ove it Then it is easy to see that
the remainder of the grid is determined in the sense that
for each diagonal in the grid there is an E fA B C g such that all
p oints on this diagonal are instances of E
horizontal successors of p oints in A are always in C of p oints in C are
always in B and of p oints in B are always in A
vertical successors of p oints in A are always in B of p oints in B are
always in C and of p oints in C are always in A
With these observations C expresses the horizontal and vertical matching
D
conditions
With the intuition given ab ove it is not hard to show that a tiling system D
has a compatible tiling i E is satis able
D
This concept is to make it easier to understand longer than necessary The sub concept
C can b e omitted b ecause it is subsumed by C u C
diag D prim
Chapter Expressive Number Restrictions
C X u X X u X X
2
C A t B t C u t D u u D
prim i j
i m
j m j
i
C A X B u X C u
diag
B X A u X C u
C X A u X B
C u A u D X C u t D u
D i j
i m
D D H
i j
X B u t D u
j
D D V
i j
B u D X A u t D u
i j
D D H
i j
X C u t D u
j
D D V
i j
C u D X B u t D u
i j
D D H
i j
X A u t D
j
D D V
i j
E X u X A u X C u C u C u C
2 prim diag D
D
where A B and C are disjoint concepts They are abbreviations for
A A B A u A C A u A
Figure Concepts used in the pro of of Theorem
Theorem Satis ability and thus also subsumption of concepts is unde
cidable for ALC N
A decidable extension
In this section a tableau based algorithm for deciding satis ability of ALCN
concepts is presented The algorithm and the pro of of its correctness are very
similar to existing algorithms and pro ofs for languages with number restrictions
on atomic roles Hollunder et al Hollunder Baader Please recall
that the presence of number restrictions on role chains has as a consequence
the loss of the tree mo del prop erty
Nevertheless the mo dels generated by this algorithm are very similar to tree
mo dels When started with a concept C the algorithm generates a mo del of
C where every element can b e reached from an initial ro ot element which is
an instance of C via role chains Furthermore the ro ot do es not have a role
predecessor and every role chain from the ro ot to an element has the same
length even though there may exist more than one such chain This fact will
Number Restrictions on Complex Roles
b ecome imp ortant in the pro of of termination
As usual without loss of generality all concepts are supp osed to b e in negation
normal form NNF see Section Similar to the algorithm presented in
Figure the new algorithm works on constraints but since it has to handle
number restrictions on role chains we need an additional kind of constraints
and an extended de nition of the notion a clash
De nition Let fx y z g b e a countably in nite set of individual
variables A constraint is either of the form
xR y where R is a role name in N and x y
R
x D for some ALCN concept D in NNF and some x or
x y for x y
As in De nition a constraint system is a set of constraints and
S
denotes the individual variables o ccurring in a constraint system S
An interpretation I is a model of a constraint system S i there is a mapping
I
such that I satisfy each constraint in S i e
S
I
x y R for all xR y S
I
x D for all x D S
x y for all x y S
Similar to De nition y is said to b e an R R successors of x in S if
m
there exist y y with
m S
xR y S
for all i m y R y S and
i i i
y y
m
S contains a clash i fx A x Ag S for some concept name A and some
variable x or x n R R S and x has n R R
S m m
successors y y in S such that for all i j we have y y S A
i j
constraint system S is called complete i none of the completion rules given in
Figure can b e applied to S In these rules the constraint system S y y
is obtained from S by substituting each o ccurrence of y in S by y
We consider such inequalities as b eing symmetric i e if x y b elongs to a constraint
system then y x implicitly b elongs to it as well
Chapter Expressive Number Restrictions
Figure introduces the completion rules that are used to test ALCN
concepts for satis ability Similar to the algorithm presented in Figure
the completion algorithm works on a tree where each no de is lab elled with a
constraint system It starts with the tree consisting of a ro ot lab elled with
S fx C g where C is the ALCN concept in NNF to b e tested for
satis ability A rule can only b e applied to a leaf lab elled with a clash free
constraint system Applying a rule S S for i n to such a leaf
i
leads to the creation of n new successors of this no de each lab elled with one
of the constraint systems S The algorithm terminates if none of the rules
i
can b e applied to any of the leaves In this situation it answers with C is
satis able i one of the leaves is lab elled with a clash free constraint system
Soundness and completeness of this algorithm is an immediate consequence of
the following facts
Lemma Let C b e an ALCN concept in NNF and let S b e a con
straint system obtained by applying the completion rules to fx C g Then
For each completion rule R that can b e applied to S and for each inter
pretation I i and ii are equivalent
i I is a mo del of S
ii I is a mo del of one of the systems S obtained by applying R
i
If S is a complete and clash free constraint system then S has a mo del
If S contains a clash then S do es not have a mo del
The completion algorithm terminates when applied to fx C g
Since Lemma is the same as Lemma b esides the di erent logics the
arguments which were used to show that Lemma implies decidability of
satis ability of ALC concepts now yield decidability of ALCN
Theorem Subsumption and satis ability of ALCN concepts is decid
able
It remains to prove Lemma
Number Restrictions on Complex Roles
Conjunction If x C u C S and x C S or x C S then
S S fx C x C g
Disjunction If x C t C S and x C S and x C S then
S S S fx C g
S S S fx C g
Value restriction If x R C S for a role name R y is an
R successor of x in S and y C S then
S S fy C g
Existential restriction If x R C S for a role name R and
there is no R successor y of x in S with y C S then
S S fxR z z C g for a new variable z n
S
Number restriction If x n R R S for role names
m
R R and x has less than n R R successors in S then
m m
S S fxR y y R z g fy R y j i m g
m m i i i
fz w j w is an R R successor of x in S g
m
where z y are new variables in n
i S
Number restriction If x n R R S x has more than
m
n R R successors in S and there are R R successors y y
m m
of x in S with y y S then
S S S y y
y y
for all pairs y y of R R successors of x with y y S
m
Figure The completion rules for ALCN
Pro of of Part of Lemma We consider only the rules concerned
with number restrictions since the pro of for Rules is just as for ALC
Number restriction Assume that the rule is applied to the constraint
x n R R and that its application yields
m
S S fxR y y R z g fy R y j i m g
m m i i i
fz w j w is an R R successor of x in S g
m
Since S is a subset of S any mo del of S is also a mo del of S
I
Conversely assume that I is a mo del of S and let b e the S
Chapter Expressive Number Restrictions
I
corresp onding mapping of individual variables to elements of On
the one hand since I satis es x n R R x has at least n
m
R R successors in I On the other hand since Rule is applicable
m
to x n R R x has less than n R R successors in S Thus
m m
there exists an R R successor b of x in I such that b w for
m
I
all R R successors w of x in S Let b b b e such that
m m
I I I I
0
x b R b b R b b R We de ne
m S
m
by y y for all y y b for all i i m and
S i i
z b Obviously I satisfy S
Number restriction Assume that the rule can b e applied to x n R
R S and let I together with b e a mo del of S On the one hand
m
since the rule is applicable x has more than n R R successors
m
in S On the other hand I satisfy x m R R S and
m
thus there are two di erent R R successors y y of x in S such
m
that y y Obviously this implies that y y S Hence
S S y y is one of the constraint systems obtained by applying
y y
Rule to x n R R In addition since y y I
m
satisfy S
y y
Conversely assume that S S y y is obtained from S by applying
y y
Rule and let I together with the valuation b e a mo del of S If
y y
we take a valuation that coincides with on the variables in
S
y y
and satis es y y then I obviously satisfy S
Pro of of Part of Lemma Let S b e a complete and clash free con
straint system that is obtained by applying the completion rules to fx C g
We de ne a canonical mo del I of S as follows
I
S
I
for all A N x A i x A S
C
I
for all R N x y R i xR y S
R
I
In addition let b e the identity on It remains to show that
S S
I indeed satisfy each constraint in S
By de nition of I a role constraint of the form xR y is satis ed by I i
xR y S More generally y is an R R successor of x in S i y is an
m
R R successor of x in I Next all constraints of the form x y are
m
obviously satis ed since is the identity
We show by induction on the structure of the concept C that every concept
constraint x C S is satis ed by I The induction base and the treatment
Number Restrictions on Complex Roles
of the prop ositional constructors and the value restrictions is just as for ALC
I satis es all constraints of the form x A by de nition Since S is clash free
all constraints of the form x A are satis ed By induction completeness of S
implies that S satis es all constraints of the form x C u D x C t D x R C
and x R C Again we go into detail only for constraints involving number
restrictions
Consider x n R R S Since S is complete Rule cannot
m
b e applied to x n R R and thus x has at least n R R
m m
successors in S which are also R R successors of x in I This
m
shows that I satisfy x n R R
m
Constraints of the form x n R R S are satis ed b ecause
m
S is clash free and complete In fact assume that x has more than n
R R successors in I Then x also has more than n R R
m m
successors in S If S contained inequality constraints y y for all
i j
these successors then we would have a clash Otherwise Rule could
b e applied
Pro of of Part of Lemma Assume that S contains a clash If
fx A x Ag S then it is clear that no interpretation can satisfy b oth
constraints Thus assume that x n R R S and x has n
m
R R successors y y in S with y y S for all i j
m i j
Obviously this implies that in any mo del I of S x has n distinct
R R successors y y in I which shows that I cannot
m
satisfy x n R R
m
Pro of of Part of Lemma In the following we consider only
constraint systems S that are obtained by applying the completion rules to
fx C g For a concept C we de ne its and or size jC j as the number of
u t
o ccurrences of conjunction and disjunction constructors in C The maximal
role depth depth C of C is de ned as follows
depth A depth A for A N
C
depth C u C maxfdepth C depth C g
depth C t C maxfdepth C depth C g
depth R C depth R C depth C
depth n R R m
m
depth n R R m
m
Chapter Expressive Number Restrictions
For the termination pro of the following observations which are an easy con
sequence of the de nition of the completion rules are imp ortant
Lemma Let S b e a constraint system obtained by application of the
completion algorithm to fx C g
Every concept C that may o ccur in S is a sub concept of C
Every variable x x that o ccurs in S is an R R successor of x
m
for some role chain of length m In addition every other role chain
that connects x with x has the same length
If x can b e reached in S by a role chain of length m from x then for
each constraint x C in S the maximal role depth of C is b ounded by
the maximal role depth of C minus m Consequently m is b ounded by
the maximal role depth of C
Pro of
is obvious
Supp ose the algorithm is started with fx C g Then each variable in
tro duced by the algorithm is generated by Rule or and thus connected
to its predecessor via some role The second fact is due to Rule which
identi es only those variables that are b oth R R successors of
m
some y Lemma follows then by induction
A constraint x C can only b e present in S if it was added by Rules
If x C was added by Rule or then the corresp onding sup erconcept
x C u D resp ectively x C t D is also present in S and it is of the same
depth If it was added by Rule or it implies the presence of some
w R C or w R C in S for some role name R and some R predecessor
i i
w of x in S In b oth cases the role depth of the concepts o ccurring in
i
constraints on w are at least the role depth of C Hence the maximal
i
role depth of all constraints on all role predecessors of x is at least
the maximal role depth of all constraints on x Finally all variables in
are role successor of C and together with Lemma this yields
S
Lemma
Number Restrictions on Complex Roles
Let m b e the maximal role depth of C Because of the second fact every
individual x in a constraint system S reached from fx C g by applying
completion rules has a unique role level level x which is its distance from
the ro ot no de x i e the unique length of the role chains that connect x with
x Because of the third fact the level of each individual is an integer b etween
and m
In the following we de ne a mapping of constraint systems S to m
tuples of nonnegative integers such that S S implies S S where
denotes the lexicographic ordering on m tuples Since the lexicographic
ordering is well founded this implies termination of our algorithm In fact if
the algorithm did not terminate then there would exist an in nite sequence
S S and this would yield an in nite descending chain of tuples
in contradiction to the well foundedness of
Thus let S b e a constraint system that can b e reached from fx C g by
applying completion rules We de ne
S
m m
where k k k k k and the comp onents k are obtained as
i
follows
k is the number of individual variables x in S with level x
k is the sum of the and or sizes jC j of all constraints x C S such
u t
that level x and the conjunction or disjunction rule is applicable to
x C in S
For a constraint x n R R let k b e the maximal cardinality of
m
all sets M of R R successors of x for which y y S for all pairs
m i j
of distinct elements y y of M We asso ciate with x n R R
i j m
the number r n k if n k and r otherwise k sums up all
the numbers r asso ciated with constraints of the form x n R R
m
for variables x with level x
k is the number of all constraints x R C S such that level x
and the existential restriction rule is applicable to x R C in S
k is the number of all pairs of constraints x R C xR y S such
that level x and the value restriction rule is applicable to x R C
xR y in S
In the following we show for each of the rules of Figure that S S implies
S S
Chapter Expressive Number Restrictions
Conjunction Assume that the rule is applied to the constraint x C u C
and let S b e the system obtained from S by its application Let
level x
First we compare and the tuples resp ectively asso ciated with the
level of x in S and S Obviously the rst components of and
agree since the number of individuals and their levels are not changed
The second component of is strictly smal ler than the second comp o
nent of jC u C j is removed from the sum and replaced by a
u t
number that is not larger than jC j jC j dep ending on whether
u t u t
the top constructor of C and C is disjunction or conjunction or some
other constructor Since tuples are compared with the lexicographic
ordering a decrease in this comp onent makes sure that it is irrelevant
what happ ens in later comp onents
For the same reason we need not consider tuples for m Thus
m
assume that m In such a tuple the rst three comp onents are
not changed by application of the rule whereas the remaining two com
p onents remain unchanged or decrease Such a decrease can happ en if
level y m and S contains constraints y R x y R C or y R C
i i
for some i f g
Disjunction This rule can b e treated similar to the conjunction rule
Value restriction Assume that the rule is applied to the constraints
x R C xR y and let S b e the system obtained from S by its ap
plication Let level x Obviously this implies that level y
level x
On level the rst three comp onents of remain unchanged the
fourth remains the same or decreases if S contains constraints z S y
and z S C for an individual z with level z and the fth de
creases by at least one since the constraints x R C xR y are no longer
counted It may decrease by more than one if S contains constraints z S y
and z S C for an individual z with level z
Because of this decrease at level the tuples at larger levels in particu
lar the one for level level x where there might b e an increase need
not b e considered
The tuples of levels smaller than are not changed by application of the
rule In particular the third comp onent of such a tuple do es not change
since no role constraints or inequality constraints are added or removed
Existential restriction Assume that the rule is applied to the constraint
x R C and let S S fxR y y C g b e the system obtained from
S by its application Let level x Obviously this implies that
Number Restrictions on Complex Roles
level y level x
The rst two comp onents of obviously remain unchanged The third
comp onent may decrease if y is the rst successor which is introduced
for an at least restriction n R or it stays the same Since the fourth
comp onent decreases the p ossible increase of the fth comp onent is ir
relevant
For the same reason the increase of the rst comp onent of is irrel
evant
Tuples of a level smaller than are not increased by the application of
this rule All comp onents of such a tuple remain unchanged with the
p ossible exception of the third comp onent which may decrease
Number restriction Assume that the rule is applied to the constraint
x n R R S let S b e the system obtained by rule application
m
and let level x
As for Rule the rst two comp onents of remain the same In
addition there is a decrease in the third comp onent of since the new
individual z can now b e added to the maximal sets of explicitly distinct
R R successors of x Note that these sets were previously smaller
m
than n b ecause even the set of all R R successors of x was smaller
m
than n
For this reason the p ossible increase in the fth comp onent of and
in the rst comp onents of tuples of levels larger than are irrelevant
Tuples of a level smaller than are either unchanged by application of
the rule or their third comp onent decreases
Number restriction Assume that the rule is applied to the constraint
x n R R S let S S b e the system obtained by rule
m y y
application and let level x
On level m the rst comp onent of the tuple decreases Thus
m
p ossible increases in the other comp onents of this tuple are irrelevant
Tuples asso ciated with smaller levels remain unchanged or decrease In
fact since y in S has all its old constraints and the constraints of y in
S some value restrictions or existential restrictions for individuals of the
level immediately ab ove level m may b ecome satis ed in the sense
that the corresp onding rule no longer applies Since no constraints are
removed previously satis ed value restrictions or existential restrictions
remain satis ed The third comp onent of tuples of smaller level cannot
increase since the individuals y y that have b een identi ed were not
related by inequality constraints
Chapter Expressive Number Restrictions
For languages where number restrictions may contain in addition to com
p osition union or intersection of roles an imp ortant prop erty used in the
ab ove termination pro of is no longer given It is not p ossible to asso ciate each
individual generated by a tableau based algorithm with a unique role level
which is its distance to the ro ot individual x i e the instance x of C
generated by the tableau algorithm Indeed in the concept
C R R A u R t R R
the number restriction enforces that an R successor of an instance of C is
also an R R successor of this instance For this reason an R successor of
the ro ot individual must b e b oth on level and on level and thus the
relatively simple termination argument that was used ab ove is not available
for these larger languages However as we shall show in the next section this
termination argument can still b e used if union and intersection are restricted
to role chains of the same length Without this restriction satis ability may
b ecome undecidable in Section we have shown that satis ability is in
fact undecidable for ALCN u For ALCN t decidability of satis ability
is still an op en problem
An extension of the decidability result
The algorithm given in Section can b e extended such that it can also treat
union and intersection of role chains that have the same length The pro of of
soundness completeness and termination of this extended algorithm is very
similar to the one for the basic algorithm and will thus only b e sketched
In the remainder of this section a complex role is
a role chain R R R or
n
the intersection R R R u S S of two role chains of the
n n
same length or
the union R R R t S S of two role chains of the same
n n
length
The satis ability algorithm is extended by adding two new rules a and b to
handle number restrictions n R for complex roles with union or intersec
tion and by substituting rule by a new rule that is able to handle the
new types of complex roles To formulate the new rules we must extend the
notion of a role successor in a constraint system appropriately Building up on
the notion of a role successor for a role chain see De nition we de ne
Number Restrictions on Complex Roles
a Number restriction If x n R t S S S and
m m
x has less than n R R t S S successors in S then
m m
S S S fxR y y R z g fy R y j i m g
m m i i i
fz w j w is an R R t S S successor of x in S g
m m
S S S fxS y y S z g fy S y j i m g
m m i i i
fz w j w is an R R t S S successor of x in S g
m m
where z y are new variables in n
i S
b Number restriction If x n R R u S S S and
m m
x has less than n R R u S S successors in S then
m m
S S fxR y xS y y R z y S z g
m m m
m
fy R y y S y j i m g
i i i i
i i
fz w j w is an R R u S S successor of x in S g
m m
where z y y are new variables in n
i S
i
Number restriction If x n R S for some complex role R
x has more than n R successors in S and there are R successors y y of
x in S with y y S then
S S S y y
y y
for all pairs y y of R successors of x with y y S
Figure The additional completion rules
y is an R R t S S successor of x in S i y is an R R
n n n
successor or an S S successor of x in S and
n
y is an R R u S S successor of x in S i y is an R R
n n n
successor and an S S successor of x in S
n
Obviously this de nition is such that role successors in S are also role succes
sors in every mo del of S if I satisfy S and y is an R successor of x in S
for a complex role R then y is an R successor of x in I
The new rules are describ ed in Figure The rules a b are added to the
completion rules whereas rule substitutes rule in Figure To show
that the new algorithm obtained this way decides satis ability of concepts for
the extended language we must pro of that all four parts of Lemma still
hold The observations presented in Lemma still hold only in the part
of Lemma one has to substitute due to Rule which identi es only those
Chapter Expressive Number Restrictions
variables that are b oth R R successors of some y by due to Rule
m
which identi es only those variables that are b oth successors of some y with
resp ect to role chains having the same length
Local correctness of the rules a b and can b e shown as in the pro of
of Part of Lemma ab ove
The canonical model induced by a complete and clash free constraint
system is de ned as in the pro of of Part of Lemma The pro of
that this canonical mo del really satis es the constraint system is also
similar to the one given there Note that the notion of an R successor of
a complex role R in a constraint system was de ned such that it coincides
with the notion of an R successor in the canonical mo del I induced by
the constraint system
The pro of that a constraint system containing a clash is unsatis able is
the same as the one given ab ove Note that this dep ends on the fact
that role successors in a constraint system are also role successors in
every mo del of the constraint system
The pro of of termination is also very similar to the one given ab ove The
de nition of the depth of a concept is extended in the obvious way to
concepts with number restrictions on complex roles
depth n R R u S S m
m m
depth n R R t S S m
m m
depth n R R u S S m
m m
depth n R R t S S m
m m
Because the role chains in complex roles are of the same length it is
easy to see that Lemma still holds Thus we can de ne the same
measure S as ab ove for all constraint systems obtained by applying
the extended completion rules to fx C g It is easy to see that the pro of
that S S implies S S can b e extended to the new rules It
should b e noted that the pro of given ab ove is already formulated in a
more general way than it were necessary for ALCN In fact we have
only used the fact that all role chains connecting two individuals have the
same length which is still satis ed for the extended language and the
fact that these role chains also have the same name was not mentioned
which is only satis ed for ALCN
The following theorem is an immediate consequence of these observations
Symbolic Number Restrictions
Theorem Subsumption and satis ability is decidable for the language
that extends ALCN by number restrictions on union and intersection of role
chains of the same length
Given this decidability result a natural question arising here is whether con
sistency of ALCN ABoxes is also decidable In Section we already
mentioned that a tableau based algorithm deciding satis ability of concepts is
likely to b e extendible such that it decides ABox consistency Unfortunately
this is not true for ALCN ABoxes In Molitor the failure of several at
tempts to design a decision pro cedure for the consistency of ALCN ABoxes
is describ ed All these attempts led either to non termination or incomplete
ness A decidability result could only b e obtained for ABoxes of a severely
restricted form Roughly sp oken consistency of ALCN ABoxes can only b e
decided for ABoxes which do not contain two individuals which are connected
via two roles chains of di erent lengths
Symbolic Number Restrictions
As in the last section we rst give the undecidability result This yields a
S
b etter insight into the expressive p ower of ALCN and it motivates the inves
S
tigation of the restricted language ALUEN
An undecidability result
As in Section undecidability of satis ability is shown by a reduction of
S
the domino problem to concept satis ability For ALCN however the pro of
is easier if we take another variant of the domino problem Instead of asking
for a compatible tiling of the rst quadrant of the plane we ask now for a
compatible tiling of the second eighth
IN IN f a b j a b IN and a bg
of the plane As a consequence the compatibility condition of a tiling is less
strict than the one for a tiling of the rst quadrant of the plane More precisely
a tiling t IN IN IN IN D is now called compatible i for all
m n IN
m n t m n t m n H and t m n t m n V
Chapter Expressive Number Restrictions
As such a tiling yields compatible tilings of arbitrarily large nite rectangles
it also yields a compatible tiling of the plane Knuth
In contrast to the reduction given in Section in this reduction the indi
viduals representing p oints in the grid are not related to each other by roles
there is nothing comparable to the horizontal and vertical roles X and Y
S
Intuitively the new reduction works as follows First we de ne an ALCN
I
concept C such that for each mo del of C with x C there is a natural
IN IN
IN
relationship b etween tuples a b IN IN and S successors y of x The
a b
p oint a b is represented by an S successor of x having a L successors and b
R successors Second for a given tiling system D we construct a concept C
D
that
is subsumed by C
IN
ensures that every y is asso ciated to exactly one domino type and
a b
ensures that the horizontal and vertical matching conditions are satis ed
The formal de nition of C is given in the upp er part of Figure Let I b e a
IN
I
mo del of C with x C Now C expresses that for every nonnegative integer
IN
IN
a x has an S successor having exactly a L successors The precondition of C
makes sure that a is smaller than b and thus the whole implication says that
for each pair a b of nonnegative integers if a b then x has an S successor
having exactly a L successors and b R successors there can b e more than one
such S successor Finally C says that whenever an S successor of x has a
L successors and b R successors we have a b Thus there is an obvious
corresp ondence b etween S successors of x and p oints in the second eighth of
the plane every S successor corresp onds to a p oint in IN IN and vice
versa To b e more precise we will formally prove the following observations
I I
concerning C For a role name R and some x x denotes the number
IN R
of role llers of x with resp ect to R in I i e
I I I
x fy j x y R g
R
Lemma C is satis able
IN
I
Let I b e a mo del of C with x C and let
IN
IN
I I
Y fy j x y S g
x
I
i For each a b IN IN there exists y Y with y a
a b x a b L
I
and y b
a b R
Symbolic Number Restrictions
C C u C u C where
IN
C S L
C S L u L S L u R
C S L u R L
Given a tiling system D fD D g H V and the sub concepts
m
C C C of C as de ned ab ove let
IN
D u C C u S t D u u
j D IN i
i m
j j m
i
u S L u R u D
i
i m
S L t R t D u
i
u
S L u R t D u
j
D D H
i j
S L u R t D
j
D D V
i j
where the following abbreviations are used
S L u R u L
u S L t L
Figure Reduction concepts used in the pro of of Theorem
I I
ii If y Y and y a and y b then a b IN IN
x L R
I
If x C then there is an injective mapping IN IN Y from
IN x
the second eighth of the plane to the set of S successors of x
I I
Pro of De ne I and x as follows
I
fxg fy j a b IN IN g fl r j a b IN g
a b a b
I
S f x y j a b IN IN g
a b
I
0
j a b IN IN and a ag L f y l
a b a
I
0
j a b IN IN and b bg R f y r
a b b
S
I I
I is an ALCN interpretation and L R are de ned such that for all a b
I I
I
IN IN we have y a and y b We show that x C
a b L a b R IN
Chapter Expressive Number Restrictions
I
I I
x C i for all a b IN x C a b x C a b and
IN
I
x C a b If a b IN then
I I
0
S for some even all b with b a x C a b since x y
a b
I I
x C a b If x S a L u b L then a b and
I I
x y S which implies x S a L u b R
a b
I I I
x C a b Let x y S If y a L u b R then
I
y y with a b which implies y b L
a b
i The sub concept C makes sure that for each a IN there exists some
I I
y Y with y a C ensures for all a b IN that if y Y with y a
x L x L
I
b which holds for all b with a b then there is some y Y with and y
x L
I I
y a and y b
L R
I
ii The third sub concept C enforces for all y Y that y a and
x L
I I
y b implies y b which yields a b
R L
I
For a b IN IN there can b e more than one y Y with y a and
x L
I
y b Hence in order to de ne an injective mapping we assume that Y is
R x
linearly ordered by some ordering Then is de ned as follows
IN IN Y
x
I I
a b min fy Y j y a and y bg
x L R
where for M Y min M denotes the minimum of M with resp ect to
x
is injective by de nition and it is a total mapping as a direct consequence of
The de nition of the concept C asso ciated with a tiling system D is also given
D
in Figure In the de nition two abbreviations and are used In the
context of the concept C these abbreviations really express the relation
IN
I
and the successor relation on natural numbers For x C we have
IN
I
x a b i a b
as an immediate consequence of Lemma Furthermore we have
I
x a b i a b
Symbolic Number Restrictions
b ecause x has some S successor having a L successors for each a IN The
de nitions of the relation and the successor relation on natural numbers are
so easy b ecause we de ned C in such a way that Y corresp onds to IN IN
IN x
in the way mentioned in Lemma namely b ecause each S successor y of x
I
has at most y L successors Replacing IN IN by IN IN thus using the
R
rst quadrant of the plane instead of the second o ctant would have made the
de nition of the successor relation on natural numbers far more complicated
The rst line in the de nition of C makes sure that C subsumes C and that
D IN D
every S successor of an instance x of C has exactly one domino type In the
D
remainder of the de nition we consider an S successor y of x with domino
a b
type D that has a L and b R successors Now the sub concept of C
i D
ensures that every S successor with the same number of L and R successors
as y has the same domino type D takes care of the horizontal match
a b i
ing condition The conjunct is found in b ecause the horizontal
matching condition asks only for matching right neighbours of p oints a b
with a b whereas the domino type asso ciated to a p oint a a do es not
imp ose any constraints on the domino type asso ciated to a a Finally
takes care of the vertical matching condition
Lemma C is satis able i D allows for a compatible tiling
D
Pro of From the de nition of C it follows immediately that C is subsumed
D D
by C
IN
I
Given a mo del I of C with x C we de ne a mapping t IN
D D
IN D as follows
I
t a b D i x S a L u b R u D
i i
First we show that t is well de ned Let a b IN Since
I
x S t D u u D
i j
i m
j m j
i
each S successor of x is an instance of exactly one D D Furthermore
i
for each a b IN IN and for each D D
i
I
x S a L u b R u D S a L t b R t D
i i
hence all S successors of x having the same number of L successors and
the same number of R successors are instances of the same D D
i
Finally as a consequence of Lemma x has an S successor y with
Chapter Expressive Number Restrictions
I I
y a and y b for each a b IN IN Thus t is well de ned
L R
and it remains to b e shown that t is indeed compatible
I
Let a b IN with a b let y b e an S successor of x with y a and
L
I
I
it b and let y D hence we have t a b D From x C y
i i D R
follows that
I
x a b u a S L u b R u D
j
for some D with D D H Hence x S a L u
j i j
I
b R u D which implies that t a b D with D D H
j j i j
The same arguments apply to the vertical matching condition which
shows that t is indeed a compatible tiling
I I
Given a compatible tiling t a mo del I of C can b e de ned
D
like the one for C in the pro of of Lemma i e
IN
I
fxg fy j a b IN and a bg fl r j a b INg
a b a b
I
S f x y j a b IN and a bg
a b
I
0
j a a b IN and a a bg L f y l
a b a
I
0
j a b b IN and a b and b bg R f y r
a b b
In addition
I
fy j t a b D g D
a b i
i
Each S successor of x is instance of exactly one D D hence
i
I
x S t D u u D
i j
i m
j m j
i
I
The pro of of Lemma shows that x C Now let a b g IN Then
IN
I
x S L u R u D a b i a b and t a b D
i i
I
If x S L u R u D a b then
i
I
x S L t R t D a b since x has only a sin
i
I
gle S successor y having a L successors and b R successors
a b
and according to the assumption y D
a b i
I
if x u a b g then a b and
a g and the de nition of I implies that x S L u
I
R u D b g for some D with D D H Hence
j j i j
x
u S L u R u
I
t D a b g
j H D j i
Symbolic Number Restrictions
I
if x b g then b g and the de nition
I
of I implies that x S L u R u D a g for
j
some D with D D V Hence
j i j
x
S L u R u
I
t D a b g
j
j V D
i
Summing up we have x C
D
Thus undecidability of the domino problem yields undecidability of the satis
S
ability problem for ALCN concepts Since C is unsatis able i C v A u A
this implies undecidability of subsumption
S
Theorem Satis ability and subsumption of ALCN concepts are unde
cidable
A decidability result
S
In this section it will b e shown that satis ability of ALUEN concepts is de
cidable In order to simplify our investigation of the satis ability problem for
S
ALUEN concepts we will restrict our attention to concepts in variable normal
form VNF this is to concepts where each variable is b ound at most once
S
by Obviously each ALUEN concept can b e transformed to an equivalent
concept in VNF by renaming of variables which are b ound more than once
S
Decidability of satis ability of ALUEN concepts will b e shown by presenting
S
a tableau based algorithm and showing that for each ALUEN concept C this
algorithm is sound complete and terminating Similar to the algorithm pre
S
sented in Section the algorithm works on constraints but for ALUEN
concepts we need additional variables Supp ose we have the constraint
x
y R C Then for each R successor x of y we need a variable that
x
stand for in the context of x
De nition We assume that we have a countably in nite set
fx y z g of individual variables and for each pair x N a new nu
V
merical variable which may o ccur free in concepts Constraints constraint
x
systems and clashes are de ned as for ALC in De nition
An interpretation I is a mo del of a constraint system S i there is a mapping
I
and a mapping N IN such that I satisfy each
V
constraint in S i e we have
Chapter Expressive Number Restrictions
I
x y R for all xR y S
I
x D for all x D S
where D is obtained from D by replacing each variable by its image
y
y
In the sequel rel denotes any relation in f g A constraint system S is said
to b e numerically consistent i the conjunction of all numerical constraints in
S i e
S x rel n x rel
num R R y
x rel n R S x rel R S
y
x R N n IN x y R N N
R R V
is satis able in IN where x are interpreted as variables for nonnega
R y
tive integers
A constraint system S is called complete i none of the completion rules of
Figure can b e applied to S
The algorithm works exactly like the tableau based algorithm presented for
ALC in Section with the only di erences that
it uses the completion rules given in Figure and
it answers C is satis able only if it has generated a complete clash
free and numerically consistent constraint system The ALC algorithm
asked only for a complete and clash free constraint system
Before showing that this completion algorithm yields a decision pro cedure for
S
satis ability of ALUEN concepts let us make some comments on the rules
First note that each of the completion rules adds constraints when applied
to a constraint system none of the rules removes constraints and individual
variables x are never identi ed nor substituted With resp ect to this last
prop erty our algorithm di ers from the tableau based algorithms for ALCN
describ ed in Donini et al a and for ALCN presented in the previous
section Unlike our Rule these algorithms introduce for each constraint of
the form x R C a new R successor of x If x also has a constraint of the
form x n R and more than n R successors have b een introduced then
some of these individuals are identi ed Our Rule avoids identi cation by
guessing the number of allowed R successors of x b efore introducing these
Symbolic Number Restrictions
successors In fact since we have numerical variables b eside explicit numbers
and since restrictions on numerical variables in constraints x R can
y y
derive from di erent parts of the constraint system a similar identi cation on
demand would either lead to non termination or incorrectness The second new
feature is Rule Given a constraint x D we substitute a new numerical
variable for to make sure that the semantics of the existential quanti er
x
is ob eyed i e that the valuation for dep ends on x If we would just use
the di erence b etween R D and R D would not b e captured
S
Decidability of satis ability of ALUEN concepts is an easy consequence of the
following lemma
S
Lemma Let C b e an ALUEN concept in VNF and let S b e a constraint
system obtained by applying the completion rules to fx C g Then
The completion algorithm terminates when applied to fx C g
For each completion rule R that can b e applied to S and for each inter
pretation I i and ii are equivalent
i I is a mo del of S
ii I is a mo del of one of the systems S obtained by applying R
i
If S is a clash free numerically consistent and complete constraint sys
tem then S has a mo del
If S contains a clash or is not numerically consistent then S do es not
have a mo del
Pro of
The termination pro of is similar to the one for the tableau based algo
rithm for ALCN Donini et al a In fact termination is a conse
quence of the following prop erties of the completion rules
As stated ab ove neither individual variables are substituted nor
constraints are removed by the rules The application of a comple
tion rule strictly increases a constraint system
All concepts o ccurring in S are sub concepts of C
The maximal depth of constraints on an individual x is b ounded
S
by the maximal depth of C minus the length of the unique role
path from x to x
Chapter Expressive Number Restrictions
Intersection If x C u C S and x C S or x C S
S S fx C x C g
u
Union If x C t C S and x C S and x C S
S S S fx C g
t
S S S fx C g
t
Numerical Existential Quanti cation If x D S
and x D S
x
S S fx D g
x
New Ob jects
If xR y S for all y and m k are maximal such that
fx R E x R E x R D x R D g S and Rules
m k
cannot b e applied to S then for each n with n m and for each
n Partition P P f mg of m let S b e de ned as follows
i n i P
S S S fxR y j i ng fx n R g
R P i
fy E j i n j P g fy D j i n j k g
i j i i j
where y are new variables i e variables not o ccurring in S
i
Prophylactic new ob jects
If xR y S for all y and x R S and k maximal with
x R D S for i k x rel N R S for N IN or N
i y
for some y N and Rules cannot b e applied to S then S S
V
are de ned as follows
S S S fx R g
n
S S S fxR y g fy D j i k g fx R g
n i
where y is a new variables i e a variable not o ccurring in S
S
Figure The completion rules for ALUEN
Finally Rules and are applied at most once to each individual
and each role name and they introduce only nitely many direct
role successors for each individual
We only consider Rules and since Rules and are obvious If S
is generated by the application of a completion rule to S then S S
hence each mo del of S is clearly a mo del of S Thus we only have to
Symbolic Number Restrictions
consider the other direction
Numerical Existential Quanti cation Application of this rule adds
a constraint x C to S if x C is contained in S If I satisfy
x
I
S then we know that there exists an n IN such that x C n
Since the variable do es not o ccur in S by our assumption that every
x
variable is b ound only once in the input concept we can assume without
loss of generality that n and thus I satisfy x C
x x
New Ob jects Let x R k m b e as sp eci ed in the precondition of
Rule and let I satisfy S Then there exist some m and z z
I
such that
I
x z R for all i with i
i
I
0
E and for all j m there is some j f g with z
j j
I
for all j k for all i it holds that z D
i j
Since m k are maximal and Rules cannot b e applied to S the con
straints mentioned ab ove on R successors of x are the only ones implied
I
0 0
E In the i z by S Let P b e the partition of m with j P
j j j
corresp onding constraint system S new variables y are introduced
P i
Let y z for i Then
i i
I
x y R for all i
i
I
I
0 0
E R and y for each x R E S holds x y
j j j j
0
for j P
j
I
y D for all i j k and
i j
I
x
R
Hence I satis es S
P
Prophylactic New Ob jects Let x R k b e as sp eci ed in the pre
condition of rule and let I satisfy S Two cases are to b e distin
I I
guished If x then clearly I satis es S Now let x with
R R
I I
x z R for some z If we de ne y z then I satis es
S S fxR y g fx g fy D j i k g
R i
As usual we construct the canonical interpretation I induced by S
S
I I
S S
consists of the individual variables o ccurring in S x y R i
I
S
xR y S and x A i x A S This yields a tree like interpretation
which needs not to b e a mo del of S since some number restrictions might
not b e satis ed for the following reasons Either a an individual do es
not have any role successors but their existence is implied by number
Chapter Expressive Number Restrictions
restrictions or b it has some but not su ciently many role successors
Note that exact numerical restrictions on the number of role successors
are given by a solution in IN of the numerical constraints which are
satis able since S is numerically consistent In the rst case S do es
not contain any constraints on such role successors and we can simply
generate an appropriate number of them In the second case the idea is
to add su ciently many copies of some already existing role successor y
together with y s role successors Pro ceeding like this from the leaves to
the ro ot we end up with a mo del of S More precisely we start with an
interpretation I de ned as follows
I
S
I I I
j xR y S g for role names R f x y R
I I
A fx j x A S g for concept names A
Since S is clash free and complete I obviously satis es all constraints
b eside at least number restrictions This can b e shown by induction on
the structure of concepts similar to the pro of of Part of Lemma
I I
I I
Let and x y b e a solution of S A mo del of S is
num
x y
R R
then recursively de ned as follows
a Let
R I
m maxfn j x nR S g f j x R S g
y
x y
R I
j
then I is obtained from I by adding If x and m
j j R
x
R R I
j
for i m m new individuals a with x a R
xRi xRi
x x
b If x and I satis es all number restrictions on all role successors
S j
R R I
j
as k for some k and m of x x and m
R
x x
de ned ab ove then we choose an R successor y of x First we make
k copies of y and all its role successors Then the copies of y are
I
j
called a and I is obtained from I by adding all copies to
xRi j j
and by de ning
I I
j j
R R x a
xRi
i k
If neither a nor b applies to I then I is obviously a mo del of S All
n n
constraints that are not at least number restrictions are already satis ed
by I and they are still satis ed after the execution of Step a b ecause S
is complete and thus do es not imp ose any constraints on the individuals
added by Step a otherwise Rule could b e applied to S They are
Symbolic Number Restrictions
also still satis ed after the execution of Step b The individuals added
by Step b are copies of already existing R successors which already
satisfy all constraints that are imp osed by S on R successors Finally
I satis es also all at least number restrictions in S
n
S
Step b is so simple b ecause ALUEN has the tree mo del prop erty and
b ecause the usage of a solution for S makes sure that we do not intro
num
duce more role successors than allowed by at most number restrictions
If S contains a clash then it is obviously unsatis able If S is complete
and I is a mo del of S then we can deduce a solution for S in IN
num
as follows We de ne
I
x x
R R
x
x
Since I satis es all constraints in S this obviously yields a solution for
S and thus S is numerically consistent
num
S
Theorem Satis ability of ALUEN concepts is decidable
Pro of Again since Lemma is the same as Lemma b esides the
di erent logics the same arguments which were used on in Remark to
show that Lemma implies decidability of satis ability of ALC concepts now
S
yield decidability of satis ability of ALUEN concepts
S
Unfortunately since ALUEN is not closed under negation subsumption can
not b e reduced to satis ability A closer lo ok at the sp eci c form of the concept
C introduced in Figure reveals that it can b e written as C D u D
D D
S
for two ALUEN concepts D D In fact D is the rst conjunct of C and
D
D is the negation of the remainder of C Note that D do es not contain
D
numerical variables Furthermore all numerical variables o ccurring in the re
mainder of C are universally quanti ed which shows that D contains only
D
existential quanti cation of numerical variables Since D u D is unsatis able
i D v D this implies
S
Theorem Subsumption of ALUEN concepts is undecidable
Chapter Expressive Number Restrictions
Related work
Certain Mo dal Logics and Description Logics can b e translated into rst order
logic such that only two di erent variable names o ccur in the formulae ob
tained by this translation see Remark Thus decidability of subsumption
and other inference problems for these languages follows from the known de
cidability result for L i e rst order logic with two variables and without
function symbols Mortimer Recently this decidability result has b een
extended to C i e rst order logic with variables and counting quanti ers
Gr adel et al Indep endently it has b een proved in Pacholski et al
that satis ability of C formulae can b e decided in nondeterministic doubly
exp onential time As an immediate consequence satis ability and subsump
tion for ALCN t u the extension of ALC by number restrictions with
inversion and Bo olean op erators on roles is still decidable It should b e noted
however that expressing comp osition of roles in predicate logic requires more
than two variables
Using sophisticated techniques for translating Description Logic concepts into
formulae of Prop ositional Dynamic Logics it has b een shown in De Giacomo
Lenzerini that deciding satis ability and subsumption for a very expres
sive extension of ALC N is ExpTime complete This extension allows for
reg
qualifying number restrictions on atomic and inverse roles The number restric
tions on inverse roles together with value restrictions involving the transitive
closure of roles lead to the loss of the nite mo del prop erty as the following
example shows
f u f u f f u f
It is obviously satis able for example together with f interpreted as the
successor function on the nonnegative integers is surely an instance of this
concept but it do es not admit a nite mo del
To our knowledge the only constructor available in other logics that has some
of the expressive p ower of existential symbolic number restrictions is in nite
disjunction As a consequence the decidability result presented in this chapter
are not subsumed and do not subsume other results
Chapter
Transitive relations in
Description Logics
As argued in Chapter transitive relations play an imp ortant r ole in the
representation of aggregated ob jects In the following three di erent ways of
extending Description Logics by transitive roles are investigated with resp ect
to their computational complexity Finally it is discussed in how far these
approaches are suitable for the representation of aggregated ob jects and which
of the demands made in Chapter are satis ed by these approaches
Three extensions of the concept language ALC by di erent kinds of transitivity
are discussed in this section
Transitive closure of roles In ALC the op erator can b e applied
to role names The role R is then interpreted as the smal lest transitive
relation containing R
certain roles must b e interpreted as tran Transitive roles In ALC
R
sitive roles but without the p ossibility to relate them to a generating
role as in the rst extension
Transitive orbits of roles In ALC the op erator can b e applied to
role names The role R is then interpreted as some not necessarily the
smallest transitive relation containing R
As a consequence of the results given in Fischer Ladner Pratt
the basic inference problems for ALC i e ALC extended by the transitive
closure of roles are ExpTime complete whereas these problems are PSpace
complete for pure ALC Since one might not b e able or willing to pay this
Chapter Transitive relations in Description Logics
high price for transitivity lo oking for alternatives to the transitive closure is a
quite natural step to undertake Using results from Mo dal Logic Ladner
Halp ern Moses we show that these problems remain PSpace complete
Finally we prove that for ALC these problems are as hard as for for ALC
R
ALC namely ExpTime complete
ALC extended by the Transitive Closure of
Roles
In Section syntax and semantics of ALC have already b een introduced
and in Section the expressive p ower added by the transitive closure
op erator is describ ed We recall that ALC do es no longer have the nite
tree mo del prop erty but the nite mo del and the tree mo del prop erty As
mentioned ab ove and in Section satis ability and subsumption of ALC
concepts are ExpTime complete These problems are in PSpace for ALC
hence the source of this high computational complexity is the transitive closure
op erator
This increase in computational complexity is due to the fact that using uni
versal value restrictions on transitive roles one can propagate constraints on
all role chain successors For example the concept
C A u R A u R R A
makes sure that each individual reachable over an R path of arbitrary but nite
length has itself an R successor To prevent a tableau based algorithm from
generating all these R successors and thus lo oping cycle detection mecha
nisms are necessary In Section these cycle detection mechanisms were
already motivated We recall that one has to check for each individual that is
generated by the algorithm whether its constraints are the same as those on
one of its role predecessor If this is the case one has furthermore to distin
guish b etween cases where constraints on individuals propagated along some
role chain are simply re generated but satis ed and cases where the satisfac
tion of constraints is always p ostp oned Beside others these cycle detection
mechanisms require the storage of all constraints on all role predecessors of an
individual whose constraints are tested for satis ability When started with
the constraint system fx C g the constraints on a role successor of x can
involve any subset of the sub concepts of C Let n b e the length of C Then
the number of sub concepts of C is linear in n but the number of subsets
of sub concepts of C is exp onential in n Intuitively ExpTime hardness of
ALC extended by Transitive Roles
ALC is due to the fact that there are satis able ALC concepts where every
mo del has a role chain of length exp onential in the length of the input concept
that the storage of the constraints along this chain is necessary and that this
storage needs space exp onential in the length of the input concept For ex
ample replacing by in the ALC concept D de ned in Section yields
an ALC concepts D whose smallest mo dels have an R path whose length is
exp onential in the length of D
In Sections and it will b e shown that the source of ExpTime hardness
is not transitivity of roles p er se but the interaction b etween an ordinary
relation R and a transitive sup errelation here R
ALC extended by Transitive Roles
When the corresp ondence b etween Mo dal Logics and Description Logics was
discovered results from the eld of Mo dal Logic gave new insight into problems
concerning Description Logics De Giacomo Lenzerini a b Schild
De Giacomo For example it is well known Schild that ALC
is a notational variant of the prop ositional Multi Mo dal Logic K Results for
n
the Mo dal Logic K which is a Multi Mo dal Logic with n so called agents
n
extending prop ositional logic gave the imp etus to lo ok closer at transitive
roles as a di erent and hop efully cheaper way to extend Description Logics
by transitive relations The results will b e given in this section
If ALC interpretations are restricted to those where all role names are inter
preted as transitive relations then there is a corresp ondence b etween K
n
formulae and ALC concepts such that a K formula is satis able i its trans
n
lation into an ALC concept is satis able with resp ect to the restricted seman
tics However this restricted semantics is to o restricted As we have argued in
Chapter adequate representation of aggregated ob jects requires transitive
relations but this do es not mean that al l relations should b e transitive In
this Section we will investigate ALC extended with transitive roles b eside
ordinary roles
In Halp ern Moses it is shown that satis ability of K formulae is
n
PSpace complete We will extend this result to mixed K K formula or
m
in Description Logics vocabulary to ALC with transitive and ordinary roles
This extension is straightforward but not trivial For example the Mo dal Logic
of one equivalence relation S is NP complete whereas the combination of
two of these logics S S is PSpace complete
Chapter Transitive relations in Description Logics
is an extension of ALC obtained by allowing the use of De nition ALC
R
transitive roles inside concepts The set of role names N is a disjoint union of
R
role names N fP P g and role names N fR R g In addition
P
I I
to what holds for ALC interpretations an interpretation I has to
interpret role names R N as transitive roles i e an interpretation has to
i
satisfy the additional condition
I I I
if d e R and e f R then d f R
i i i
for each role R N
i
In this section a tableau based algorithm is presented which tests for the
concepts The algorithm extends and combines those satis ability of ALC
R
presented in Halp ern Moses for Multi Mo dal Logics in order to deal
with the simultaneous use of b oth ordinary and transitive roles It will b e
shown that this algorithm uses space p olynomial in the length of the input
concept
The tableau based algorithm given b elow is similar to the one given in Sec
tion except for the mo di cations that were necessary for the correct han
dling of transitive roles and the mo di cations that came in b ecause this al
gorithm should use only p olynomial space One di erence can b e found in
the nondeterminism of Rule which handles disjunction Another di erence
is that the role successorship b etween two variables x y is no longer repre
sented by a constraint xR y but by the fact that a no de y which contains
all constraints concerning y is an R successor of the no de x which contains
all constraints concerning x The canonical mo del of a satis able concept is
constructed by using the whole tree generated by the completion algorithm
instead of just using a clash free leaf Again all concepts are supp osed to
b e in negation normal form
concept C in De nition Constraints are of the form x C for an ALC
R
concept C the NNF and a variable x When started with an ALC
R
completion algorithm starts with a tree consisting of a single no de the ro ot
x lab elled with the constraint system fx C g Using the rules given in
Figure the completion algorithm expands this tree by adding either new
no des lab elled with constraint systems or new constraints to lab els of no des
Such a tree namely a tree where each no de x is lab elled with a set of constraints
S and whose edges are lab elled with role names in N is called a completion
x R
tree A no de y that is a successor of a no de x is called a P successor of x if
the edge b etween them is lab elled with P For a transitive role R N y is
ALC extended by Transitive Roles
called an R successor of x if there is a path from x to y where each edge is
lab elled with R A no de x is called an ancestor of a no de y if there is a path
from x to y regardless of the lab els of its edges The set of variables o ccurring
in a completion tree T is the same as the set of its no des and is denoted by
T
An interpretation I is a model of a completion tree T i there is a mapping
I I
that maps all no des of T to individuals in such that
T
I satis es all constraints in all constraint systems that lab el no des in T
I
i e x C for all no des x and all constraints x C S and
x
I
x y R for all x y where y is an R successor of x
T
The rules given in Figure are applied only to leafs lab elled with clash free
constraint systems The completion algorithm terminates if no more rules can
b e applied A tree is called complete if no more rules can b e applied to it and
clash free if all no des are lab elled with clash free constraint systems When
started with a constraint system fx C g the completion algorithm answers
with C is satis able if and only if the rules can b e applied in such a way
that they generate a complete and clash free tree
To present the rules in a uni ed way the following abbreviations are introduced
De nition Let x y P N and R N b e transitive role names
P
and let S b e the lab el of a no de in a completion tree Then the x consequences
x
S P y resp S R y for y via P resp R are de ned as follows
x x
S P y fy C j x P C S g
x x
S R y fy C j x R C S g fy R C j x R C S g
x x x
Similar to Lemma the pro of of soundness and completeness can b e split
into subtasks
concept in NNF let T b e a tree obtained Lemma Let C b e an ALC
R
by applying the completion rules to fx C g Then
For each completion rule R that can b e applied to T and for each
interpretation I i and ii are equivalent
i I is a mo del of T
ii R can b e applied in such a way that it yields some T satis ed by I
Chapter Transitive relations in Description Logics
Conjunction If x C u C S and x C S or x C S then
x x x
S S fx C x C g
x x
Disjunction If x C t C S and x C S and x C S then
x x x
S S fx D g for D fC C g
x x
Value and existential restriction
If Rule and Rule cannot b e applied to S then
x
for each x P C S with P N create a P successor y
x P
lab elled with
S fy C g S P y for a new y
y x
for each x R C S with R N if there is no ancestor y
x
of x where fy C g S R y S then create an R successor y
x y
lab elled with
S fy C g S R y for a new y
y x
Figure The completion rules for ALC
R
If T is a complete and clash free completion tree then T has a mo del
If T contains a clash then T do es not have a mo del
The completion algorithm terminates when applied to fx C g
Pro of of Lemma The if direction All rules expand a completion
tree hence if I is a mo del of the tree T obtained from T by applying R then
it is clearly also a mo del of T
The only if direction Let I b e a mo del of T If Rule can b e applied to a
leaf x then x C u C S and I clearly satis es T If Rule is applicable
x
to a leaf x then x C t C S and I satis es either x C or x C hence
x
Rule can b e applied in such a way that I is a mo del of the tree T generated
by Rule from T Now let Rule b e applicable to a leaf x in T It has to b e
shown that I satis es all constraint systems S generated by Rule
y
I
Let b e the corresp onding mapping of individual variables to
T
I
elements of and let S fy C g S P y b e the lab el of one of the new
y x
no des generated by Rule Then x P C S and x has at least one x
ALC extended by Transitive Roles
I I I
P successor b with b C and b D for all D with x P D S
x
Hence I with y b clearly satis es S
y
Now let S fy C g S R y for some R N b e the lab el of one of the
y x
new no des generated by Rule Then x R C S and x has at least one
x
I I
R successor c with c C Since I satis es T all constraints of the form
I I
x R D are also satis ed and c D for all D with x R D S Finally R
x
is a transitive relation Hence if c has an R successor d this individual is also
I
an R successor of x and thus d D for all D with x R D S Hence
x
I also satis es all constraints of the form y R D S for x R D S and
y x
thus satis es the whole constraint system S
y
As a consequence I satis es each constraint system that lab el a no de added
by Rule Since the lab el S of a new no de y contains only constraints
y
concerning y namely those of the form y C the extension of the mapping
to y is indep endent from the extension of to other no des y and thus I
satis es all constraint systems lab elling no des added by Rule Finally Rule
do es not change the lab els of no des in T Summing up we have that I satis es
the tree T generated by Rule
Pro of of Lemma Let T b e a complete and clash free completion tree
with variables and let r b e the transitive closure of a binary relation r
T
i e
n
r r
n IN n
We de ne a mo del I as follows For concept names A and role names P N
P
and R N we de ne
I
T
I I
A fx j x is a no de in T and x A S g
x
I I I
P f x y j y is an P successor of x in T g
I I I
R f x y j y is an R successor of x in T g
I I
f x y j x R C S for some C and y is an an
x
cestor of x in T with fy C g S R y S g
x y
T is a completion tree Hence all constraint systems lab elling no des in I are
on di erent variables It remains to show by induction on the structure of
concepts that I is indeed a mo del of T
First by de nition I satis es all constraints of the form x A in T for
concept names A
Chapter Transitive relations in Description Logics
T is clash free hence I satis es by de nition all constraints of the form
x A
Since T is complete it follows by induction on the structure of concepts
that constraints of the form x C u C x C t C are also satis ed
I
Next I obviously interprets all P successors in T as P successors in I
Transitive role names R N are interpreted by de nition by transitive
relations i e by the transitive closure of a binary relation
For role names P N it follows immediately from the completeness of
P
T and by induction that I satis es all constraints of the form x P C
Furthermore value restrictions of the form x P D are taken into ac
count by S P y and since neither Rule nor Rule can b e applied
when a P successor y of x is generated by Rule all value restrictions
concerning P successors of x are taken care of
We now consider existential value restrictions on transitive roles If an
R successor y was generated for some x R C then x R C is surely
satis ed by I If no R successor was generated for x R C then there
exists an ancestor y of x with fy C g S R y S According to the
x y
I I
de nition of R x y R for such an R successor In b oth cases I
satis es constraints of the form x R C by induction
Finally supp ose there exists some x such that I do es not satisfy
T
a constraint x R D S This is only p ossible if there exists some
I I I I
y with x y R and y D According to the de nition of R
this y is either an R successor of x in T or an ancestor of x with
fy C g S R y S or an R successor of such an ancestor In all
x y
three cases due to the de nition of Rule y D S By induction
y
I
y D in contradiction to the assumption
Pro of of Lemma If T contains a clash there is some leaf x in T with
fx A x Ag S Hence no interpretation can satisfy S and thus T is
x x unsatis able
ALC extended by Transitive Roles
Pro of of Lemma For a constraint system S the maximum role depth
x
of S depth S is the maximum of nested R C R C concepts of the
x x
concepts o ccurring in S i e
x
depth S maxfdepth C j C S g
where depth C is already de ned in the pro of of Part of Lemma
Furthermore sub C denotes the set of sub concepts of C
sub A fAg for all concept names A N
C
sub C f C g sub C
sub C u D fC u D g sub C sub D
sub C t D fC t D g sub C sub D
sub R C f R C g sub C
sub R C f R C g sub C
We recall that the number of sub concepts of C is linear in the length of C In
the following we assume that sub C is linearly ordered and that fP P
R R g is the set of role names o ccurring in C where P is a role name
n i
in N and R a role name in N
P i
Now let m jsub C j Then depth S m for all no des x in T Since all
x
no des in T are lab elled with subsets of sub C jS j m for all no des x
x
Besides showing termination an upp er b ound for the space needed by the
algorithm will b e given hence the depth of the tree constructed is investigated
more closely
Fact If y is a P successor of x then depth S depth S If y is an
i y x
R successor of x C S and C is not of the form R C then
i y i
depth C depth S
x
Fact If z is a P or an R successor of y y is a P or an R successor of x
i i j j
for i j then depth S depth S
z x
Fact The only way that the depth of the lab els do es not decrease is along
some R path Let x x b e no des on such a path lab elled with clash
i k
free constraint systems S such that each x is an R successor of x
x j i j
j
Then each S can b e divided into three parts S S and S
x j j j
j
The rst S consists of S R x The second is S fx C g
j x i j j j
j
where x R C led to the creation of x The third consists of all
j i j
those constraints that were added to S by Rule or
x
j
Chapter Transitive relations in Description Logics
By de nition we have S S and S for all j k If
j j j
S fx C g then for all ancestors w of x thus for all x with
j j j
j we have
S S x w S R w fw C g S
j j j x w
j
by de nition of Rule There are at most m di erent choices for S and
j
at most m di erent choices for S along this R path Rule tests only
j i
whether S S is not contained in the lab elling of an ancestor
j j
of x b efore creating a new no de x Hence the number of di erent
j j
choices for subsets S do not contribute to the length of this R path
j i
and we thus have k m
Collecting these facts we have that along one path in the completion tree
the depth of the constraints lab elling no des on this path may not decrease for
at most m consecutive no des and that the depth can decrease at most m
times Hence the length of paths in the completion tree is b ounded by m
Furthermore it is of b ounded out degree Hence its construction terminates
As a consequence of the facts in the pro of of Lemma the completion algo
concepts rithm yields not only a decision pro cedure for satis ability of ALC
R
but also a PSpace decision pro cedure
concepts is PSpace complete Theorem Satis ability of ALC
R
Pro of As a consequence of results in Savitch for showing that a prob
lem is in PSpace it su ces to give a nondeterministic decision pro cedure
that uses only p olynomial space The completion algorithm handles disjunc
tion in a nondeterministic way In contrast to other completion algorithms
presented within this thesis which test b oth p ossibilities for constraints of the
form x C t D it chooses nondeterministically b etween x C and x D
As stated in the pro of of Lemma the tree T constructed by the tableau
construction algorithm for C is of depth at most m where jsub C j m
Once this algorithm has visited all successors of a no de without detecting a
clash it can forget ab out the subtree b elow this no de and reuse the space
where it was memorised Each S is a subset of sub C hence each S can
x x
b e stored in m bits given that we have a table that can b e used to asso ciate
An upp er b ound for the outdegree is m
ALC extended by Transitive Roles
binary m vectors with subsets of sub C Such a table can b e stored easily in
m times the space that is needed to store C There are less than m concepts
of the form R C in sub C hence there are less than m subtrees directly
b elow a no de x and we can memorise in m bits which of them still have to b e
investigated
Summing up at each moment the algorithm is running it has to store the
following information for its actual no de x at depth h
S
x
which of the subtrees b elow x still have to b e investigated and
these two pieces of information for each of its h ancestors
This information can b e stored in m m h m m h m bits The
ab ove mentioned table can b e stored in m bits Since h m the tableau
construction algorithm needs at most c m m m bits of storage for
some constant c
concepts is in PSpace Finally As a consequence satis ability of ALC
R
is an extension of ALC PSpace hardness follows from the facts that ALC
R
and that satis ability of ALC was shown to b e PSpace complete in Schmidt
Schau Smolka
is lower than the one of As we have seen worst case complexity of ALC
R
ALC The price in expressive p ower one has to pay for this lower complexity
is illustrated by the following example Supp ose devices signal lines and pip e
connections are three disjoint concepts and we want to describ e devices whose
direct parts are either signal lines or devices Furthermore these devices must
have as a p ossibly indirect part a pip e connection In ALC these devices
can b e describ ed by the following concept
V Device Device u
has part Device t Signal Line u
part Pipe Connection has
some part is These devices cannot b e describ ed if only a transitive role has
available without the p ossibility to relate this transitive role to a p ossible
we can only refer to parts without the non transitive subrole In ALC
R
p ossibility to distinguish b etween direct and indirect parts Replacing in the
ab ove example has part and has part by has some part yields an unsatis
able concept
Chapter Transitive relations in Description Logics
ALC extended by Transitive Orbits
The gap in expressive p ower and in computational complexity separating ALC
motivated the search for a compromise b etween these two ways of and ALC
R
extending Description Logics by transitive roles As it turned out there is one
natural candidate for this compromise which will b e called transitive orbits
It allows one to relate a role to a transitive sup errole called the transitive
orbit of the role Since the relationship b etween a role and its transitive orbit
is less strong than the one b etween a role and its transitive closure one might
hop e that transitive orbits lead to a lower computational complexity than the
transitive closure of roles
De nition ALC is the extension of ALC obtained by allowing the use of
transitive orbits of roles inside concepts The transitive orbit of a role R is
I
denoted R and interpreted as a transitive role containing R i e an ALC
interpretation has to satisfy the additional conditions
I I I
R R and R is transitive
A small example is given to highlight the di erence b etween ALC and ALC
concepts Let
Carc Device Device u has part Carcinogenic
I
Device Then d Carc Device even if there and let I b e a mo del of Carc
I I
is no has part chain from d to some c Carcinogenic For d b eing an
I
instance of Carc Device it is su cient that there is some c Carcinogenic
I
with d c has part
As a consequence transitive orbits are easier to handle by tableau based al
gorithms than the transitive closure of roles When trying to build a mo del
value restrictions of the form R C can b e treated by simply introducing
n
an R successor y which do es not need to b e an R successor for any n Hence
for y b eside C we have to take care only of universal value restrictions on R
and not of universal value restrictions on R In contrast to this for value re
strictions of the form R C in principle we have to guess an n for that this
n
successor and we have thus to take value restriction is satis ed by some R
care also of universal value restrictions on R For transitive orbits it is neither
necessary to guess such an n nor to take care of universal value restrictions on
R
In general each mo del of an ALC concept D is also a mo del of its ALC
counterpart which is obtained by replacing each R in D by R If C v D
ALC extended by Transitive Orbits
holds for two ALC concepts C D then clearly C v D holds for their ALC
counterparts C D The converse do es not hold
R A u A v R A u A
I
holds if is replaced by if x has no R successors then it has clearly no
I
R successors but it do es not hold if is replaced by x can have an
I
I
R successor without having an R successor
Now the computational complexity of ALC is investigated The completion
algorithm given in Section can easily b e mo di ed to handle ALC concepts
It su ces to mo dify the de nition of the x consequences S R y resp S R y
x x
for y via R resp R
S R y fy C j x R C S g
x
fy C j x R C S g fy R C j x R C S g
S R y fy C j x R C S g fy R C j x R C S g
x
and to distinguish b etween R and R successors where the rst ones are
generated for constraints of the form x R C and the latter ones for constraints
of the form x R C Furthermore the test whether there already exists some
predecessor lab elled with a sup erset of the constraints to b e generated has to
b e p erformed b efore creating any successor
It can easily b e seen that these mo di cation yield soundness and completeness
for ALC concepts An imp ortant p oint to note is that R successors generated
by Rule are lab elled with consequences from value restrictions on R as well
as on R whereas R successors are pure R successors This is to say that
their lab els do not include any constraints y C for value restrictions x R C
n
This is p ossible b ecause an R successor do es not need to b e an R successor
for any n
concepts the depth In contrast to the depth of the trees constructed for ALC
R
of trees constructed by this mo di ed algorithm can no longer b e b ounded
p olynomially in the length of the concept For example if A is de ned as
i
given b elow each mo del of the concept
D R A u A u A u
n
R R u A u A u u A
n
n
has paths of length The individuals on this path can b e viewed as the
n
representations of the binary enco ding of the numbers from to Let I
I
b e an interpretation with x D The concept D is de ned such that
Chapter Transitive relations in Description Logics
each R successor of x can b e viewed as the representation of a number
n
b etween to
n
no R successor of x can represent b oth k and for k
x has an R successor representing and
for each of its R successors y if y represents k then it has an R
n
successor representing k mo d
In order to ensure this each A expresses the correct switching of the i th bit
i
I
n k
when counting from up to If y is an R successor of x we have
I
that y A i the i th bit in the binary enco ding of k is equal to More
i
precisely
A A u R A t A u R A
A u A u A u R A t A u R A t
i j i i i i
j i
u A u A u R A t A u R A
j i i i i
j i
I
The length of D is quadratic in n whereas each mo del I of D has an R path
n
of length in O
Hence there are ALC concepts whose smallest mo dels are of a relatively
large depth namely exp onential in the size of the concept This prop erty is
a go o d hint that satis ability of ALC concepts cannot b e decided using only
p olynomial space As the following theorem shows this is indeed true
Theorem Satis ability of ALC concepts is ExpTime complete
Pro of Satis ability of ALC concepts is in ExpTime b ecause it can b e
decided by the mo di ed tableau construction algorithm It is easy to see that
for an ALC concept C this mo di ed algorithm creates a tree whose depth
is exp onentially b ounded by the length of C b ecause a successor of a no de x
lab elled with S is only generated if no ancestor is lab elled with a sup erset of
y
S
y
To show that satis ability of ALC concepts is indeed ExpTime hard we can
mo dify the pro of ExpTime hardness of the satis ability of PDL formulae given
in Fischer Ladner The pro of gives for an alternating Turing Machine
M and an input word x a PDL formula f such that f is satis able i
M x M x
x is accepted by a simpli ed trace of M A translation of PDL formulae into
ALC concepts can b e found in Schild This translation is satis ability
Representation of part whole relations
preserving and yields a concept whose length is linear in the length of the input
concept If this translation is applied to f it yields an ALC concept D
M x M x
using a single role R and its transitive closure R This concept is of the form
D C u C u R C
M x
where R is the only role name o ccurring in C Furthermore R o ccurs
M x
neither in C nor in C Because of this sp ecial form D is satis able i
M x
its ALC counterpart D C u C u R C is satis able
M x
Each mo del of D is clearly a mo del of D Now let I b e a mo del of
M x
M x
0
0
0 0
I
I
I I
D with x D Then x C u C and for all y with x y R
M x M x
0
I
it holds that y C Let I b e an interpretation of D which is equal to I
M x
I
for concept and role names in C C and where R is the transitive closure of
0
0
I I
I I
R Hence we have that x y R implies x y R for all x y
I
I I
It follows that y C for all x y R which yields x D Hence I
M x
is a mo del of D
M x
Representation of part whole relations
In this section the general observations concerning part whole relations made
in Chapter are related to the results concerning expressive p ower and compu
tational complexity of transitive relations in Description Logics in this chapter
As discussed in Chapter the minimal prerequisite for the adequate repre
sentation of part whole relations is the p ossibility to express that a certain
relation the general part whole relation is transitive This p ossibility is
given in all extensions of ALC investigated in this chapter Hence if only
this general part whole relation is needed the cheapest kind of transitivity
namely transitive roles is su cient If ALC is chosen as the basic Description
is the appropriate extension for the application Logic this means that ALC
R
under consideration
In case the application asks for the distinction b etween di erent part whole
relations the question arises which Description Logic has enough expressive
p ower for an adequate representation of these part whole relations At the
same time the according inference problems should b e of a computational
complexity that is as low as p ossible We will rst consider integral part
whole relations Given an integrity condition Int the according integral
part whole relation was de ned in Section as
Int
def
x y x y Int x Int
Chapter Transitive relations in Description Logics
given a transitive role part In ALC of integral part whole relations can b e
R
represented as follows First introduce a role name intpw and a concept Int
for the integral part whole relation Then the following substitutions will
Int
force all value restrictions to b e interpreted in the appropriate way namely to
address exactly those parts which are integral with resp ect to Int
of C intpw C ; Int u part
intpw C ; Int t part of C
In this solution the concept Int introduced for an integrity condition can
b e atomic or complex Hence concepts representing integrity conditions can
b e ordered with resp ect to the subsumption relation which induces a natural
hierarchy on the corresp onding integral part whole relations For example
let If the corresp onding integrity conditions are mo deled in an
Int
Int
appropriate way i e Int v Int it follows immediately that intpw C v
intpw C Furthermore integral part whole relations inherit transitivity
from the general part whole relation This is to say that
intpw C v intpw intpw C
which is a translation of the axiom for transitive frames in Mo dal Logic
see Halp ern Moses
is also expressive enough for the adequate representa As a consequence ALC
R
tion of the general part whole relation together with a hierarchically structured
set of integral part whole relations provided that the integrity conditions can
Hence ALC can b e extended by the general part whole b e expressed in ALC
R
relation together with an imp ortant class of part whole relations without losing
PSpace completeness of the relevant inference problems
In contrast to this comp osed part whole relations as subrelations of the general
part whole relation ask for more expressive p ower In Section these
comp osed part whole relations were de ned as
def
x y x y x CompRel y
Comp
To represent a single comp osed part whole relation it su ces to use as
Comp
the general part whole role the transitive orbit CompRel of a role CompRel
representing the additional condition CompRel
If more than one comp osed part whole relation must b e represented transi
tive orbits or the transitive closure are no longer su cient b ecause they allow
only to relate one transitive role to one subrole To overcome this problem one
could use ALC instead of ALC and represent the general part whole relation
reg
Representation of part whole relations
as the transitive closure of the union of all its sub part whole relations Alter
a Description Logic introduced in Horro cks natively one could use ALCH
R
Gough which was designed to overcome the ab ove mentioned lack in ex
by role hierarchies that is the p ossibility pressive p ower It extends ALC
R
to sp ecify inclusion axioms R v S on roles regardless of whether they are
transitive or not For example to represent two non transitive comp osed part
whole relations one introduces two non transitive role names
Comp
Comp
of and adds the inclu CompRel CompRel and one transitive role name part
sion axioms
CompRel v part of and CompRel v part of
A mo del of these axioms interprets b oth and as subroles of the
Comp
Comp
of Although the extension of ALC by inclusion axioms transitive role part
on p ossibly transitive roles leads to ExpTime hardness of the corresp onding
inference problems since transitive orbits can b e expressed the authors of
Horro cks Gough claim that the algorithm b ehaves quite well in prac
tice that it is much simpler than the algorithms for ALC and that it is
amenable to optimisation techniques
Summing up there are mainly two alternatives for the representation of part
whole relations in Description Logics
If the application do es not ask for the representation of comp osed part
is expressive enough even if hierarchically whole relations then ALC
R
structured integral part whole relations are needed In this case the
inference problems are still in PSpace
If the application asks for comp osed part whole relations b esides the gen
are the least expressive eral part whole relation then ALC or ALCH
R
prop ositionally closed Description Logic able to represent arbitrary roles
together with transitive sup erroles Unfortunately even for ALC the
relevant inference problems are ExpTime hard
Chapter
Conclusion
This thesis rep orts the investigations that were made to nd out which knowl
edge representation system based on Description Logics is b est suited for sup
p orting the design of pro cess mo dels To b e well suited for the pro cess engi
neering application a Description Logic has to provide the expressive p ower
to describ e relevant prop erties of the building blo cks pro cess mo dels consist of
Furthermore this Description Logic has to supp ort the structured storage of
these building blo cks To satisfy the latter requirement the relevant inference
problems should b e of an acceptable computational complexity namely at least
decidable The goal of these investigations was not to implement a complete
DL system but to extend already existing Description Logics such that they
provide the expressive p ower required for the pro cess engineering application
and to investigate the computational prop erties of these Description Logics
As a matter of fact no such b est suited DL system could b e chosen This is
not only due to the limited amount of time available for these investigations
but also due to the fact that such a b est suited DL system do es not need
to exist Roughly sp oken the pro cess engineering application asks for a DL
system with rather high expressive p ower and decidable inference problems
Now the expressive p ower of a DL system is prop ortional to the computational
complexity of its relevant inference problems Additionally expressive p ower
is a multi dimensional prop erty and thus there are several candidates for a
b est suited DL system The most imp ortant of these candidates were de ned
and investigated with resp ect to the computational complexity of the relevant
inference problems and expressive p ower
To cop e with the high complexity of pro cess mo dels aggregated ob jects are
widely used while building pro cess mo dels As a consequence the part whole
relation which relates parts to aggregated ob jects plays an imp ortant r ole for
the description of aggregated ob jects in pro cess mo dels The part whole re
lation has some inherent prop erties which ask for an adequate representation
the most imp ortant of which is transitivity Beside the general part whole re
lation it seems natural to use various sub part whole relations whose meaning
is mostly only given by intuitive examples in literature For the adequate rep
resentation of the general part whole relation and its sub part whole relations
exact de nitions of their prop erties and the way in which they interact are
necessary Since these exact de nitions were to our knowledge lacking we
established a taxonomy of part whole relations which gives a precise scheme
for the classi cation of sub part whole relations This scheme implies b oth the
main prop erties of the sub part whole relations de ned within this scheme as
well as the way in which they interact With this taxonomy in mind three
ways of extending Description Logics by transitive relations were investigated
namely transitive roles transitive orbits and the transitive closure of roles
Transitive roles p er se without referring to an underlying subrole are algo
rithmically easier to handle than the transitive closure or transitive orbits of
roles Hence when using a Description Logic system for an application that
asks for the adequate representation of aggregated ob jects it is worth ask
ing which kind of sub part whole relations are imp ortant in this application
We could show that if b eside the general part whole relation only integral
part whole relations are needed then transitive roles are expressive enough
However the adequate representation of comp osed part whole relations asks
for transitive orbits or hierarchies of p ossibly transitive roles which b oth in
crease the complexity of the relevant inference problems All solutions enable
us to de ne concepts or describ e individuals by referring to parts at a level of
decomp osition not known and not b ounded in advance As a by product all
solutions provide the expressive p ower to refer to ancestors friends or relatives
in any generation or in any degree of relationship
The complexity results hold for the case where ALC is extended by transitive
roles the transitive closure or the transitive orbits An interesting question
arising from these observations is whether these results hold for extensions of
other Description Logics as well but its investigation go es b eyond the scop e
of this work
Expressive number restrictions Since a pro cess mo del strongly dep ends
on the way in which its building blo cks are interconnected the pro cess engi
neering application asks for a Description Logic that has the expressive p ower
to describ e this structural information Expressive number restrictions can
provide this expressive p ower This can b e seen in the fact that for exam
ple ALCN do es not have the tree mo del prop erty Within this thesis two
Chapter Conclusion
approaches to make number restrictions more expressive were investigated
Symbolic number restrictions turned out to b e very expressive For exam
ple they allow to describ e symmetry conditions or the dep endency b etween
the number of an ob jects connections and the number of the connections of
its parts Unfortunately this high amount of expressive p ower makes all rele
vant inference problems undecidable However a slight restriction of this logic
could b e shown to have a decidable satis ability problem even though the
according subsumption problem is still undecidable
Number restrictions on complex roles proved also to b e very expressive
but in contrast to symbolic number restrictions it was p ossible to design a
new expressive and decidable Description Logic namely ALCN This do es
not hold for other extensions by either more complex roles inside number re
strictions than those built using comp osition only or by a more expressive
underlying Description Logic namely ALC Within this thesis a variety of
op en problems could b e solved concerning the e ects complex roles inside num
b er restrictions have on the computational complexity of the relevant inference
problems
In order to evaluate the suitability of DL system as a means to structure the
storage of pro cess mo del building blo cks the DL system Crack was integrated
into the pro cess mo deling to ol ModKit This integration is realized in such
a way that browsing the class hierarchy de ning new classes and migrating
mo di ed prototypes are supp orted by the system services provided by Crack
The fact that the expressive p ower of Crack is high enough for providing this
supp ort is due to the fact that all information concerning ob jects and classes in
ModKit that can b e deduced automatically from their sp eci cations can b e
expressed in Crack A pro cess mo deling to ol where more of this information
could b e deduced could pro t from the integration of a more expressive DL
system
Consequences of the complexity results for
the pro cess mo deling application
The results of the investigations of the computational complexity of Descrip
tion Logics can b e summarised as follows a Complex roles inside number
restrictions lead to the undecidability of the relevant inference problems for
some combinations of the role forming constructors that include comp osition
whereas the basic Description Logic ALC extended by comp osition inside num
Consequences for the application
b er restrictions still has decidable inference problems for an overview see Fig
ure b If ALC is extended by symbolic number restrictions of the form
motivated by the examples it still has a decidable satis ability problem Un
fortunately subsumption of this extension by symbolic number restrictions
turned out to b e undecidable c Transitive relations which are needed for
the adequate representation of aggregated ob jects can b e added to ALC with
out strongly increasing the computational complexity of the relevant inference
problems In contrast di erent ways of using a transitive sup errole together
with a p ossibly non transitive subrole strongly increasing this complexity
For the representation of pro cess mo del building blo cks this means that we
have gained a b etter insight into di cult prop erties those that necessitate
constructors which lead to a dramatical increase of the computational com
plexity of building blo cks we would like to describ e and well behaved ones
those that can b e expressed using constructors which lead to an acceptable
computational complexity Two examples for di cult prop erties and exam
ples of well behaved ones will follow
If we de ne a class by restricting the number of ob jects related to its in
stances via complex roles involving conjunction and comp osition of roles such
as for example in the concept connected to has phenomena u
has phenomena then either the computation of the taxonomy runs into an
in nite lo op or this information is simply ignored Hence this prop erty is an
example for a di cult prop erty A second example for di cult prop erties are
symmetry prop erties that must hold for instances of a sp eci c class which can
only b e expressed using symbolic number restrictions Then this information
concerning symmetry can b e taken into account for testing whether this class
can ever b e instantiated but it has to b e ignored for the computation of the
taxonomy This shortcoming of the inference services of a DL system are not
due to an unwilling software developer or a slow computer but they were
proved to b e a consequence of the theory of computation
It turned out that prop erties of aggregated ob jects that are describ ed by re
ferring to its parts can b e expressed using a Description Logic for which the
complexity of the relevant inference problems is rather low provided that
we are not to o demanding with resp ect to the di erent kinds of part whole
relations we would like to use to refer to these parts Hence these prop er
ties b elong to the well behaved ones even though their description may refer
to parts that are in a level of decomp osition neither known nor b ounded in advance
App endix A Syntax rules
Complex concepts and p ossibly complex roles of the di erent concept languages
referred to in this pap er are built according to the following syntax rules where
C C C denote concepts A stands for a concept name or a number restriction
R R R denote roles n stands for a nonnegative integer and is a numerical
variable
ALCN
C C u C j C t C j C j
R C j R C j n R j n R
ALC
C C u C j C t C j C j
R C j R C
R R j R
ALC
reg
C C u C j C t C j C j
R C j R C
R R t R j R R j R
S
ALCN
C C u C j C t C j C j
R C j R C j C j
n R j n R j R j R
S
ALUEN
C C u C j C t C j A j
R C j R C j C j
n R j n R j R j R
ALCN M
C C u C j C t C j C j
R C j R C j R is a role name
n R j n R j R is built according to M
ALC N M
C C u C j C t C j C j
R C j R C j R is a regular role
n R j n R j R is built according to M
Semantics
The following table contains a short description of the semantics of the most
imp ortant role and concept forming constructors for an interpretation I
I I
The letter n stands for a nonnegative integer
Syntax Semantics
I I I
Role names R R
I I
Concept names C C
I
I I
C n C
I I
C u D C D
I I
C t D C D
I I I I
R C fd j For some e d e R and e C g
I I I I
R C fd j For all e if d e R then e C g
I I
n R fd j There are at least n elements e
I
with d e R g
I I
n R fd j There are at most n elements e
I
with d e R g
I I I I I I
R R R R f d f j There exists e
I I
with d e R and e f R g
I I I
I I i I n
R R where R R R R
i
z
n times
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Index
interpretation ALC
ALCN M
mereology
ALCN
mo del
ALC
reg
S
of a completion tree
ALCN
of a constraint system
ALC
ALC N M
number restrictions
ALC
R
qualifying
with comp osition
ABox
with intersection
atomic roles
with inversion
atomicity
with union
building blo cks
op en world assumption
clash
part whole relation
complete
comp osed
completion algorithm
integral
completion rules
prop er part principle
completion tree
clash free
regular roles
complete
role hierarchies
concept
consistent
satis able
constraint
sound
constraint system
structured storage
Constraints
sub concepts
cyclic
subsumption
successor
domino problem
taxonomy
nite mo del prop erty
TBox
nite tree mo del prop erty
transitive orbit
transitive role individual
tree mo del prop erty instance
INDEX
unfolding
unique name assumption
unravelling
variable normal form
variables
for individuals
numerical
List of Figures
Flowsheet of the ethylene glycol pro cess
The ethylene glycol pro cess in ModKit representation
Decomp osition of the reactor of the ethylene glycol pro cess into
building blo cks
Decomp osition of the gas phase material balance equation
A taxonomy of part whole relations
Example of a collection y and members x
i
A semantic network
Example TBox
Example ABox
The completion rules for ALC
A non terminating application of the completion rules
A ModKit screenshot showing the catalogue and the structure
of the ethylene glycol pro cess
A ModKit screenshot showing the dialogue for the prop erty
sp eci cation of a reactor
The architecture of the integration of Crack into ModKit
A set of domino types and a rst part of a tiling
Concepts used in the pro of of Theorem
Un decidability results for extension of ALCN
LIST OF FIGURES
Un decidability results for extension of ALC N
Visualisation of the grid as enforced by the ALC N reduction
concept
Concepts used in the pro of of Theorem
The completion rules for ALCN
The additional completion rules
Reduction concepts used in the pro of of Theorem
S
The completion rules for ALUEN
The completion rules for ALC R