J LOGIC PROGRAMMING to app ear
Constraint Logic Programming A Survey
Joxan Jaffar and Michael J Maher
Constraint Logic Programming CLP is a merger of two declarativepara
digms constraint solving and logic programming Though a relatively new
eld CLP has progressed in several quite di erent directions In particu
lar the early fundamental concepts have b een adapted to b etter servein
di erent areas of applications In this survey of CLP a primary goal is to
give a systematic description of the ma jor trends in terms of common fun
damental concepts The three main parts cover the theory implementation
issues and programming for applications
Intro duction
Constraint Logic Programming CLP b egan as a natural merger of two declarative
paradigms constraint solving and logic programming This combination helps
make CLP programs b oth expressive and exible and in some cases more e cient
than other kinds of programs Though a relatively new eld CLP has progressed in
several and quite di erent directions In particular the early fundamental concepts
have b een adapted to b etter serve in di erent areas of applications In this survey
of CLP a primary goal is to give a systematic description of the ma jor trends in
terms of common fundamental concepts
Consider rst an example program in order to identify some crucial CLP con
cepts The program b elow de nes the relation sumto n n for natural
numb ers n
sumto
sumto N S N N S sumto N S N
The query S sumto N S gives rise to three answers N S
Address correspondenceto Joxan Ja ar and Michael Maher IBM Thomas J Watson Research
Center PO Box Yorktown Heights NY USA Email fjoxan mjmg watson ibm com
FINAL DRAFT Comments esp ecially ab out errors are solicited
N S and N S and terminates The computation sequence
of states for the third answer for example is
S sumto N S
S N N S S N N S
sumto N S N
S N N S S N N S
N N S N S N N S
N S N sumto
S N N S S N N S
N N S N S N N S
N S N
The constraints in the nal state imply the answer N S Termination is
reasoned as follows Any in nite computation must use only the second program
rule for state transitions This means that its rst three states mustbeasshown
ab ove and its fourth state must b e
S N N S S N N S
N N S N S N N S
N N S N S N N S
sumto
We note nowthatthiscontains an unsatis able set of constraints and in CLP no
further reductions are allowed
This example shows the following key features in CLP
Constraints are used to sp ecify the query as well as the answers
During execution new variables and constraints are created
The collection of constraints in every state is tested as a whole for satis a
bility b efore execution pro ceeds further
In summary constraints are used for input output dynamically generated and
globally tested in order to control execution
Constraint Languages
Considerable work on constraint programming languages preceded logic program
ming and constraint logic programming Weno w brie y survey some imp ortant
works with a view toward the following features Are constraints used for in
put output Can new variables and or constraints b e dynamically generated Are
constraints used for control What is the constraint solving algorithm and to what
extent is it complete What follows is adapted from the survey in
SKETCHPAD was p erhaps the earliest work that one could classify as a
constraint language It was in fact an interactive drawing system allowing the user
to build geometric ob jects from language primitives and certain constraints The
constraints are static and were solved by lo cal propagation and relaxation tech
niques See chapter in for an intro duction to these and related techniques
Subsequent related work was THINGLAB whose language to ok an ob ject
oriented avor While lo cal propagation and relaxation were also used to deal with
the essentially static constraints the system considered constraint solving in two
di erent phases When a graphical ob ject is manipulated a plan was generated
for quickly re solving the appropriate constraints for changed part of the ob ject
This plan was then rep eatedly executed while the manipulation continued Works
following the THINGLAB tradition included the Filters Pro ject and Animus
Another graphical system this one fo cusing on geometrical layout was JUNO
The constraints were constructed as in THINGLAB by text or graphical
primitives and the geometric ob ject could b e manipulated A di erence from the
ab ovementioned works is that constraint solving was p erformed numerically using
a Newton Raphson solver
Another collection of early works arose from MIT motivated by applications in
electrical circuit analysis and synthesis gave rise to languages for general problem
solving In the CONSTRAINTS language variables and constraints are static
and constraint solving was limited to using lo cal propagation An extension of this
work provided a more sophisticated environment for constraint programming
including explanation facilities Some other related systems EL ARS and
SYN used the constraintsolver MACSYMA toavoid the restrictions of
that there was a conceptual lo cal propagation It was noted at this p erio d
corresp ondence b etween the constrainttechniques and logic programming
The REF ARF system was also designed for problem solving One comp o
nent REF was essentially a pro cedural language but with nondeterminism b ecause
of constraints used in conditional statements The constraints are static They are
in fact linear integer constraints and all variables are b ounded ab ove and b elow
The constraint solver ARF used backtracking
The Bertrand system was designed as a meta language for the building
of constraint solvers It is itself a constraint language based on term rewriting
Constraints are dynamic here and are used in control All constructs of the lan
guage are based on augmented rewrite rules and the programmer adds rules for
the sp eci c constraint solving algorithm to b e implemented
Post CLP there have b een a numberofworks which are able to deal with dy
namic constraints The language LP is describ ed to b e a CLP language with
a C like syntax for representing and solving combinatorial problems Obtaining
parallel execution is one of the main ob jectives of this work The commercial lan
guage CHARME also based on a pro cedural framework arose from the work on
CHIP by essentially omitting the logic programming part of CHIP The subse
quentwork ILOG which is also commercial is a library of constraint algorithms
designed to work with C programs Using a pro cedural language as a basis
intro duced Constraint Imp erative Programming which has explicit constraints in
the usual way and also a new kind of constraints obtained by considering variable
assignments suchasx x as time stamp ed Such assignments are treatable
as constraints of the form x x Finally we mention ConstraintFunc
i i
tional Programming whose goal is the amalgamation of the ideas of functional
programming found in the HOPE language with constraints
There is work on languages and systems which are not generally regarded as
constraint languages but are nevertheless related to CLP languages The develop
ment of symb olic algebra systems suchasMACSYMA concentrated on the
solving of di cult algebraic problems The programming language asp ects are less
develop ed Languages for linear programming provide little more than a prim
itive do cumentation facility for the array of co e cients whichisinputinto a linear
programming mo dule
In parallel with the development of these constraint languages muchwork was
done on the mo delling of combinatorial problems as Constraint Satisfaction Prob
lems CSPs and the development of techniques for solving such problems The
work was generally indep endentofany host language A p ossible exception is
ALICE which provided a wide variety of primitives to implement di erent
search techniques One imp ortantdevelopmentwas the de nition and study of
several notions of consistency Thiswork had a signi cant in uence on the later
development of the CLP language CHIP We refer the reader to foranintro
duction to the basic techniques and results concerning CSPs Finally we mention
the survey which deals not just with constraint programming languages but
constraint based programming techniques
Logic Programming
Next we consider conventional logic programming LP and argue by example that
hanges to LP systems the p ower of CLP cannot b e obtained by making simple c
The question at hand is whether predicates in a logic program can b e meaningfully
regarded as constraints That is is a predicate with the same declarative semantics
as a constraint a su cient implementation of the constraint as p er CLP Consider
for example the logic program
add N N
add s N M s K add N M K
where natural numb ers n are represented by s s with n o ccurrences of
s Clearly the meaning of the predicate add n m k coincides with the relation
n m k However the query add N M K add N M s K which is clearly
unsatis able runs forever in a conventional LP system The imp ortant p oint here
is that a global test for the satis abilityofthetwo add constraints is not done by
the underlying LP machinery
In this example the problem is that the add predicate is not invoked with a
representation of the add constraints collected so far and neither do es it return such
a representation after having dealt with one more constraint More concretely
the second subgoal of the query ab ove is not given a representation of the fact that
N M K
A partial solution to this problem is the use of a delaymechanism Roughly the
idea is that in vo cation of the predicate is delayed until its arguments are su ciently
instantiated For example if invo cation of add is systematically delayed until its
rst argument is instantiated then add behaves as in CLP when the rst argument
is ground Thus the query N s s s add N M K add N M
s K fails as desired However the original query add N M K add N M
s K will b e delayed forever
A total solution could in principle b e obtained by simply adding two extra
arguments to the predicate One would b e used for the input representation and
one for the output This would mean that each time a constraint is dealt with
a representation of the entire set of constraints accumulated must b e manipulated
and a new representation constructed But this is tantamount to a meta level
implementation of CLP in LP Furthermore this approach raises new challenges to
e cient implementation
Since LP is an instance of CLP inwhich constraints are equations over terms
its solver also requires a representation of accumulated constraints It happ ens
however that there is no need for an explicit representation such as the extra
arguments discussed ab ove This is b ecause the accumulated constraints can b e
represented by an mgu and this of course is globally available via a simple binding
mechanism
CLP Languages
Viewing the sub ject rather broadly constraint logic programming can b e said to
involve the incorp oration of constraints and constraint solving metho ds in a logic
based language This characterization suggests the p ossibilityofmanyinteresting
languages based on di erent constraints and di erent logics However to this
p oint work on CLP has almost exclusively b een devoted to languages based on
Horn clauses Wenow brie y describ e these languages concentrating on those
that have received substantial development e ort
ts are equations Prolog can b e said to b e a CLP language where the constrain
over the algebra of terms also called the algebra of nite trees or the Herbrand
domain The equations are implicit in the use of uni cation Almost every lan
guage we discuss incorp orates Prolog like terms in addition to other terms and
constraints so we will not discuss this asp ect further Prolog I I employs equa
tions and disequations over rational trees an extension of the nite trees of
Prolog to cyclic structures It was the rst logic language explicitly describ ed as
using constraints
CLP R has linear arithmetic constraints and computes over the real num
b ers Nonlinear constraints are ignored delayed until they b ecome e ectively
linear CHIP and Prolog I I I compute over several domains Both compute
over Bo olean domains Prolog I I I over the well known valued Bo olean algebra
and CHIP over a larger Bo olean algebra that contains symbolic values Both CHIP
and Prolog I I I p erform linear arithmetic over the rational numb ers Separately
domains cannot b e mixed CHIP also p erforms linear arithmetic over b ounded
subsets of the integers known as nite domains Prolog I I I also computes over
a domain of strings There are now several languages which compute over nite
domains in the manner of CHIP including clp FD Echidna and Flang
cc FD isessentially a second generation CHIP system
LOGIN and LIFE compute over an order sorted domain of feature trees
This domain provides a limited notion of ob ject in the ob ject oriented sense The
languages supp ort a term syn tax which is not rst order although every term can b e
interpreted through rst order constraints Unlike other CLP languages domains
We note however some work combining constraints and resolution in rst order automated
theorem proving
The language Absys whichwas very similar to Prolog used equations explicitly making
it more obviously a CLP language
Prolog like trees are essentially part of this domain instead of b eing built on top
of the domain CIL computes over a domain similar to feature trees
BNR Prolog computes over three domains the valued Bo olean algebra
nite domains and arithmetic over the real numb ers In contrast to other CLP
languages over arithmetic domains it computes solutions numerically instead of
symb olically Trilogy computes over strings integers and real numbers
Although its syntax is closer to that of C LP can b e considered to b e a
CLP language p ermitting only a subset of Horn clauses It computes with linear
constraints over integers and real numb ers
CAL computes over two domains the real numb ers where constraints are
equations b etween p olynomials and a Bo olean algebra with symbolic values where
equalitybetween Bo olean formulas expresses equivalence in the algebra Instead
of delaying non linear constraints CAL makes partial use of these constraints dur
ing computation In the exp erimental system RISC CLP Real non linear
constraints are fully involved in the computation
L andElf are derived from Prolog and compute over the
values of closed typ ed lamb da expressions These languages are not based on Horn
clauses they include a universal quanti er and were not originally describ ed as
er it is argued in that their op erational b ehavior is CLP languages Howev
b est understo o d as the b ehavior of a CLP language An earlier language Le Fun
also computed over this domain and can b e viewed as a CLP language with a
weak constraint solver
Synopsis
The remainder of this pap er is organized into three main parts In part one we
provide a formal framework for CLP Particular attention will b e paid to op era
tional semantics and op erational mo dels As wehave seen in examples it is the
op erational interpretation of constraints rather than the declarativeinterpretation
which distinguishes CLP from LP Inparttwo algorithm and data structure con
siderations are discussed A crucial prop ertyofany CLP implementation is that
its constraint handling algorithms are incremental In this lightwe review several
imp ortantsolvers and their algorithms for the satis ability entailment and delay
ing of constraints We will also discuss the requirements of an inference engine for
CLP In part three we consider CLP applications In particular we discuss two
rather di erent programming paradigms one suited for the mo delling of complex
problems and one for the solution of combinatorial problems
In this survey we concentrate on the issues raised bytheintro duction of con
straints to LP Consequently we will ignore or pass over quickly those issues inher
entinLP We assume the reader is somewhat familiar with LP and basic rst order
logic Appropriate background can b e obtained from for LP and for logic
For intro ductory pap ers on constraint logic programming and CLP languages we
refer the reader to For further reading on CLP we suggest other
surveys some collections of pap ers and some b o oks
More generally pap ers on CLP app ear in various journals and confer
ence pro ceedings devoted to computational logic constraint pro cessing or symb olic computation
Notation and Terminology
This pap er will hop efully keep to the following conventions Upp er case letters
generally denote collections of ob jects while lower case letters generally denote
individual ob jects u v w x y z will denote variables s t will denote terms p q
will denote predicate symb ols f g will denote function symb ols a will denote a
constant a b h will denote atoms A will denote a collection of atoms will
denote substitutions c will denote a constraint C S will denote collections of
constraints r will denote a rule P Q will denote programs G will denote a goal
D will denote a structure D will denote its set of elements and d will denote an
elementofD These symb ols may b e subscripted or haveanover tilde x denotes
a sequence of distinct variables x x x for an appropriate n s denotes a
n
sequence of not necessarily distinct terms s s s for an appropriate n s t
n
s t s t denotes the existential closure abbreviates s t
n n x
of the formula except for the variablesx which remain unquanti ed denotes
the full existential closure of the formula
A signature de nes a set of function and predicate symb ols and asso ciates an arity
with each symbol If is a signature a structure D consists of a set D and an
assignment of functions and relations on D to the symb ols of whichrespectsthe
arities of the symb ols A rst order formula is built from variables function and
predicate symb ols of the logical connectives and quanti ers over
variables in the usual way A formula is closed if all variable o ccurrences
ariable A theory in the formula are within the scop e of a quanti er over the v
is a collection of closed formulas A model of a theory T is a structure D
such that all formulas of T evaluate to tr ue under the interpretation provided by
D AD model of a theory T is a mo del of T extending D this requires that the
signature of D be contained in the signature of T We write T Dj to denote
that the formula is valid in all D mo dels of T
In this pap er the set of function and predicate symb ols de ned in the constraint
domain is denoted by and the set of predicate symb ols de nable by a program
is denoted by Aprimitive constraint has the form p t t where t t
n n
are terms and p is a predicate symbol Every constraint is a rst order
formula built from primitive constraints The class of constraints will vary butwe
will generally consider only a subset of formulas to b e constraints An atom has
the form p t t where t t are terms and p A CLP program is a
n n
collection of rules of the form a b b where a is an atom and the b s are
n i
atoms or constraints a is called the head of the rule and b b is called the
n
body Sometimes we represent the rule by a c B where c is the conjunction of
constraints in the b o dy and B is the collection of atoms in the b o dy and sometimes
we represent the rule by a B where B is the collection of atoms and constraints
in the b o dy In one subsection we will also consider programs with negated atoms
in the b o dy Agoal or query G is a conjunction of constraints and atoms A fact
we will identify conjunction and is a rule a c where c is a constraint Finally
multiset union
To simplify the exp osition we assume that the rules are in a standard form
where all arguments in atoms are variables and eachvariable o ccurs in at most one
In a many sorted language this would include asso ciating a sort with each argument and the
result of eachsymb ol However we will not discuss such details in this survey
atom This involves no loss of generality since a rule suchasp t t C q s s
can b e replaced by the equivalentrulep x x x t x t y s y
s C q y y We also assume that all rules de ning the same predicate havethe
same head and that no two rules haveany other variables in common this is simply
a matter of renaming variables However in examples we relax these restrictions
Programs will b e presented in teletype font and will generally follow the Ed
inburgh syntax In particular program variables b egin with an upp er case letter
HeadjT ail denotes a list with head Head and tail Tail and denotes an empty
list In one variation from this standard we allow subscripts on program variables
to improve readability
Part I
The Semantics of CLP Languages
Many languages based on de nite clauses have quite similar semantics The crucial
insight of the CLP Scheme and the earlier scheme of was
that a logic based programming language its op erational semantics its declarative
semantics and the relationships b etween these semantics could all b e parameterized
byachoice of domain of computation and constraints The resulting scheme de nes
the class of languages CLP X obtained by instantiating the parameter X
We take the view that the parameter X stands for a tuple D L T Here
is a signature D is a structure L is a class of formulas and T is a rst order
theory Intuitively determines the prede ned predicate and function symb ols and
their arities D is the structure over which computation is to b e p erformed L is the
class of constraints which can b e expressed and T is an axiomatization of some
prop erties of D In the following section we de ne some imp ortant relationships
between the elements of the tuple and give some examples of constraint domains
We then give declarative and op erational semantics for CLP programs para
meterized by X The declarativesemantics are quite similar to the corresp onding
semantics of logic programs and wecover them quickly There are manyvariations
of the resolution based op erational semantics and we present the main ones We
also present the main soundness and completeness results that relate the twostyles
of semantics Finally we discuss some linguistic features that have b een prop osed
as extensions to the basic CLP language
Constraint Domains
For any signature let D b e a structure the domain of computation and L be
a class of formulas the constraints We call the pair D L aconstraint domain
In a slight abuse of notation we will sometimes denote the constraint domain by D
We will makeseveral assumptions none of which is strictly necessary to simplify
the exp osition We assume
The terms and constraints in L come from a rst order language
Without this assumption some of the results we cite are not applicable since there can b e
no appropriate rst order theory T The remaining assumptions can b e omitted at the exp ense
The binary predicate symb ol is contained in and is interpreted as iden
tityin D
There are constraints in L which are resp ectively identically true and iden
tically false in D
The class of constraints L is closed under variable renaming conjunction
and existential quanti cation
We will denote the smallest set of constraints which satis es these assumptions
and contains all primitive constraints the constrain ts generated by the primitive
constraints b y L In general L may b e strictly larger than L since for example
universal quanti ers or disjunction are p ermitted in L it also maybesmaller as
in example b elow However we will usually take L L On o ccasion wewill
consider an extension of and L to and L resp ectively so that there is a
constantin for every elementofD
Wenow present some example constraint domains In practice these are not
always fully implemented but weleave discussion of that until later Most gen
eral purp ose CLP languages incorp orate some arithmetic domain including BNR
Prolog CAL CHIP CLP R Prolog I I I RISC CLP Real
Example Let contain the constants and the binary function symb ols
and and the binary predicate symb ols and LetD b e the set of real
numb ers and let D interpret the symb ols of as usual i e is interpreted as
addition etc Let L b e the constraints generated by the primitive constraints
Then D L is the constraint domain of arithmetic over the real numb ers If
we omit from the symbol then the corresp onding constraint domain
Lin
D L is the constraint domain of linear arithmetic over the real numbers If
the domain is further restricted to the rational numb ers then wehave a further
constraint domain Q In constraints in and Q we will write terms
Lin Lin Lin
suchas and x as abbreviations for and x x x x x resp ectively
Thus y x y z y is a constraintin and Q whereas
Lin Lin
x x y is a constraintonlyin Ifweextend L to allow negated equations
we will use the symbol then the resulting constraint domains and
Lin
p ermit constraints suchas x y x y Finally ifwe restrict to Q
Lin
ts f g we obtain the constraintdomain where the only constrain
LinE q n
are linear equations
and Q and and Q are essentially the same constraintdomain
Lin Lin
Lin Lin
they have the same language of constraints and the two structures are elementarily
equivalent In particular a constraintsolver for one is also a constraint solver
for the other
of a messier reformulation of de nitions and results
This assumption is unnecessary when terms have a most general uni er in D as o ccurs in
Prolog Otherwise is needed to express parameter passing
Other syntactic sugar such as the unary and binary minus symbol are allowed Rational
numb er co e cients can b e used all terms in the sugared constraintneedonlybemultiplied by
an appropriate numb er to reduce the co e cients to integers
Sometimes called disequations
Prolog and standard logic programming can b e viewed as constraint logic pro
gramming over the constraint domain of nite trees
Example Let contain a collection of constant and function symb ols and
the binary predicate symbol Let D b e the set of nite trees where each
no de of each tree is lab elled by a constant or function symb ol the number of
children of each no de is the arityofthelabelofthenode andthechildren are
ordered Let D interpret the function symb ols of as tree constructors where
each f ofarity n maps n trees to a tree whose ro ot is lab elled by f and
whose subtrees are the arguments of the mapping The primitive constraints
are equations b etween terms and let L b e the constraints generated by these
primitive constraints Then FT D L is the Herbrand constraint domain as
used in Prolog Typical constraints are x g y and zx f z z y g z
It is unnecessary to write a quanti er in Prolog programs b ecause all variables
that app ear only in constraints are implicitly existentially quanti ed
It was p ointed out in that complete i e alw ays terminating uni cation which
omits the o ccurs checksolves equations over the rational trees
Example Wetake andL as in example D is the set of rational trees
see for a de nition and the function symb ols are interpreted as tree con
structors as b efore Then RT D L is the constraint domain of rational
trees
If we take the set of in nite trees instead of rational trees then we obtain a constraint
domain that is essentially the same as RT inthesameway that and Q
Lin Lin
are essentially the same they have the same language of constraints and the two
structures are elementarily equivalent
The next domain contains ob jects similar to the previous domains but has a
di erent signature and constraint language which results in slightly di erent
expressivepower It can b e viewed as the restriction of domains over whichLO
GIN and LIFE compute when all sorts are disjoint The close relationship
between the constraints and terms is emphasized by a syntactic sugaring of
the constraints
is a set of unary predicate symb ols Example Let f g S F where S
sorts and F is a set of binary predicate symb ols features Let D b e the set of
nite or in nite trees where eachnodeofeach tree is lab elled by a sort each
edge of each tree is lab elled by a feature and no no de has two outb ound edges
with the same lab el Such trees are called featuretrees LetD interpret each
sort s as the set of feature trees whose ro ot is lab elled by s andinterpret each
feature f as the set of pairs t t of feature trees such that t is the subtree
of t that is reached by the edge lab elled by f Ift has no edge lab elled by
f then there is no pair t t in the set Thus features are essentially partial
functions The domain of feature trees FEAT D L A typical constraintis
w ine x y r eg ion x y r uther g l en y ycolor x y red y but there is
Avariant of this domain with a slightly di erent signature is used in
also a sugared syntax whichwould represent this constraintas x wine region
r uther g l en col or red
The next constraint domain takes strings as the basic ob jects It is used in Prolog
III
Example Let contain the binary predicate symb ol the binary function
symbol a constant andanumb er of other constants D is the set of nite
strings of the constants The symbol is interpreted in D as string concatenation
and is interpreted as the empty string L is the set of constraints generated by
equations b etween terms Then WE D L is the constraint domain of equa
tions on strings sometimes called the domain of word equations An example
constraintisx a b x
The constraint domain of Bo olean values and functions is used in BNR Prolog
CAL CHIP and Prolog I I I CAL and CHIP employ a more general
constraint domain which includes symb olic Bo olean values
Example Let contain the constants and the unary function symbol
and the binary predicate symbol the binary function symb ols
Let D be the set ftr ue f al seg and let D interpret the symb ols of as
the usual Bo olean functions i e is interpreted as conjunction is exclu
sive or etc Let L b e the constraints generated by the primitive constraints
Then BOOL D L is the two valued Bo olean constraint domain An ex
ample constraintis x y y In a slight abuse of notation we allowa
constraint t to b e written simply as t so that for example x y y
denotes the constraint x y y For the more general constraintdomain
let fa a g where the a are constants Let L b e the con
i i
straints generated by the primitive constraints and let D b e the free Bo olean
algebra generated by fa a g Then BOOL D L is the Bo olean
i
constraint domain with in nitely manysymbolic values A constraint c x a is
satis able in BOOL i Dj x yc x y
The nite domains of CHIP are b est viewed as having the integers as the underlying
structure with a limitation on the language of constraints
Example Let D Z and ff m n g g For every pair of
m n
m x n integers m and n theinterval constraint x m n denotes that
The other symbols in have their usual meaning Let L b e the constraints c
generated by the primitive constraints restricted so that every variable in c is
sub ject to an interval constraint Then FD D L is the constraint domain
referred to as nite domains The domain of a variable x is the nite set of
values which satisfy all unary constraints involving x Atypical constraintin
Only nitely many constants are used in any one program so it can b e argued that a nite
Bo olean algebra is a more appropriate domain of computation However the two alternatives agree
on satis ability and constraintentailment although not if an expanded language of constraints
is p ermitted and it is preferable to view the constraint domain as indep endent of the program
Currently it is not clear whether the alternatives agree on other constraint op erations
FD is x y x x y x y The domain of x is
f g
There are several other constraint domains of interest that we cannot exemplify
here for lack of space They include pseudo Bo olean constraints for example
which are intermediate b etween Bo olean and integer constraints order sorted
feature algebras domains consisting of regular sets of strings domains of
nite sets domains of CLP Fun D which employ a function variable
domains of functions expressed by expressions etc
It is also p ossible to form a constraint domain directly from ob jects and op er
ations in an application instead of more general purp ose domains suchasthose
ab ove This p ossibility has only b een pursued in a limited form where a general
purp ose domain is extended by the ad ho c addition of primitive constraints For
example in some uses of CHIP the nite domain is extended with a predicate sym
bol el ement The relation el ement x l t expresses that t is the x th elemen t
in the list l We discuss such extensions further in Section
These constraint domains are exp ected to supp ort p erhaps in a weakened form
the following tests and op erations on constraints whichhave a ma jor imp ortance in
CLP languages The rst op eration is the most imp ortant it is almost obligatory
while the others might not b e used in some CLP languages
The rst is a test for consistency or satis ability Dj c
The second is the implication or entailment of one constraintby another
Dj c c More generally wemay ask whether a disjunction of con
W
n
straints is implied by a constraint Dj c c
i
i
The third is the projection of a constraint c onto variablesx to obtain a
ys p ossible to take c to constraint c such that Dj c c It is alwa
x
be c but the aim is to compute the simplest c with fewest quanti ers
x
In general it is not p ossible to eliminate all uses of the existential quanti er
The fourth is the detection that given a constraint c there is only one
value that a variable x can take that is consistent with c Thatis Dj
c x z c y w x y or equivalently Dj z x yc x y x z We
say x is determined or grounded by c
In Section we will discuss problems and techniques which arise when imple
epoint out here that some menting these op erations in a CLP system However w
implementations of these op erators in particular the test for satis abilit y are
incomplete In some cases it has b een argued that although an algo
rithm is incomplete with resp ect to the desired constraint domain it is complete
with resp ect to another arti cially constructed constraint domain
Wenow turn to some prop erties of constraint domains which will b e used later
The rst two solution compactness and satisfaction completeness w ere intro
duced as part of the CLP Scheme
De nition Let d range over elements of D and c c range over constraints
i
in L and let I b e a p ossibly in nite index set A constraintdomain D L is
solution compact if it satis es the following conditions
V
c x SC d fc g s t Dj xx d
i i i I
i I
W
c x SC c fc g s t Dj x c x
i i i I
i I
Roughly sp eaking SC is satis ed i every element d of D can b e de ned bya
p ossibly in nite conjunction of constraints and SC is satis ed i the complement
of each constraint c in L can b e describ ed by a p ossibly in nite disjunction of
constraints
The de nition of SC in is not quite equivalent to the de nition in which
we paraphrase ab ove see It turns out that SC is not necessary for the results
we present we include it only for historical accuracy There is no known natural
constraint domain for which SC do es not hold There are however some arti cial
constraint domains for which it fails
Example Let denote the constraint domain obtained from by
Lin
Lin
adding the unary primitive constraint x The negation of this constraint
i e x cannot b e represented as a disjunction of constraints in Thus
Lin
is not solution compact
Lin
The theory T in the parameter of the CLP scheme is intended to axiomatize some
of the prop erties of D We place some conditions on D and T to ensure that
T re ects D su ciently The rst two conditions ensure that D and T agree on
satis ability of constraints while the addition of the third condition guarantees
that every unsatis abilityin D is also detected by T The theory T and these
conditions mainly play a role in the completeness results of Section
De nition For a given signature let D L b e a constraint domain with
signature and T b e a theory WesaythatD and T correspond on L if
D is a mo del of T and
for every constraint c L Dj c i Tj c
Wesay T is satisfaction complete with resp ect to L if for every constraint c L
c or Tj c either Tj
Satisfaction completeness is a weakening of the notion of a complete theory
Thus for example the theory of the real closed elds corresp onds and is
satisfaction complete with resp ect to since the domain is a mo del of this theory
and the theory is complete Clark s axiomatization of uni cation de nes a
satisfaction complete theory with resp ect to FT which is not complete when there
are only nitely many function symb ols
e constraints plays a signi cantroleincon The notion of indep endence of negativ
straint logic programming In Colmerauer used indep endence of inequations
to simplify the test for satis ability of equations and inequations on the rational
It is also closely related to the mo del theoretic prop erties that led to an interest in Horn
formulas
trees The indep endence of inequations states if a conjunction of p ositive and neg
ative equational constraints are inconsistent then one of the negative constraints
is inconsistent with the p ositive constraints Indep endence of negative constraints
has b een investigated in greater generalityin The prop erty has b een shown
to hold for several classes of constraints including equations on nite rational and
in nite trees linear real arithmetic constraints where only equa
tions may b e negated sort and feature constraints on feature trees and
in nite Bo olean algebras with p ositive constraints among others We
consider a restricted form of indep endence of negative constraints
De nition A constraint domain D L has the Indep endence of Negated Con
straints prop erty if for all constraints c c c L
n
Dj c c c i Dj c c for i n
n i
The fact that L is assumed to b e closed under conjunction and existential quan
ti cation is an imp ortant restriction in the ab ove de nition For example Colmer
auer s work is not applicable in this setting since that dealt only with primitive
y of the other results cited ab ove at least not in their constraints Neither are man
full generality However there are still several useful constraint domains known to
have this prop erty including the algebras of nite rational and in nite trees with
equational constraints when there are in nitely many function symb ols
feature trees with in nitely many sorts and features linear arithmetic equa
tions over the rational or real numb ers and in nite Bo olean algebras with p ositive
constraints
Example In the Herbrand constraint domain FT with only two function
symb ols a constant a and a unary function f it is easily seen that the fol
lowing statements are true FT j x y z x f y y a y f z
FT j x y x f y y a FT j x y z x f y y f z This
is an example of the indep endence of inequations for FT However when we
consider the full class of constraints of FT wehave the following facts The
statement FT j x y x f y y a zy f z is not true since
every nite tree y is either the constant a or has the form f z for some
nite tree z On the other hand b oth FT j x y x f y y a and
FT j x y z x f y zy f z are true Thus for these function
symb ols and it is easy to see ho w to extend this example to any nite set of
function symb ols the indep endence of negated constrain ts do es not hold
As is clear from constraint domains and constraints are closely related to
t his domain the information systems and their elements used by Scott to presen
theory Information systems co dify notions of consistency and entailmentamong
elements which can b e interpreted as satis ability and implication of constraints
on a single variable Saraswat extended the notion of information system
to constraint systems which allow manyvariables and showed that some of the
motivating prop erties of information systems continue to hold
although do es not treat consistency onlyentailment
Constraint systems we will not give a formal de nition here can b e viewed as
abstractions of constraint domains which eliminate consideration of a particular
structure D the relation Dj c c c among constraints c c c
n n
is abstracted to the relation c c c and the satis ability relation Dj
n
c c among constraints can b e abstracted to a set Con of all consistent
n
nite sets of constraints fc c g Many of the essential semantic
n
details of a constraint domain are still present in the corresp onding constraint
system although prop erties such as solution compactness and indep endence of
negated constraints cannot b e expressed without more detail than a constraint
system provides
Logical Semantics
There are two common logical semantics of CLP programs over a constraint domain
D L The rst interprets a rule
p x b b
n
as the logic formula
x yp x b b
n
wherex y is the set of all free variables in the rule The collection of all such
formulas corresp onding to rules of P gives a theory also denoted by P
The second logical semantics asso ciates a logic formula to eachpredicatein
If the set of all rules of P with p in the head is
p x B
p x B
p x B
n
then the formula asso ciated with p is
xp x y B
y B
y B
n n
wherey is the set of variables in B except for v ariables inx Ifp do es not o ccur
i i
in the head of a rule of P then the formula is
x p x
The collection of all suchformulas is called the Clark completion of P andis
denoted by P
A valuation is a mapping from variables to D and the natural extension which
maps terms to D and formulas to closed L formulas If X is a set of facts then
X fv a j a c X Dj v c g AD interpretation of a formula is an D
interpretation of the formula with the same domain as D and the same interpre
tation for the symb ols in as D It can b e represented as a subset of B where
D
k
B fp d j p d D g AD mo del of a closed formula is a D interpretation
D
which is a mo del of the formula
Let T denote a satisfaction complete theory for D L The usual logical seman
tics are based on the D mo dels of P and the mo dels of P T The least D mo del
of a formula Q under the subset ordering is denoted by lm Q D and the great
est is denoted by gm Q D A solution to a query G is a valuation v suchthat
v G lm P D
tics Fixedp oint Seman
The xedp oint semantics we present are based on one step consequence functions
D D D D
T and S and the closure op erator P generated by T The functions T and
P P P P
P map over D interpretations The set of D interpretations forms a complete
lattice under the subset ordering and these functions are continuous on B
D
D
T I fp d j p x c b b is a rule of P a I i n
n i
P
v is a valuation on D such that
i ng Dj v c v x d and v b a
i i
D
It represents a deductive closure based P is the closure op erator generated by T
P
on the rules of P Let Id b e the identity function and de ne f g x f x g x
D
Id greater than I and the least xedp oint Then P I is the least xedp ointofT
P
D
of T
P I
D
The function S is de ned on sets of facts which form a complete lattice under
P
D
the subset ordering We denote the closure op erator generated from S by hhP ii
P
Both these functions are continuous
D
I fp x c j p S x c b b is a rule of P
n
P
a c I i n the rule and facts renamed apart
i i
V
n
c a b g Dj c c
i i i
i
We denote the least xedp oint of a function f by lf p f and the greatest xedp oint
by gf p f These xedp oints exists for the functions of interest since they are mo
notonic functions on complete lattices For a function f mapping D interpretations
to D interpretations we de ne the upward and downward iteration of f as follows
f
f f f
S
f f if is a limit ordinal
f B
D
f f f
T
f f if is a limit ordinal
D D
We can takeassemantics lf p S orlf p T The two functions involved are
P P
D D D
related in the following way S I T I Consequently lf p S
D D D
P P P
D D
lf p T lf p S corresp onds to the s semantics for languages with constraints
P P
Fixedp oint semantics based on sets of clauses also extend easily to CLP
languages
Based largely on the facts that the D mo dels of P are the xedp oints of P and
D
the D mo dels of P are the xedp oints of T wehave the following connections
P
between the logical and xedp ointsemantics just as in standard logic programming
Proposition Let P P P beCLPprograms and Q a set of facts over a con
straint domain D with corresponding theory T Then
D D D
T lf p T lf p S P
D
P P P
Dj h c g fh c j P Tj h c g lm P D fh c j P
D D
D
lm P D lm P D lf p T
P
D
gm P D gf p T
P
P Q P Q lm P Q D
D
D
hhP ii Q hhP Q ii lf p S
P Q
Dj P P i P P
We will need the following terminology later P is said to b e D L canonical i
D D
gf p T T Canonical logic programs but not constraint logic programs
P P
were rst studied by who showed that every logic program is equivalent wrt
the success and nite failure sets to a canonical logic program The pro of here
was not constructive but subsequently provided an algorithm to generate
the canonical logic program Like many other kinds of results in traditional logic
programming these results are likely to extend to CLP in a straightforward way
Top down Execution
The phrase top down execution covers a multitude of op erational mo dels We
will present a fairly general framework for op erational semantics in whichwecan
describ e the op erational semantics of some ma jor CLP systems
We will present the op erational semantics as a transition system on states tuples
hA C S i where A isamultiset of atoms and constraints and C and S are multisets
of constraints The constraints C and S are referred to as the constraint stor e and
in implementations are acted up on by a constraintsolver Intuitively A is a
collection of as yet unseen atoms and constraints C is the collection of constraints
which are playing an active role or are awake and S is a collection of constraints
playing a passive role or are asleep There is one other state denoted by fail To
express more details of an op erational semantics it can b e necessary to represent
the collections of atoms and constraints more precisely For example to express
the left to right Prolog execution order wemight use a sequence of atoms rather
than a multiset However we will not b e concerned with such details here
This pro of was p erformed in the more general class of logic programs with negation
We will assume as given a computation rule which selects a transition typ e and
an appropriate elementof A if necessary for each state The transition system
is also parameterized by a predicate consistent and a function inf er whichwe will
discuss later An initial goal G for execution is represented as a state by hG i
The transitions in the transition system are
hA a C S i hA B C S a h i
r
if a is selected by the computation rule a is an atom h B isaruleofP renamed
to new variables and h and a have the same predicate symb ol The expression
a h is an abbreviation for the conjunction of equations b etween corresp onding
arguments of a and h Wesay a is rewritten in this transition
hA a C S i f ail
r
if a is selected by the computation rule a is an atom and for every rule h B of
P h and a have di erent predicate symb ols
hA c C S i hA C S ci
c
if c is selected by the computation rule and c is a constraint
hA C S i hA C S i
i
if C S inf er C S
hA C S i hA C S i
s
if consistent C
hA C S i f ail
s
if consistent C
The transitions arise from resolution transitions intro duce constraints
r c
into the constraint solver transitions test whether the active constraints are
s
transitions infer more active constraints and p erhaps mo dify consistent and
i
the passive constraints from the current collection of constraints We write to
refer to a transition of arbitrary typ e
The predicate consistent C expresses a test for consistency of C Usually it
is de ned by consistent C i Dj C that is a complete consistency test
However systems may employ a conservative but incomplete or partial test if
Dj C then consistent C holds but sometimes consistent C holds although
Dj C One example of such a system is CAL which computes over the
domain of real numb ers but tests consistency over the domain of complex numbers
The function inf er C S computes from the current sets of constraints a new set
of active constraints C and passive constraints S Generally it can b e understo o d
as abstracting from S or relaxing S in the presence of C to obtain more active
constraints These are added to C to form C and S is simpli ed to S We require
A computation rule is a convenient ction that abstracts some of the b ehavior of a CLP
system To b e realistic a computation rule should also dep end on factors other than the state
for example the history of the computation We ignore these p ossibilities for simplicity
that Dj C S C S so that information is neither lost nor guessed by
inf er The role that inf er plays varies widely from system to system In Prolog
there are no passive constraints and we can de ne inf er C S C S In
CLP R non linear constraints are passive and inf er simply passes the linearized
version of a constraint from S to C when the constraint b ecomes linear in the
context of C and deletes the constraint from S For example if S is x y
z z y and C is x z then inf er C S C S where C is
x z y z and S is z y
In a language like CHIP inf er p erforms less obvious inferences For example if
S is x y and C is x y then inf er C S C S where C
is x y and S S Note that we could also formulate the nite
domain constraint solving of CHIP as having no passive constraints but having an
incomplete test for consistency However the formulation wegive seems to re ect
the systems more closely Similarly in languages employing interval arithmetic
over the real numb ers such as BNR Prolog intervals are active constraints and
other constraints are passive In this case inf er rep eatedly computes smaller in
tervals for each of the variables based on the constraints in S terminating when
no smaller interval can b e derived mo dulo the precision of the arithmetic Exe
cution of language constructs such as the cardinality op erator constructive
disjunction and sp ecial purp ose constructs for example in can also
be understood as transitions where these constructs are viewed as part of the
i
language of constraints
Generally the active constraints are determined syntactically As examples in
Prolog all equations are active in CLP R all linear constraints are active on the
nite domains of CHIP all unary constraints i e constraints on just one variable
suchasx or x are active andintheinterval arithmetic of BNR Prolog
only intervals are active
The stronger the collection of active constraints the earlier failure will b e de
tected and the less searching is necessary With this in mind we mightwish inf er
to b e as strong as p ossible for every active constraint c ifinf er C S C S
and Dj C S c then Dj C c However this is not always p ossible
Even if it were p ossible it is generally not preferred since the computational cost
of a p owerful inf er function can b e greater than the savings achieved by limiting
search
A CLP system is determined by the constraint domain and a detailed op erational
semantics The latter involves a computation rule and de nitions for consistent
and inf er Wenow de ne some signi cant prop erties of CLP systems We distin
guish the class of systems in which passive constraints playnoroleandtheglobal
consistency test is complete These systems corresp ond to the systems treated in
De nition Let and Wesay that a CLP
ris r i s cis c i s
system is quick checking if its op erational semantics can b e describ ed by and
ris
A CLP system is progressive if for every state with a nonempty collection
cis
of atoms every derivation from that state either fails contains a transition
r
For example in CLP R where linear constraints are active and nonlinear constraints are
passive if S is y x x then we can take c to b e y kx k for any k There is no nite
collection C of active constraints which implies all these constraints and is not stronger than S
or contains a transition A CLP system is ideal if it is quick checking
c
progressive inf er is de ned by inf er C S C S and consistent C
holds i Dj C
In a quick checking system inference of new active constraints is p erformed and a
test for consistency is made each time the collection of constraints in the constraint
solver is changed Thus within the limits of consistent and inf er it nds incon
sistency as so on as p ossible A progressive system will never in nitely ignore the
collection of atoms and constraints in the rst part of a state during execution All
ma jor implemented CLP systems are quick checking and progressive but most are
not ideal
A derivation is a sequence of transitions hA C S i hA C S i
i i i
A state which cannot b e rewritten further is called a nal state A derivation is
successful if it is nite and the nal state has the form h C Si Let G be a
goal with free variablesx which initiates a derivation and pro duces a nal state
h C Si Then C S is called the answer constraint of the derivation
x
A derivation is failed if it is nite and the nal state is fail A derivation is fair
if it is failed or for every i and every a A a is rewritten in a later transition A
i
computation rule is fair if it gives rise only to fair derivations A goal G is nitely
failed if for any one fair computation rule every derivation from G in an ideal CLP
system is failed It can b e shown that if a goal is nitely failed then every fair
derivation in an ideal CLP system is failed A derivation ounders if it is nite and
the nal state has the form hA C S i where A
The computation tree of a goal G for a program P in a CLP system is a tree
or such that with no des lab elled by states and edges lab elled by
s r c i
the ro ot is lab elled by hG i for every no de all outgoing edges have the same
lab el if a no de lab elled by a state S has an outgoing edge lab elled by or
c i
then the no de has exactly one child and the state lab elling that child can b e
s
obtained from S via a transition or resp ectively if a no de lab elled by
c i s
a state S has an outgoing edge lab elled by then the no de has a child for each
r
rule in P and the state lab elling eachchild is the state obtained from S bythe
transition for that rule for each and edge the corresp onding transition
r r c
uses the atom or constraint selected by the computation rule
Every branch of a computation tree is a derivation and given a computation
rule every derivation following that rule is a branch of the corresp onding computa
tion tree Di erent computation rules can give rise to computation trees of radically
di erent sizes Existing CLP languages use computation rules based on the Prolog
left to right computation rule which is not fair We will discuss linguistic features
intended to improve on this rule in Section
The problem of nding answers to a query can b e seen as the problem of searc h
ing a computation tree Most CLP languages employ a depth rst search with
chronological backtracking as in Prolog although there have b een suggestions to
use dep endency directed backtracking Since depth rst search is incomplete
on in nite trees not all answers are computed The depth rst search can b e incor
p orated in the semantics in the same way as is done for Prolog see for example
but we will not go into details here In Section we will discuss a class of
CLP languages which use a top down execution similar to the one outlined ab ove
but do not use backtracking
Consider the transition
hA C S i hA C i
g
where C is a set of equations in L such that Dj C C S and for every
variable x o ccurring in C or S C contains an equation x d for some constant d
Thus grounds all variables in the constraintsolver We also have the transitions
g
hA C S i f ail
g
if no such C exists i e C S is unsatis able in D A ground derivation is a
derivation comp osed of and
r g c g
Wenow de ne three sets that crystallize three asp ects of the op erational se
ts to simple goals mantics The success set SS P collects the answer constrain
p x The nite failure set FF P collects the set of simple goals which are nitely
failed The ground nite failure set GF F P collects the set of grounded atoms
all of whose fair ground derivations are failed
SS P fp x c jhp x i h c c i Dj c c c g
x
FF P fp x c j for every fair derivation hp x c i failg
d i failg GF F P fp d j for every fair ground derivation hp
Soundness and Completeness Results
Wenow present the main relationships b etween the declarative semantics and the
top down op erational semantics Tokeep things simple we consider only ideal CLP
systems However many of the results hold much more generally The soundness
results hold for any CLP system b ecause of restrictions we place on consistent and
inf er Completeness results for successful derivations require only that the CLP
system b e progressive
Theorem Consider a program P in the CLP language determinedbya tuple
D L T where D and T correspond on L and executing on an ideal CLP
system Then
D
and SS P lm P D SS P lf p S
D
P
If the goal G has a successful derivation with answer constraint c then
P Tj c G
Supp ose T is satisfaction complete wrt L IfG has a nite computation tree
with answer constraints c c then P Tj G c c
n n
If P Tj c G then there are derivations for the goal G with answer
W
n
constraints c c such that Tj c c If in addition D L
n i
i
has indep endence of negated constraints then the result holds for n i e
without disjunction
Supp ose T is satisfaction complete wrt L IfP Tj G c c then
n
G has a computation tree with answer constraints c c and p ossibly
m
others such that Tj c c c c
n
m
Supp ose T is satisfaction complete wrt L
The goal G is nitely failed for P i P Tj G
gm P D B GF F P
D
D
Supp ose D L is solution compact T B FF P
D D
P
Supp ose D L is solution compact
P is D L canonical i FF P fh c j P Dj h c g
D D
Most of these results are from but there are also some from
Results and of the ab ove theorem which are equivalent are the only results
of those listed ab ove which require solution compactness In fact the prop erties
shown are equivalentto SC the second condition of solution compactness
SC is not needed In soundness results and half of as mentioned earlier
T can b e replaced by D Ifwe omit our assumption of a rst order language of
constraints see Section then only results and the soundness half
of replacing T by D where necessary continue to hold
The strong form of completeness of successful derivations result provides
an interesting departure from the conventional logic programming theory Itshows
that in CLP it is necessary in general to consider and combine several successful
derivations and their answers to establish that c G holds whereas only one
successful derivation is necessary in standard logic programming The other results
in this theorem are more direct liftings of results in the logic programming theory
The CLP Scheme provides a framework in which the lifting of results from LP
to CLP is almost trivial By replacing the Herbrand universe by an arbitrary con
straint domain D uni abilityby constraint satisfaction Clark s equality theory
by a corresp onding satisfaction complete theory etc most results and even their
pro ofs lift from LP to CLP The lifting is discussed in greater detail in Fur
thermore most op erational asp ects of LP and Prolog can b e interpreted as logical
op erations and consequently these op erations although not their implementations
also lift to CLP One early example is matching which is used in various LP sys
tems e g GHC NU Prolog as a basis for a ecting the computation rule the
corresp onding op eration in CLP is constraintentailment
The philosophy of the CLP Scheme gives primacy to the structure D over
ve which computation is p erformed and less prominence to the theory T Weha
followed this approach However it is also p ossible to start with a satisfaction
complete theory T see for example without reference to a structure We
can arbitrarily cho ose a mo del of T as the structure D and the same results ap
ply Another variation considers a collection D of structures and de nes
consistent C to hold i for some structure D D wehave Dj C Weaker
forms of the soundness and completeness of successful derivations apply in this case
Bottom Up Execution
Bottom up execution has its main use in database applications The set at a time
pro cessing limits the numb er of accesses to secondary storage in comparison to
tuple at a time pro cessing as in top down execution and the simple semantics
gives great scop e for query optimization
Bottom up execution is also formalized as a transition system For every rule r
of the form h c b b in P and every set A of facts there is a transition
n
n
A A fh c j a c i n are elements of A Dj c c c b a g
i i i i i
i
D
In brief then wehave A A S A for every set A and every rule r in P
r
D
S was de ned in Section An execution is a sequence of transitions It is
P
fair if each rule is applied in nitely often The limit of ground instances of sets
generated by fair executions is indep endentoftheorderinwhich transitions are
applied and is regarded as the result of the b ottom up execution If Q is an initial
set of facts and P is a program and A is the result of a fair b ottom up execution
then A SS P Q hhP ii Q and P Q A
D D
An execution Q X X X terminates if for some m and every
i
i m X X Wesay P is nitary if for every nite initial set of facts Q and
i m
every fair execution there is a k suchthat X X for all i k However
i D k D
execution can b e non terminating even when the program is nitary and the initial
set is nite
Example Consider the following program P on the constraint domain
Lin
p X p X
p X X
p X X
Straightforward b ottom up computation gives fp x x p x x
p x x g and p x x g fp x x p x x
D D
fp d j d g T do es not terminate Wealsohave lf p T
P P
A necessary technique is to test whether a new fact is subsumed by the current set
of facts and accumulate only unsubsumed facts A fact p x c is subsumed by
W
n
c The the facts p x c i n with resp ect to D L if Dj c
i i
i
transitions in the mo di ed b ottom up execution mo del are
D
A A A A r educe S
P
where r educe X Y eliminates from X all elements subsumed by Y Under this
execution mo del every nitary program terminates on every nite initial set Q
Unfortunately checking subsumption is computationally exp ensive in general
If the constraint domain D L do es not satisfy the indep endence of negated con
straints then the problem of showing that a new fact is not subsumed is at least
NP hard see for the pro of in one constraint domain In constraint domains
with indep endence of negated constraints the problem is not so bad the new fact
only needs to b e checked against one fact at a time Classical database
optimizations are also more di cult without indep endence of negated constraints
A pragmatic approach to the problem of subsumption in is given
Lin
in Some work avoids the problem of subsumption by allowing only ground
facts in the database and intermediate computations
Even with subsumption there is still the problem that execution mightnot
terminate for example if P is not nitary The approach of is to restrict
the constraint domains D to those which only p ermit the computation of nitely
representable relations from nitely representable relations This requirementis
slightly weaker than requiring that all programs are nitary but it is not clear that
there is a practical di erence Regrettably very few constraint domains satisfy this
condition and those whichdohave limited expressivepower
The alternative is to takeadvantage of P and a sp eci c query or a class of
queries A transformation technique such as magic templates pro duces a
mg
program P that is equivalenttoP for the sp eci c query Other techniques
attempt to further limit execution by placing constraints at
appropriate p oints in the program Analyses can b e used to check that execution
of the resulting program terminates although most work has ignored
the capability of using constraints in the answers
Comparatively little work has b een done on the nuts and b olts of implementing
b ottom up execution for CLP programs with all the work addressing the constraint
domain suggested the use of intervals computed as the pro jection of a
Lin
collection of constraints as the basis for indexing on constrained variables Several
di erent data structures originally develop ed for spatial databases or computa
tional geometry have b een prop osed as appropriate for indexing A
new data structure was presented in which minimizes accesses to secondary
storage A sort join algorithm for joins on constrained variables is given in
That pap er also provides a query optimization metho dology for conjunctive queries
that can balance the cost of constraint manipulation against the cost of traditional
database op erations
Concurrent Constraint Logic Programming
Concurrent programming languages are languages which allow the description of
collections of pro cesses whichmayinteract with each other In concurrentcon
straint logic programming CCLP languages communication and synchronization
are p erformed by asserting and testing for constraints The op erational semantics
of these languages are quite similar to the top down execution describ ed in Sec
ever the di erentcontext in which they are used results in a lesser tion How
imp ortance of the corresp onding logical semantics
For this discussion we will consider only the at ask tell CCLP languages which
were de ned in based on ideas from We further restrict our atten
tion to languages with only committed choice nondeterminism sometimes called
don t care nondeterminism more general languages will b e discussed in Section
For more details of CCLP languages see
Just as Prolog can b e viewed as a kind of CLP language obtained by a particular
choice of constraint domain so most concurrent logic languages can b e viewed as
concurrent CLP languages
A program rule takes the form
h ask tel l j B
where h is an atom B is a collection of atoms and ask and tel l are constraints
Many treatments of concurrent constraint languages employ a language based on a
pro cess algebra involving ask and tell primitives but weusethesyntax ab ove
to emphasize the similarities to other CLP languages
For the sake of brevity wepresent a simpler transition system to describ e the op
erational semantics than the transition system in Section However implemented
languages can make the same pragmatic compromises on testing consistency and
Concurrent Prolog is not an ask tell language but shows how it can b e t inside
the CCLP framework
implication as re ected in that transition system The states in this transition
system have the form hA C i where A is a collection of atoms and C is a collection
of constraints Any state can b e an initial state The transitions in the transition
system are
hA a C i hA B C a h ask tel l i
r
if h ask tel l j B is a rule of P renamed to new variablesx h and a havethe
same predicate symb ol Dj C xa h ask and Dj C a h ask tel l
Roughly sp eaking a transition can o ccur with such a rule provided the accumulated
constraints imply the ask constraint and do not contradict the tell constraint Some
languages use only the ask constraint for synchronization It is shown in that
such languages are strictly less expressive than ask tell languages
An op erational semantics such as the ab ove is not completely faithful to a real
execution of the language since it is p ossible for two atoms to b e rewritten si
multaneously in an execution environment with concurrency The ab ove semantics
only allows rewritings to b e interleaved A true concurrency semantics based on
graph rewriting is given in
All ask tell CCLP programs have the following monotonicity orstability
prop erty If hA C i hA C i and Dj C C then hA C i hA C i This
r r
prop erty provides for simple solutions to some problems in distributed computing
related to reliability Whenlooked at in a more general framework stability
seems to b e one advantage of CCLP languages over other languages most programs
in conventional languages for concurrency are not stable It is interesting to note
that a notion of global failure as represented in Section by the state fail destroys
stability Of course there are also pragmatic reasons for wanting to a void this
notion in a concurrent language A framework which p ermits non monotonic CCLP
languages is discussed in
A program is determinate if every reachable state is determinate where a state
is determinate if every selected atom gives rise to at most one transition Con
r
sequently for every initial state every fair derivation rewrites the same atoms
with the same rules or every derivation fails Thus non failed derivations byde
terminate programs from an initial state di er from each other only in the order
of rewriting and the renaming of rules Substantial parts of many programs are
determinate The interest in determinate programs arises from an elegant seman
tics for such programs based up on closure op erators For every collection
of atoms A the semantics of A is given by the function P C C where
A x
hA C i hA C i x is the free variables of hA C i andhA C i is a nal state
r
This semantics is extended in to a comp ositional and fully abstract semantics
of arbitrary programs A semantics based on traces is given in
For determinate programs we also have a clean application of the classical logical
semantics of a program If hA C i hA C i then P Dj A C
r
A C wherex is the free variables of hA C i In cases where execution can b e
x
For the programs we consider determinate programs can b e characterized syntactically
by the following condition for every pair of rules renamed apart except for identical heads
h ask tell j B and h ask tell j B in the program wehave Dj ask ask tell
or Dj ask ask tell In languages where pro cedures can b e hidden as in manyprocess
algebra formulations or there is a restriction on the initial states the class of determinate programs
is larger but is not so easily characterized
guaranteed not to susp end any atom inde nitely the soundness and completeness
results for success and failure hold see Section
Linguistic Extensions
We discuss in this section some additional linguistic features for top down CLP
languages
Shrinking the Computation Tree
The aim of transitions is to extract as much information as is reasonable from
i
the passive constraints so that the branching of transitions is reduced There
r
are several other techniques used or prop osed for achieving this result
In it is suggested that information can also b e extracted from the atoms in
a state The constraint extracted would b e an approximation of the answers to the
atom This op eration can b e expressed by an additional transition rule
hA a C S i hA a C S ci
x
where extr act a C c Here extr act is a function satisfying P Dj a C c
The evaluation of extr act p erformed at run time involves an abstract or approx
imate execution of ha C i For example if P de nes p with the facts p and
p then the constraint extracted by extr act p x y mightbe y x
A more widespread technique is to mo dify the order in which atoms are selected
Most CLP systems employ the Prolog left to right computation rule This improves
the programmability byproviding a predictable owofcontrol However when
an appropriate owofcontrol is data dep endentorvery complex for example in
combinatorial search problems greater exibility is required
One solution to this problem is to incorp orate a data dep endent computation
rule in the language The Andorra principle involves selecting determinate
atoms if p ossible A determinate atom is an atom which only gives rise to one
ris
transition A second approachistoallow the programmer to annotate parts of the
program atoms predicates clauses to provide a more exible computation
rule that is nonetheless programmed This approachwas pioneered in Prolog
I I and MU Prolog The automatic annotation of programs brings
this approach closer to the rst A third approachistointro duce constructs from
concurrent logic programming into the language There are basically twovarieties
of this approach guarded rules and guarded atoms The former intro duces a
committed choice asp ect into the language whereas the latter is a variantofthe
second approach All these approaches originated for conventional logic programs
but the ideas lift to constraint logic programs and there are nowseveral prop osals
based on these ideas
One p otential problem with using guarded rules is that the completeness of the
op erational semantics with resp ect to the logical semantics of the program can b e
lost This incompleteness was shown to b e avoided in ALPS mo dulo in nitely
delayed atoms but that work was heavily reliant on determinacy Smolka
discusses a language of guarded rules which extends ALPS and a metho dology for
extending a predicate de nition with new guarded rules such that completeness can
b e retained even though execution would involve indeterminate committed choice
The Andorra Kernel Language AKL also combines the Andorra principle
with guarded rules There the interest is in providing a language which subsumes
the expressivepower of CCLP languages and CLP languages
Guarded atoms and more generally guarded goals take the form c G where c
is a constraint and G is a goal G is available for execution only when c is implied
by the current active constraints Wecallc the guardconstraint and G the delayed
goal Although the underlying mechanisms are very similar guarded atoms and
guarded rules di er substantially as linguistic features since guarded atoms can b e
combined conjunctively whereas guards in guarded rules are combined disjunctively
Complex Constraints
Several language constructs that can b e said simply to b e complex constraints have
b een added to CLP languages We can classify them as follows those whichim
plement Bo olean combinations of generally simple constraints and those which
describ e an ad ho c often application sp eci c relation Falling into the rst cate
gory are some implementations of constraint disjunction sometimes called
constructive disjunction and the cardinality op erator Into the second cat
constraint and the cumulative constraint of among egory fall the el ement
others These constraints are already accounted for in the op erational semantics
of Section since they can b e considered passive constraints in L However it
also can b e useful to view them as additions to a b etter known constraint domain
indeed this is how they arose
The cardinality op erator can b e used to express any Bo olean combination of
constraints A use of this combinator has the form L c c U where the
n
c are constraints and L and U are variables This use expresses that the number
i
of constraints c that are true lies b etween the value of L and the value of U lower
i
and upp er b ound resp ectively By constraining L the combinator represents
the disjunction of the constraints by constraining U the combinator represents
the conjunction of the negations of the constraints The cardinality combinator
is implemented by testing whether the constraints are entailed by or inconsistent
with the constraint store and comparing the numb ers of entailed and inconsistent
constraints with the values of L and U When L and U are not ground the
cardinality constraint can pro duce a constraint on these variables For example
after one constraint is found to b e inconsistent U can b e constrained by U n
In constraint languages without disjunction an intended disjunction c x c x
must b e represented by a pair of clauses
p x c x
p x c x
In a simple CLP language this representation forces a choicetobemade between
the two disjuncts Constructive disjunction refers to the direct use of a disjunctive
constraint without immediately making a choice Instead an active constraintis
computed which is a safe approximation to the disjunction in the context of the
current constraintstoreC In the constraint domain FD suggests two p ossible
We also p ermit the constraint g r ound x
approximations one based on approximating each constraint C c using the domain
i
of eachvariable and the other less accurately approximating each constraint using
the interval constraints for eachvariable The disjunctions of these approximations
is easily expressed as an active constraint For linear arithmetic suggests the use
of the convex hull of the regions de ned bythetwo constraints as the approximation
Note that the constructive disjunction b ehavior could b e obtained from the clauses
for p using the metho ds of
In the second category we mention two constructs used with the nite domain
solver of CHIP el ement X L T expresses that T is the X th element in the list
L Op erationally it allows constraints on either the index X or element T of the
list to b e re ected by constraints on the other For example if X is constrained
so that X f g then el ement X T can constrain T so that
T f g and similarly if T is constrained so that T f g then X is
constrained so that X f g Declaratively the cumul ativ e constraintof
expresses a collection of linear inequalities on its arguments Several problems
that can b e expressed as integer programming problems can b e expressed with
cumul ativ e Op erationally itbehaves somewhat di erently from the way CHIP
would usually treat the inequalities
User De ned Constraints
Complex constraints are generally built in to the language There are prop osals
to extend CLP languages to allow the user to de ne new constraints together with
inference rules sp ecifying how the new constraints react with the constraint store
A basic approach is to use guarded clauses The new constraint predicate is
de ned with guarded clauses where the guards sp ecify the cases in which the con
straint is to b e simpli ed and the b o dy is an equivalent conjunction of constraints
Using g r ound x or a similar construct as a guard constraint it is straightfor
ward to implement lo cal propagation i e propagation of ground values Wegive
an example of this use in Section and has other examples Some more
general forms of propagation can also b e expressed with guarded clauses
The w ork can b e seen as an extension of this metho d The new constraints
o ccur as predicates and guarded rules called constraint simpli cation rules are
used to simplify the new constraints However the guarded rules mayhavetwo or
more atoms in the head Execution matches the head with a collection of constraint
atoms in the goal and reduces to an equivalent conjunction of constraints This
metho d app ears able to express more p owerful solving metho ds than the guarded
clauses For example transitivity of the user de ned constraint leq can b e sp eci ed
by the rule
leq X Y leq Y Z true leq X Z
whereas it is not clear how to express this in a one atom p er head guarded clause
Adrawbackofhaving multiple atoms however is ine ciency In particular it is
not clear whether constraint simpli cation rules can pro duce incremental in the
sense de ned in Section constraintsolvers except in simple cases
A di erentapproach prop oses the intro duction of indexical terms which
refer to asp ects of the state of the constraint solver thus providing a limited form
of re ection Constraints containing these terms are called indexical constraints
and from these indexical constraints user de ned constraints are built Sp eci cally
discusses a language over nite domains which can access the current domain
and upp er and lower b ounds on the value of a variable using the indexical terms
dom X max X andmin X resp ectively Indexical constraints have an op era
tional semantics each constraint de nes a metho d of propagation For example
the constraint Yin max X continually enforces the upp er b ound of Y to b e
less than or equal to the upp er b ound of X This same b ehavior can b e obtained
in a nite domain solver with the constraint Y X buttheadvantage of indexical
constraints is that there is greater control over propagation with Y X we also
propagate changes in the lower b ound of Y to X whereas wecanavoid this with
indexical constraints A discussion of an implementation of indexical constraints
is given in One application of this work is a constraint solver for Bo olean
ts we describ e this application in Section constrain
Negation
Treatments of negation in logic programming lift readily to constraint logic pro
gramming with only minor adjustments necessary Indeed many of the semantics
for programs with negation are essentially prop ositional b eing based up on the col
lection of ground instances of program rules The p erfect mo del
well founded mo del stable mo del and Fitting xedp ointsemantics
to name but a few fall into this category The grounding of variables in CLP
rules by all elements of the domain i e by all terms in L and the deletion of
all grounded rules whose constraints evaluate to false pro duces the desired prop o
sitional rules see for example
Other declarativesemantics based on Clark s completion P of the program
also extend to CLP Thecounterpart of comp P is T P where T is
satisfaction complete Interestingly it is necessary to consider the complete theory
T of the domain if the equivalence of three valued logical consequences of T P
and consequences of nite iterations of Fitting s op erator as shown by Kunen
continues to hold for CLP programs
SLDNF resolution and its variants are also relatively untouched by the lifting
to CLP programs although of course they must use a consistency test instead of
uni cation The other main mo di cation is that groundness must b e replaced by
the concept of a variable b eing determined by the current constraints see Section
For example a safe computation rule may select a non ground negated
atom provided all the variables in the atom are determined by the current collec
tion of constraints Similarly the de nition of an allowed rule for a CLP
program requires that every variable either app ear in a p ositive literal in the b o dy
or b e determined by the constraints in the b o dy With these mo di cations various
soundness and completeness results for SLDNF resolution and comp P extend eas
ily to ideal CLP systems An alternative implementation of negation constructive
negation has b een expanded and applied to CLP programs byStuckey
who gave the rst completeness result for this metho d
This approach has b een called a glass b ox approach
For example the extension to allow arbitrary rst order formulas in the b o dies of rules
Preferred Solutions
Often it is desirable to express an ordering or preference on solutions to a goal
This can provide a basis for computing only the b est solutions to the query
One approach is to adapt the approach of mathematical programming op erations
research and employ an ob jective function An optimization primitive
is added to the language to compute the optimal value of the ob jective function
CHIP and cc FD havesuch primitives but they have a non logical b ehavior
Tworecent pap ers discuss optimization primitives based up on the following
logical characterization
m is the minimum value of f x suchthatG x holds i
x G x f x m y G y f y m
Optimization primitives can b e implemented by a branch and b ound approach
pruning the computation tree of G based on the current minimum A similar
ve logical behavior can b e obtained through constructive negation using the ab o
formulation although a sp ecial purp ose implementation is more e cient
gives a completeness result for such an implementation based on Kunen s
semantics for negation
A second approach is to admit constraints which are not required to b e satis ed
by a solution but express a preference for solutions which do satisfy them Such
constraints are sometimes called soft constraints The most develop ed use of this ap
proach is in hierarchical constraint logic programming HCLP In HCLP
soft constraints have di erent strengths and the constraints accumulated during a
derivation form a constraint hierarchy based on these strengths There are many
p ossible ways to compare solutions using these constraint hierarchies
di erent metho ds b eing suitable for di erent problems The hierarchy dictates that
anynumber of weak constraints can b e over ruled by a stronger constraint Thus
for example default b ehavior can b e expressed in a program byweak constraints
which will b e over ruled by stronger constraints when non default b ehavior is re
quired The restriction to b est solutions of a constraint hierarchy can b e viewed as
a form of circumscription
Each of the ab ove approaches has some programming advantages over the other
in certain applications but b oth have problems as general purp ose metho ds While
the rst approachworks well when there is a natural choice of ob jective function
suggested by the problem in general there is no natural choice The second ap
proach provides a higher level expression of preference but it cannot b e so easily
ne tuned and it can pro duce an exp onential numberofbestanswers if not used
carefully The approaches have the advantages and disadvantages of explicit re
sp ectively implicit representations of preference In the rst approach it can b e
di cult to re ect intended preferences In the second approach it is easier to re ect
intended preferences but harder to detect inconsistency in these preferences It is
also p ossible to weight soft constraints whichprovides a combination of b oth
approaches
We discuss only minimization maximization is similar
Part I I
Implementation Issues
The main innovation required to implement a CLP system is clearly in the manipu
lation of constraints Thus the main fo cus in this part of the survey is on constraint
solver op erations describ ed in the section b elow The next section then considers
the problem of extending the LP inference engine to deal with constraints Here
the discussion is not tied down to a particular constraintdomain
It is imp ortant to note that the algorithms and data structures in this part are
presented in view of their use in top down systems and in particular systems with
backtracking At the present there is little exp erience in implementing b ottom up
CLP systems and so we do not discuss them here However some of the algorithms
we discuss can b e used p erhaps with mo di cation in b ottom up systems
Algorithms for Constraint Solving
In view of the op erational semantics presented in part I there are several op erations
involving constraints to b e implemented These include a satis ability test to
implement consistent and inf er anentailment test to implement guarded goals
and the pro jection of the constraintstoreonto a set of variables to compute the
answer constraint from a nal state The constraintsolver must also b e able to
undo the e ects of adding constraints when the inference engine backtracks In
this section we discuss the core e ciency issues in the implementation of these
op erations
Incrementality
According to the folklore of CLP algorithms for CLP implementations must b e
incremental in order to b e practical However this prescription is not totally sat
isfactory since the term incremental can b e used in two di erent senses On one
hand incremen tality is used to refer to the nature of the algorithm That is an
algorithm is incremental if it accumulates an internal state and a new input is
pro cessed in combination with the internal state Such algorithms are sometimes
called on line algorithms On the other hand incrementality is sometimes used to
refer to the performance of the algorithm This section serves to clarify the latter
notion of incrementality as a prelude to our discussion of algorithms in the following
subsections Wedonot however o er a formal de nition of incrementality
We b egin by abstracting away the inference engine from the op erational seman
tics to leave simply the constraint solver and its op erations We consider the state
of the constraint solver to consist of the constraint store C a collection of con
straints G that are to b e entailed and some backtrackpoints In the initial state
denoted by there are no constraints nor backtrack p oints The constraint solver
reacts to a sequence of op erations and results in a a new state and b a resp onse
Recall that the op erations in CLP languages are
augment C with c to obtain a new store determine whether the new store
is satis able and if so determine which constraints in G are implied bythe new store
add a new constrainttoG
set a backtrack p oint and asso ciate with it the current state of the system
backtrack to the previous backtrackpoint i e return the state of the system
to that asso ciated with the backtrack p oint
pro ject C onto a xed set of variables
Only the rst and last of these op erations can pro duce a resp onse from the con
straint solver
Consider the application of a sequence of op erations o o on a state denote
k
the up dated state by F o o and the sequence of resp onses to the op erations
k
by G o o In what follows we shall b e concerned with the average cost of
k
computing F and G Using standard de nitions this cost is parameterized bythe
distribution of sequences of op erations see for example We use average
cost assuming the true distribution the distribution that re ects what o ccurs in
practice Even though this distribution is almost always not known we often have
some hyp otheses ab out it For example one can identify typical and often o ccurring
op eration sequences and hence can approximate the true distribution accordingly
The informal de nitions b elow therefore are intended to b e a guide as opp osed to
a formal to ol for cost analysis
AV exp o to b e the For an expression exp o denoting a function ofo de ne
average value of exp o over all sequences of op erationso Note that the de nition
of average here is also dep endent on the distribution of theo For example let
cost o denote the cost of computing F o by some algorithm for each xed
sequenceo ThenAV cost o denotes the average cost of computing F o over
all o
Let b e shorthand for F o o Let A denote an algorithm which
k
applies a sequence of op erations on the initial state giving the same resp onse as
do es the constraint solver but not necessarily computing the new state That is A
is the batch or o line version of our constraintsolver In what follows we discuss
what it means for an algorithm to b e incremental relative to some algorithm A
Intuitively A represents the b est available batch algorithm for the op erations
At one extreme we consider that an algorithm for F and G is non incremental
relativetoA if the average cost of applying an extra op eration o to isnobetter
k
than the cost of the straightforward approach using A on o o We express this
k
as
AV cost o AV cost o o
k A k
At the other extreme we consider that an algorithm for F and G is p erfectly
incremen tal relativeto A if its cost is no worse than that of A In other words
no cost is incurred for the incremental nature of the algorithm We express this as
AV cost o o cost o AV cost o o
k k A k
In general any algorithm lies somewhere in b etween these two extremes For ex
ample it will not b e p erfectly incremental as indicated by the cost formula ab ove
but instead wehave
cost o o AV cost o o cost o AV cost o o extr a
k k k A k
where the additional term extr a cost o o denotes the extra cost incurred by
k
the on line algorithm over the b est batch algorithm Therefore one p ossible de n
ition of an incremental algorithm go o d enough for use in a CLP system is simply
that its extr a cost factor is negligible
In what follows we shall tacitly b ear in mind this expression to obtain a rough
de nition of incrementality Although wehave de ned incrementality for a col
lection of op erations we will review the op erations individually and discuss incre
mentality in isolation This can sometimes b e an oversimpli cation for example
has shown that the standard uni cation problem do es not remain linear when
backtracking is considered In general however it is simply to o complex in a
survey article to do otherwise
Satis ability Non incremental
We consider rst the basic problem of determining satis ability of constraints in
dep endent of the requirement for incrementality Aswe will see in the brief tour
below of our sample domains the dominant criteria used by system implementers
is not the worst case time complexity of the algorithm
For the domain FT linear time algorithms are known and for RT the
b est known algorithms are almost linear time Even so most Prolog systems
implement an algorithm for the latter b ecause the best case complexity of uni
cation in FT is also linear whereas it is often the case that uni cation in RT can
b e done without insp ecting all parts of the terms b eing uni ed Hence in practice
Prolog systems are really implementations of CLP RT rather than CLP FT
In fact many Prolog systems cho ose to use straightforward algorithms whichare
slower in the worst case than these almost linear time algorithms The reason for
er in the worst case is this choice of algorithms which are quadratic time or slow
the b elief that these algorithms are faster on average
For the arithmetic domain of the most straightforward algorithm is
LinE q n
based on Gaussian elimination and this has quadratic worst case complexity For
the more general domain p olynomial time algorithms are also known
Lin
but these algorithms are not used in practical CLP systems Instead the Sim
plex algorithm see eg despite its exp onential time worst case complexity
is used as a basis for the algorithm However since the Simplex algorithm
works over non negativenumb ers and non strict inequalities it must b e extended
for use in CLP systems While such an extension is straightforward in principle
implementations must b e carefully engineered to avoid signi cantoverhead The
main di erences b etween the Simplex based solvers in CLP systems is in the sp e
ci c realization of this basic algorithm For example the CLP R system uses a
oating p oint representation of numb ers whereas the solversofCHIPandProlog
I I I use exact precision rational numb er arithmetic As another example in the
CLP R system a ma jor design decision was to separately deal with equations and
There are similar notions found in the non CLP literature see the bibliography
This is often realized simply by omitting the o ccur check op eration from a standard uni
cation algorithm for FT Some Prolog systems p erform such an omission naively andthus obtain
an incomplete algorithm whichmay not terminate in certain cases These cases are considered
pathological and hence are ignored Other systems guarantee termination at slightly higher cost
but enjoy the new feature of cyclic data structures
inequalities enjoying a faster Gaussian elimination based algorithm for equations
but enduring a cost for communication b etween the two kinds of algorithms
Some elements of the CHIP solver are describ ed in Disequality constraints
can b e handled using entailment of the corresp onding equation discussed in Section
since an indep endence of negative constraints holds
For the domain of word equations WE an algorithm is known butno
e cient algorithm is known In fact the general problem though easily provable
to b e NP hard is not known to b e in NP The most e cient algorithm known still
has the basic structure of the Makanin algorithm but uses a far b etter b ound for
termination Systems using word equations Prolog I I I for example thus resort
to partial constraint solving using a standard delaytechnique on the lengths of word
variables Ra jasekar s string logic programs also uses a partial solution of
word equations First solutions are found for equations over the lengths of the
word variables app earing in the constraint only then is the word equation solved
As with word equations the satis ability problem in nite domains suchasFD
is almost always NP hard Partial constraint solving is once again required and
here is a typical approach Attachtoeachvariable x a data structure representing
dom x its current p ossible values Clearly dom x should b e a sup erset of the
x andmax x to b e the smallest and largest pro jection space w r t x De ne min
numb ers in dom x resp ectively Now assume that every constraint is written in so
that each inequality is of the form x y or x y each disequalityisoftheform
x y andeach equation is of the form x n x y x y z where x y z are
variables and n anumb er Clearly every constraintinFD can b e rewritten into a
conjunction of these constraints
The algorithm considers one constraint at a time and has two main phases
First it p erforms an action which is determined by the form of the constraint a
for constraints x y ensure that min x max y by mo difying dom x and or
dom y appropriately b for x y ensure that min x max y c for
x y consider three sub cases if dom x dom y then the constraint reduces
to true otherwise if dom x fng then remove n from dom y and similarly for
the case when dom y is a singleton otherwise nothing more need b e done d
for x n simple make dom x fng e for x y make dom x dom y
dom x dom y f for x y z ensure that max x min y min z and
min x max y max z If at any time during steps a through f the domain
ofavariable b ecomes empty then unsatis ability has b een detected The second
phase of this algorithm is that for each x such that dom x thatischanged bysome
action in steps a through f all constraints but the current one that gaverise
to this action that contain x are re considered for further action Termination is
of course assured simply b ecause the domains are nite
In the domain of Bo olean algebra BOOL there are a varietyoftechniques for
testing satis ability Since the problem is NP complete none of these can b e ex
p ected to p erform e ciently over all constraints An early technique pioneered by
The choice of such a data structure should dep end on the size of the nite domains For
example with small domains a characteristicvector is appropriate
In this case simply removefromdom x all elements bigger than max y and removefrom
dom y all elements smaller than min x We omit the details of similar op erations in the following
discussion
If b oth are singletons clearly the constraint reduces to either true or false
Davis and Putnam is based up on variable elimination The essential idea reduces a
normal form representation into two smaller problems each with one less variable
Binary decision diagrams provide an e cient representation One of the two
Bo olean solvers of CHIP for example uses variable elimination and these diagrams
A related technique is based on enumeration and propagation The constraints are
expressed as a conjunction of simple constraints and then lo cal propagation simpli
es the conjunction after eachenumeration step See for example The metho d
used in Prolog I I I is a mo di cation of SL resolution whose main elementisthe
elimination of redundant expressions Another technique comes from Op erations
Research Here the b o olean formula is restated in arithmetic form with variables
constrained to b e and Then standard techniques for integer programming
for example cutting planes can b e used See for a further discussion of this
technique This technique has not b een used in CLP systems A more recentde
velopment is the adaptation of Buchb erger s Gro ebner basis algorithm to Bo olean
algebras which is used in CAL Finally there is the class of algorithms which
p erform b o olean uni cation see the survey for example Here satis ability
testing is only part of the problem addressed and hence we will discuss these algo
rithms in the next section
The satis ability problem for feature trees is essentially the same as the satis
ability problem for rational trees provided that the numb er of features that may
o ccur is b ounded by a known constant Generally this b ounding constantcan
b e determined at compile time Two di erent sort constraints on the same variable
clash in the same way that di erent function symb ols on terms in RT clash An
equation b etween t wo feature tree variables of the same sort induces equations
between all the subtrees determined by the features of the variables in the same
way as o ccurs in RT The main di erence is that some sorts or features maybe
undetermined roughly unb ound in FEAT
Satis ability Incremental
As alluded to ab ove it is crucial that the algorithm that determines the satis ability
of a tentatively new constraint store b e incremental For example a linear time
algorithm for a satis ability problem is often as go o d as one can get Consider a
sequence of constraints c c of approximately equal size N A naive application
k
of this linear time algorithm to decide c then c c and nally c
onaverage In contrast a p erfectly c could incur a cost prop ortional to Nk
k
incremental algorithm as discussed in Section has a cost of O Nk on average
In practice most algorithms represent constraints in some kind of solvedform
a format in which the satis ability of the constraints is evident Thus the satis
ability problem is essentially that of reducibilityinto solved form For example
t constraints by onevariantof standard uni cation algorithms for FT represen
its mgu that is in the form x t y x t y where each t y denotes
n n i
a term structure containing variables fromy and no variable x app ears iny
i
Similarly linear equations in are often represented in parametric form
LinE q n
x le y x le y where each le y denotes a linear expression contain
n n i
ing variables fromy andnovariable x app ears iny In b oth these examples call
i
the x eliminable variables and the y parametric variables For linear inequalities
i i
ts in an n m matrix form in the Simplex algorithm represents the constrain
Lin
Ax B where A contains an n n identity submatrix de ning the basis vari
ables and all numb ers in the column vector B are nonnegative For domains based
on a unitary equality theory the standard representation is the mgu as in
the case of FT which corresp onds to the most elementary equality theory Word
equations over WE however are asso ciated with an in nitary theory andthus a
uni cation algorithm for these equations may not terminate A solved form
for word equations or any closed form solution for that matter is not known
The rst two kinds of solved form ab ove are also examples of solution forms that
is a format in which the set of all solutions of the constraints is evident Here any
instance of the variablesy determines values forx and thus gives one solution The
set of all such instances gives the set of all solutions The Simplex format however
is not in solution form eachchoice of basis variables depicts just one particular
solution
An imp ortant prop erty of solution forms and sometimes of just solved forms
is that they de ne a convenient representation of the projection of the solution
space with resp ect to any set of variables More sp eci cally eachvariable can b e
equated with a substitution expression containing only parametric variables that
is variables whose pro jections are the entire space This prop erty in turn aids
enowshow via our sample domains incrementalityasw
In each of the following examples let C b e a satis able constraintinsolved form
and let c b e the new constraint at hand For FT the substitution expression for a
variable x is simply x if x is not eliminable otherwise it is the expression equated
to x in the solved form C This mapping is generalized to terms in the obvious way
Similarly we can de ne a mapping of linear expressions by replacing the eliminable
variables therein with their substitution expressions and then collecting like terms
For the domain in which case C is in Simplex form the substitution expression
Lin
for a variable x is simply x if x is not basic otherwise it is the expression obtain
by writing the unique equation in C containing x with x as the sub ject Once
again this mapping can b e generalized to any linear expression in an obvious way
In summary a solution form de nes a mapping which can b e used to map any
expression t into an equivalentformt which is free of eliminable variables
The basic step of a satis ability algorithm using a solution form is essentially
this
Algorithm Given C a Replacethenewlyconsideredc onstraint c by c where
is the substitution de nedbyC b Then write c into equations of the form
x and this involves choosing the x and rearranging terms Unsatis ability is
detected at this stage c If the previous step succeeds use the new equations to
substitute out al l occurrences of x in C d Final ly simply add the new equations
to C to obtain a solution form for C c
Note that the nonappearance of eliminable variables in substitution expressions is
needed in b to ensure that the new equations themselves are in solved form and
in c to ensurethatC augmented with the new equations remains in solution
form
The b elief that this metho dology leads to an incremental algorithm is based up on
b elieving that the cost of dealing with c is more closely related to the size of c
which is small on average than that of C whichisvery large on average This
in turn is based up on b elieving that
the substitution expressions for the eliminable variables in c which largely
determine the size of c oftenhave a size that is indep endent of the size of
C and
the numb er of o ccurrences of the new eliminable variable x in C which
largely determines the cost of substituting out x in C is small in comparison
to the size of C
The domain FT provides a particularly go o d example of a solved form for whichthe
basic algorithm is incremental Consider a standard implementation in which
there is only one lo cation for eachvariable and all references to x are implemented
by p ointers Given C in solved form and given a new constraint c there is really
nothing to do to obtain c since the eliminable or in this case b ound variables
in c are already p ointers to their substitution expressions Nowifc is satis able
and we obtain the new equations x then just one p ointer setting of x to its
substitution expression is required and we are done In other words the cost of
this p ointer based algorithm is fo cussed on determining the satis abilityof c and
extracting the new equations in contrast step c of global substitution using the
new equations incurs no cost
For the size of c can b e large even though the nally obtain equations
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u v x v w x w u and maynotbe For example if C contained just x
c were y x x x then c is as big as C Up on rearrangement however the
nally obtained new equation is simply y Next the substitution phase using
the new equation also can enlarge the equations in C even if temp orarily and
rearrangement is needed in each equation substituted up on In general however
the b eliefs ab ove hold in practice and the algorithm b ehaves incrementally
We next consider the domain RT whose universally used solved form due to
is like that of FT with one imp ortantchange constraints are represented in
the form x t x t where each t is an arbitrary term structure Thus this
n n i
solved form di ers from that of FT in that the t can contain the variables x and
i j
hence algorithm is not directly applicable It is easy to show that a constraintis
satis able i it has suchasolved form and further the solved form is fairly explicit
representation of the set of all solutions though not as explicit as the solution
forms for FT or A straightforward satis ability algorithm is roughly
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as follows Let x stand for a variable and s and t stand for non variable terms
Now p erform the following rewrite rules until none are applicable a discard each
x x b for any x y replace x by y throughout c replace t x by x t
d replace f s s f t t n by n equations s t i n
n n i i
e replace f g by false and thus the entire collection of constraints is
unsatis able f replace every pair of equations x t x t and say t is not
bigger than t by x t t t Termination needs to b e argued but we will leave
the details to
Wenow discuss algorithms which do not t exactly with Algorithm but which
employa solved form Consider rst the Simplex algorithm for the domain
Lin
The basic step of one pivoting op eration within this algorithm is essentially the same
as Algorithm The arguments for incrementality for Algorithm thus apply The
main di erence from Algorithm is that in general several pivoting op erations are
required to pro duce the nal solved form However empirical evidence from CLP
systems has shown that often the numberofpivoting op erations is small
In the Bo olean domain Bo olean uni cation algorithms conform to the
structure of Algorithm One uni cation algorithm is essentially due to Bo ole and
weborrow the following presentation from Without loss of generality assume
the constraints are of the form t x x where the x are the variables in
n i
t Assuming n rewrite t into the form
g x x x h x x
n n n
so that the problem for t can b e reduced to that of
g x x h x x
n n
whichcontains one less variable If this equation is satis able then the assignment
g x x y x h x x
n n n n
where y is a new variable describ es all the p ossible solutions for x This reduction
n n
clearly can b e rep eatedly applied until we are left with the straightforward problem
of deciding the satis ability of equations of the form t x u where t and u
are ground The uni er desired is given simply by collecting and substituting out
all assigned variables in the assignments such as that for x ab ove
n
The key e ciency problem here is of course that the variable elimination pro cess
gives rise to larger expressions an increase whichisexponential in the number of
eliminated variables in the worst case So even though this algorithm satis es
the structure of Algorithm it do es not satisfy our assumption ab out the size of
expressions obtained after substitution and hence our general argument for incre
mentality do es not apply here Despite this and the fact that Bo ole s work dates
far back this metho d is still used for example in CHIP
Another uni cation algorithm is due to L owenhein and we adapt the presenta
tion of here Let f x x b e the equation considered Leta denote
n
a solution The uni er is then simply given by
x y f y y a i n
i i i i
where the y are new variables The basic e ciency problem is of course to determine
i
a The obtained uni ers are only slightly larger than f incontrast to Bo ole s
metho d Thus L owenhein s metho d provides a way of extending a constructive
satis abilitytestinto a satis ability test which has an incremental form However
this metho d is not to our knowledge used in CLP languages
Other algorithms for testing the satis ability of Bo olean constraints are con
siderably di erent from Algorithm The Gro ebner Basis algorithm pro duces a
basis for the space of Bo olean constraints implied by the constraint store It is a
solved form but not a solution form The remaining algorithms mentioned in the
previous subsection do not havea solved form The algorithm used in Prolog I I I
retains the set of literals implied to b e true by the constraints but the normal
form do es not guarantee solvability that must b e tested b eforehand Enumeration
algorithms have the same b ehavior they exhibit a solution and may retain some
further information but they do not compute a solved form
In the domain of feature trees FEAT equations o ccur only b etween variables
Thus Algorithm do es not address the whole problem Existing algorithms
employasolved form in which all implied equations b etween variables are explicit
hsolved forms are in fact solution forms The and there are no clashes of sort Suc
implied equations are found by congruence closure treating the features as partial
functions analogously to rule d in the algorithm for RT
In summary for this subsection an imp ortant prop erty for algorithms to decide
satis ability is that they have good average case b ehavior More imp ortant and
even crucially so is that the algorithm is incremental Toward this goal a common
technique is to use a solved form representation for satis able constraints
Entailment
Given satis able C guard constraints G such that no constraint therein is entailed
by C and a new constraint c the problem at hand is to determine the subset G of
G of constraints entailed by C c We will also consider the problem of detecting
groundness which is not strictly sp eaking an entailment problem However itis
essentially the same as the problem of detecting groundness to a sp eci c value
whichisanentailment problem In what follows the distinction is unimp ortant
We next present a rule of thumb to determine whether an entailment algorithm
is incremental in the sense discussed earlier The imp ortant factor is not the num
b er of constraints entailed after a change in the store but instead the number of
constraints not entailed That is the algorithm must b e able to ignore the latter
ber of entailed con constraints so that the costs incurred dep end only on the num
straints as opp osed to the total numb er of guard constraints As in the case of
incremental satis ability the prop erty of incremental entailment is a crucial one
for the implementation of practical CLP systems
Wenow brie y discuss mo di cations to some of the previously discussed algo
rithms for satis ability which provide for incremental entailment
Consider the domain FT and supp ose G contains only guard constraints of the
form x t where t is some ground term Add to a standard implementation of
a uni cation algorithm an index structure mapping variables x to just those guard
constraints in G whichinvolve x See for a detailed description Now add to
the pro cess of constructing a solved form a check for groundness when variables are
b ound and this is easily detectable This gives rise to an incremental algorithm
b ecause the only guard constraints that are insp ected are those x t for which x
has just b ecome ground and not the entire collection G
Just as with satis ability testing entailment is essentially the same over the
domains RT and FEAT Four recentworks have addressed this problem all in the
context of a CLP system but with sligh tly di ering constraint domains We will
discuss them all in terms of RT With some mo di cations these works can also
apply to FT
In a theoretical foundation is built then prop oses a concrete
algorithm very roughly as follows the to b e entailed constraint c is added to the
constraint store C The satis ability tester has the capability of detecting whether
c is entailed by or inconsistent with C If neither is detected then c essentially a
simpli ed form of c is stored and the e ect of adding c to C is undone Every time
a constraint is added to C that a ects c this entailment test is rep eated with c
instead of c
As mentioned ab ove this discussion will essentially apply to guard constraints of the form
g r ound x
The algorithm of has some similarities to the previous algorithm but avoids
the necessity of undoing op erations Instead op erations that might a ect C are
delayed and or p erformed on a sp ecial data structure separate from C Strong
incrementalityisclaimed ifwe replace average case complexitybyworst case com
plexity the algorithm satis es our criterion for p erfect incrementality
goesbeyond the problem of entailing equations to give an algorithm for
entailment when b oth equations and disequations are considered constraints
This algorithm has a rather di erent basis than those discussed ab ove it involves
memoization of pairs of terms entailments and disequations and the use of a
reduction of disequation entailmenttoseveral equation entailments
For and let G contain arbitrary equations e Add to the algorithm
LinE q n
which constructs the solved form a representation of eachsuch equation e in which
all eliminable variables are substituted out Note however that even though these
equations are stored with the other equations in the constraint store they are
considered as a distinct collection and they play no direct role in the question of
satis ability of the current store For example a constraint store containing x
z y z would cause the guard equation y z to b e represented as z
It is easy to show that a guard equation e is entailed i its representation reduces
to the trivial form and similarly the equation is refuted if its representation
is of the form n where n is a nonzero numb er In our example the guard
equation is entailed or refuted just in case z b ecomes ground In order to have
incrementalitywemust argue that the substitution op eration is often applied only
to very few of the guard constraints This is tantamount to the second assumption
made to argue the incrementality of Algorithm Hence we b elieve our algorithm
is incremental
Wemovenow to the domain but allow only equations in the guard con
Lin
straints G Here we can pro ceed as in the ab ove discussion for to obtain
LinE q n
an incremental algorithm but we will have the further requirementthatthecon
straint store contains all implicit equalities explicitly represented as equations It
is then still easy to show that the entailment of a guard equation e need b e checked
only when the representation of e is trivial The argument for incrementalitygiven
ab ove for essentially holds here provided that the cost of computing im
LinE q n
plicit equalities is su ciently low
There are twomainworks on the detection of implicit equalities in CLP systems
over In the existence of implicit equalities is detected by the app earance
Lin
of an equation of a sp ecial kind in the Simplex tableau at the end of the satis ability
checking pro cess Such an equation indicates some of the implicit equalities but
more pivoting which in turn can give rise to more sp ecial equations is generally
required to nd all of them An imp ortantcharacteristic of this algorithm is that
the extra cost incurred is prop ortional to the numb er of implicit equalities This
metho d is used in CLP R and Prolog I I I CHIP uses a metho d based on In
this metho d a solved form which is more restrictive than the usual Simplex solved
form is used An equation in this form do es not contribute to any implicit equality
and a whole tableau in this solved form implies that there are no implicit equalities
The basic idea is then to maintain the constraints in the solved form and when a
new constraint is encountered the search for implicit equalities can b e rst limited
These are equalities whichareentailed by the store b ecause of the presence of inequalities
For example the constraintstore x y x y entails the implicit equality x y
to variables in the new constraint One added feature of this solved form is that
directly accomo dates strict inequalities and disequations
Next consider still the domain butnow allow inequalities to b e in G Here
Lin
it is not clear how to represent a guard inequality say x in suchaway that its
entailment or refutation is detectable by some simple format in its representation
Using the Simplex tableau format as a solved form as discussed ab ove and using
the same intuition as in the discussion of guard equations we could substitute
out x in x incase x is basic However it is not clear to which format s we
should limit the resulting expression in order to avoid explicitly checking whether
x isentailed Thus an incremental algorithm for checking the entailmentof
inequalities is yet to b e found
For BOOL there seems to b e a similar problem in detecting the entailment
of Bo olean constraints However in the case of groundness entailment some of the
algorithms wehave previously discussed are p otentially incremental The Prolog I I I
algorithm in fact is designed with the detection of groundness as a criterion The
algorithm represents explicitly all variables that are grounded by the constraints
The Gro ebner basis algorithm will also contain in its basis an explicit representation
of grounded variables Finally for the uni cation algorithms the issue is clearly
the form of the uni er If the uni er is in fully simpli ed form then every ground
variable will b e asso ciated to a ground value
In summary for this subsection the problem of detecting entailment is not limited
just to the cost of determining if a particular constraintisentailed Incrementality
is crucial and this prop erty can b e de ned roughly as limiting the cost to dep end on
the numb er of guard constraints a ected by eachchange to the store In particular
dealing even brie y with the entire collection of guard constraints each time the
store changes is unacceptable
Below in Section an issue related to entailmentistaken up Here wehave
fo cussed on how to adapt the underlying satis ability algorithm to b e incremental
for determining entailment There we will consider the generic problem indep endent
of the constraint domain of managing delayed goals whichawake when certain
constraints b ecome entailed
Projection
The problem at hand is to obtain a useful representation of the pro jection of con
straints C w r t a given set of variables More formally the problem is given
target variablesx and constraints C x y involving variables fromx andy express
yC x y in the most usable form While we cannot de ne usability formally it
typically means b oth conciseness and readability An imp ortant area of use in the
output phase of a CLP system the desired output from running a goal is the pro
jection of the answer constraints with resp ect to the goal variables Here it is often
useful to have only the target variables output though dep ending on the domain
this is not always p ossible For example the output of x z y z w r t
to x and y should b e x y or some rearrangement of this but it should not
involveany other variable Another area of use is in meta programming where a de
scription of the current store maybewanted for further manipulation For example
And this can of course b e done p erhaps even e ciently but the crucial p oint is once again
that we cannot a ord to do this every time the store changes
pro jecting constraints onto a single variable x can showifx is b ounded and
Lin
if so this b ound can b e used in the program Pro jection also provides the logical
basis for eliminating variables from the accumulated set of constraints onceitis
known that they will not b e referred to again
There are few general principles that guide the design of pro jection algorithms
across the various constraint domains The primary reason is of course that these
algorithms havetobeintimately related to the domain at hand We therefore will
simply resort to brie y mentioning existing approaches for some of our sample
domains
The pro jection problem is particularly simple for the domain FT the result of
pro jection isx x where is the substitution obtained from the solved form
of C Now wehave describ ed ab ove that this solved form is simply the mgu of
C that is equations whose r h s do es not contain anyvariable on the l h s For
example x f y y f z would havethesolved form x f f z y f z
y f z are more e ciently stored internally However the equations x f y
as they are and this is done in actual implementations The solved form for x
therefore is obtained only when needed during uni cation for example by fully
dereferencing y in the term f y A direct representation of the pro jection of C on
avariable x as required in a printout for example can b e exp onential in the size
of C This happ ens for example if C is of the form x f x x x f x x
n
x f a a b ecause x would contain o ccurrences of the constant a A
n
solution would b e to present the several equations equivalenttox x suchas
the n equations in this example This however is a less explicit representation
of the pro jection for example it would not always b e obviousifavariable were
ground
Pro jection in the domain RT can b e done by simply presenting those equations
whose l h s is a target variable and recursively all equations whose l h s app ears
in anything already presented Such a straightforward presentation is in general
not the most compact For example the equation x f f x x f x x is b est
presented as x f x x In general the problem of nding the most compact
representation is roughly equivalent to the problem of minimizing states in a nite
state automaton
For the problem is only slightly more complicated Recall that equa
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tions are maintained in parametric form with eliminable and parametric variables
A relatively simple algorithm can b e obtained by using a form of Gaussian elimi
nation and is informally describ ed in Figure It assumes there is some ordering
on variables and ensures that lower priorityvariables are represented in terms of
higher priorityvariables This ordering is arbitrary except for the fact that the
target variables should b e of higher priority than other variables We remark that
a crucial p oint for e ciency is that the main lo op in Figure iterates n times and
this numb er the numb er of target variables is often far smaller than the total
number of variables in the system More details on this algorithm can b e found in
For there is a relatively simple pro jection algorithm Assume all inequalities
Lin
C denote the subset of constraints are written in a standard form Let C
x x
C in which x has only p ositive negative co e cients Let C denote those inequal
x
ities in C not containing x at all Wecannow describ e an algorithm due to Fourier
which eliminates a variable x from a given C If constraints c and c havea
p ositive and a negative co e cientofx we can de ne el im c c to b e a linear x
let x x b e the target variables
n
for i i n i i f
if x is a parameter continue
i
let e denote the equation x r h s x athand
i i
if r h s x contains a variable z of lower prioritythanx f
i i
cho ose the z of lowest priority
rewrite the equation e into the form z t
if z is a target variable mark the equation e as nal
substitute t for z in the other equations
g else mark the equation e as nal
g
return all nal equations
Figure Projection Algorithm for Linear Equations
combination of c and c which do es not contain x A Fourier step eliminates x
from a set of constraints C by computing F C fel im c c c C c C g
x x
x x
It is easy to show that xC F C Clearly rep eated applications of F eliminat
x
ing all non target variables result in an equivalent set of constraints in whichthe
only variables if any are target variables
The main problem with this algorithm is that the worst case size of F C
x
is O N where N is the numberofconstraints in C It is in fact precisely
j jC j jC j jC j jC j In principle the numb er of constraints jC
x x x x x
needed to describ e C using inequalities over variables var C fxg is far larger
than the numb er of inequalities in C In practice however the Fourier step gen
erates many redundant constraints See for discussion on such redundancy
Work by Cernikov prop osed tests on the generated constraints to detect and
eliminate some redundant constraints The output mo dule of the CLP R system
furthered these ideas as did Imb ert Imb ert also considered the
more general problem in which there are disequations All these redundancy elim
ination metho ds are correct in the following sense if fC g is the sequence of
i i
constraints generated during the elimination of variables x x from C then
i
C x x C forevery i
i i
The survey contains further p ersp ectives on the Fourier variable elimination
techique It also contains a discussion on how the essential technique of Fourier can
b e adapted to p erform pro jection in other domains such as linear integer constraints
and the b o olean domain
We nish here bymentioning the non Fourier algorithms of In some
circumstances esp ecially when the matrix representing the constraints is dense the
algorithm of can b e far more e cient It is however b elieved that typical CLP
programs pro duce sparse matrices The algorithm of has the advantageous
prop erty that it can pro duce an approximation of the pro jection if the size of the
Obtained for example bymultiplying c by m and c by m where m and m are
the co e cients of x in c and c resp ectively and then adding the resulting equations together
A constraint c C is redundantin C if C C fcg
pro jection is unmanageably large
Backtracking
The issue here is to restore the state of the constraintsolver to a previous state
or at least an equivalent state The most common technique following Prolog is
the trailing of constraints when they are mo di ed by the constraintsolver and the
restoration of these constraints up on backtracking In Prolog constraints are equa
tions b etween terms represented internally as bindings of variables Since variables
are implemented as p ointers to their b ound values backtracking can b e facili
tated by the simple mechanism of an untagged trail This identi es the set
of variables whichhave b een b ound since the last choice p oint Up on backtracking
these variables are simply reset to b ecome unb ound Thus in Prolog the only in
formation to b e trailed is whichvariables have just b ecome b ound and untrailing
simply unbinds these variables
For CLP in general it is necessary to record changes to constraints While in
Prolog a variable s expression simply b ecomes more and more instantiated during
forward execution in CLP an expression may b e completely changed from its
original form In for example a variable x mayhave an original linear
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form and subsequently another Assuming that a choice p ointisencountered just
b efore the change in x the original linear form needs to b e trailed in case of back
t in fact holds in all our sample domains with tracking This kind of requiremen
the exception of FT and RT Thus wehave our rst requirement on our trailing
mechanism the trail is a value trail that is eachvariable is trailed together with
its asso ciated expression Strictly sp eaking we need to trail constraints rather
than the expression a variable is asso ciated to However constraints are typically
represented internally as an asso ciation b etween a variable and an expression
Now the trailing of expressions is in general far more costly than the trailing of
the variables alone For this reason it is often useful to avoid trailing when there is
no choice p ointbetween the time a variable changes value from one expression to
another A standard technique facilitating this involves the use of time stamps a
variable is always time stamp ed with the time that it last changed value and every
choice p oint is also time stamp ed when created Now just b efore a variable s value
is to b e changed its time stamp n is compared with the time stamp m of the most
recentchoice p oint and if n m clearly no trailing is needed
Next consider the use of a cross reference table for solved forms such as those
discussed for the arithmetic domains which use parametric variables This is an
index structure which maps each parametric variable to a list of its o ccurrences in
the solved form Suc h a structure is particularly useful and can even b e crucial
for e ciency in the pro cess of substituting out a parametric variable step c in
algorithm However its use adds to the backtracking problem A straight
forward approach is to simply trail the entries in this table according to time
stamps However since these entries are in general quite large and since the cross
reference table is redundant from a semantic p oint of view a useful approachis
to reconstruct the table up on backtracking The details of such reconstruction are
Recall that this means that eliminable variables are not explicitly dereferenced in the r h s
of the equations in the solved form
In Prolog one can think of the stackpositionofavariable as the time stamp
straightforward but tedious and hence are omitted here see for the case of
the CLP R system A nal remark this reconstruction approach has the added
advantage of incurring cost only when backtracking actually takes place
In summary backtracking in CLP is substantially more complex than in Prolog
Some useful concepts to b e added to the Prolog machinery are as follows a value
trail and in practice a tagged trail as well b ecause most systems will accommo date
variables of di erenttyp es for example the functor and arithmetic variables in
CLP R time stamps to avoid rep eated trailing for a variable during the lifetime
of the same choice p oint and nally reconstruction of cross reference s rather than
trailing
Inference Engine
This section deals with extensions to the basic inference engine for logic program
ming needed b ecause of constraints What follows contains two main sections In
the rst we consider the problem of an incremental algorithm to manage a col
lection of delayed goals and constraints This problem discussed indep endently of
the particular constraint domain at hand reduces to the problem of determining
which of a given set of guard constraints cf Section are a ected as a result of
change to the constraint store The next section discusses extensions to the WAM
in b oth the design of instruction set as well as to the main elements of the runtime
structure Finally we give a brief discussion of work on parallel implementations
Delaying Wakeup of Goals and Constraints
The problem at hand is to determine when a delayed goal is to b e woken or when
a passive constraint b ecomes active The criteria for suchaneventisgiven by
ate the constraint when the a guard constraint that is awaken the goal or activ
guard constraintisentailed by the store In what follows we use shall use the
term delayed constrainttobesynonymous with passive constraint to emphasize the
similarities with delayed goals
The underlying implementation issue as far as the constraintsolver is concerned
is how to e ciently pro cess just those guard constraints that are a ected as a
result of a new input constraint Sp eci cally toachieve incrementality the cost of
pro cessing a change to the current collection of guard constraints should b e related
to the guard constraints a ected bythechange and not to all the guard constraints
The following two items seem necessary to achievethisend
First is a representation of what further constraints are needed so that a given
guard constraintisentailed For example consider the delayed CLP R constraint
z
pow x y z meaning x y which in general awaits the grounding of twoof
the three variables x y z In constrast the constraint pow y z only awaits the
grounding of y to a nonzero numb er or z to In general a delayed constraintis
awoken by not one but a conjunction of several input constraints When a subset of
For guarded clauses the problem is extended to determining whichclauseistobechosen
However signi cantchanges to the inference engine is needed to handle delayed goals and
guarded clauses But these issues are the same as those faced by extending logic programming
systems to implement delayed goals and or guarded clauses see for example
woken $3=# $1=#
$2=# $2=# $1=# $3=#
pow(#,$2,$3) pow($1,#,$3) pow($1,$2,#) $3=0 $3=1 $2=1 $1=0 #<>1 $2=# $1=# $3=# #<>0 #<>0,#<>1
pow($1,$2,$3)
Figure Wakeup system for pow
such input constraints has already b een encountered the runtime structure should
relate the delayed constraint to the disjunction of just the remaining kinds of
constraints which will awaken it
Second we require some index structure which allows immediate access to just
the guard constraints a ected as the result of a new input constraint The main
challenge is how to maintain such a structure in the presence of backtracking For
example if changes to the structure were trailed using some adaptation of Prolog
techniques then a cost prop ortional to the number of entries can b e incurred
even though no guard constraints are a ected
The following material is a condensation of
Wakeup Systems For the purp oses of this section we will describ e an
instance of a constraint in the form p C where p is the n ary constraint
n
symb ol at hand are distinguished variables used as templates for the
n
arguments of p and C is a constraint which determines the values of
n
wakeup degree represents a subset of the p constraints and a wakeup system A
consists of a set of wakeup degrees and further these degrees are organized into
an automaton where transitions b etween degrees are lab elled by constraints called
wakeup conditions Intuitively a transition o ccurs when a wakeup condition b e
comes entailed by the store There is a distinguished degree called woken which
represents active p constraints We pro ceed now with an example
Consider the CLP R constraint pow x y z and see gure A wakeup degree
maybespeci edby means of constraints containing for describing the
three arguments and some meta constants for describing unsp eci ed
values Thus for example sp eci es that the second argument is ground
These are templates for the guard constraints
Such a meta language can also b e used to sp ecify the wakeup conditions Thus for
example the wakeup condition attached to the b ottom
most degree in gure represents a transition of a constraint pow C
where C do es not ground into pow C c where C c do es ground
into a numb er di erent from and The wakeup condition which represents
a transition to the degree woken represents the fact that pow is an active
constraint equivalentto Similarly represents the transition to the
active constraint Note that there is no wakeup condition b ecause
pow which is equivalentto is not
active
In general there will b e certain requirements on the structure of suchanau
tomaton to ensure that it do es in fact de ne a mapping from p constraints into
wakeup degrees and that this mapping satis es certain prop erties like it de nes
a partition it maps only active constraints into woken it is consistent with the
wakeup conditions sp ecifying the transitions etc A starting p oint will b e a formal
ization of meta language used These formal asp ects are b eyond the scop e of this
survey
In summary wakeup systems are an intuitiveway to sp ecify the organization
of guard constraints The wakeup degrees represent the various di erentcasesofa
delayed constraint which should b e treated di erently for e ciency reasons Asso
ciated with each degree is a number of wakeup conditions which sp ecify when an
input constraintchanges the degree of a delayed constraint What is intended is that
eup conditions represent all the situations in which the constraint solver can the wak
e ciently up date its knowledge ab out what further constraints are needed to wake
the delayed constraint
Before embarking on the runtime structure to implement delayed constraints
suchaspow we amplify the ab ovementioned p oint ab out the similarities b etween
delayed constraints and guarded clauses Consider the guarded clause program
pow X Y Z Y X
pow X Y Z ground X X ground Y Y Z log X log Y
p
Z
pow X Y Z ground X X ground Z Y X
Z
pow X Y Z ground Y Y ground Z X Y
pow X Y Z X Y Z
pow X Y Z Z X
pow X Y Z Z X Y
This program could b e compiled into the wakeup system in Figure where the
three intermediate no des re ect sub expressions in the guards that mightbeentailed
without the entire guard b eing entailed More precisely several woken no des would
b e used one for each clause b o dy Thus wakeup systems express a central part of
the implementation of at guarded clauses Since a guarded atom can b e viewed as
a one clause guarded clause program for an anonymous predicate wakeup systems
are also applicable to implementing these constructs
Runtime Structure Here we present an implementational framework in
the context of a given wakeup system There are three ma jor op erations with de
layed goals or delayed constraints which corresp ond to the actions of delaying
awakening and backtracking
adding a goal or delayed constraint to the current collection
awakening a delayed goal or delayed constraint as the result of inputting a
new active constraint and
restoring the entire runtime structure to a previous state that is restor
ing the collection of delayed goals and delayed constraints to some earlier
collection and restoring all auxiliary structures accordingly
In what follows we concentrate on delayed constraints as mentioned ab ove the
constraint solver op erations to handle delayed goals and guarded clauses are essen
tially the same
The rst of our two ma jor structures is a stackcontaining the delayed constraints
Thus implementing op eration simply requires a push op eration Additionally
the stack contains constraints which are newer forms of constraints deep er in the
stack For example if the constraint pow x y z were in the stack and if the input
constraint y were encountered then the new constraint pow x z would
b e pushed together with a p ointer from the latter to the former In general the
collection of delayed constraints contained in the system is describ ed bythesub
ked constraints whichhavenoinb ound p ointers collection of stac
Now consider op eration In order to implement this e ciently it is necessary
to have some access structure mapping an entailed constraint to just those delayed
constraints a ected Since there are in general an in nite numb er of p ossible en
tailed constraints a nite classi cation of them is required A guard constraint or
simply guard for short is an instance of a wakeup condition obtained by renam
ing the distinguished argumentvariables into runtime variables It is used as a
i
template for describing the collection of entailed constraints its instances which
a ect the same sub collection of delayed constraints For example supp ose that the
only delayed constraintispow y z whose degree is pow with wakeup
conditions and Then only two guards need b e considered y
and z
Wenow sp ecify an index structure which maps a delayed constraintinto a doubly
linked list of o ccurence no des Eachnodecontains a p ointer to a stack elementcon
taining a delayed constraint Corresp onding to eac h o ccurrence no de is a reverse
p ointer from the stack element to the o ccurrence no de Call the list asso ciated with
adelayed constraint DW a DW list and call eachnodeinthelistaDW occurrence
node
Initially the access structure is empty The following sp eci es what is done for
the basic op erations
Delay Push the constraint C onto the stack and for eachwakeup condition asso
ciated with the degree of C create the corresp onding guard and DW list
All o ccurrence no des here are p ointed to C
Pro cess Entailment Say x isnowentailed Find all guards which are implied
by x Iftherearenone we are done Otherwise for each DW list
L corresp onding to each of these conditions and for each constraint C
p C p ointed to in L a delete all o ccurrence no des p ointing to C using
C x the reverse p ointers push the new delayed constraint C p
The total numb er of o ccurrence no des is generally larger than the numb er of delayed con
straints
...
x=# y=# z=# #<>0 #<>1 x=0 y=1 #<>0 z=0 z=1 #<>1 pow(y,x,y)
pow(x,y,z)
Figure The index structure
with a downward p ointer to C and nally c construct the new DW lists
corresp onding to C as de ned ab ove for the delay op eration
Backtrack Restoring the stack during backtracking is easy b ecause it only re
quires a series of p ops Restoring the list structure however is not so
straightforward b ecause no trailing saving of the changes was p erformed
In more detail the op eration of backtracking is the following a p op the
stack and let C denote the constraint just p opp ed b Delete all o ccurrence
no des p ointed to by C If there is no p ointer from C and so it was a con
straint that was newly delayed to another constraint deep er in the stack
then nothing more need b e done c If there is a p ointer from C to another
constraint C and so C is the reduced form of C then p erform the mo d
i cations to the access structure as though C were b eing pushed onto the
stack These mo di cations describ ed ab ove involve computing the guards
p ertinenttoC inserting o ccurrence no des and setting up reverse p ointers
Note that the index structure obtained in backtracking may not b e struc
t however turally the same as that of the previous state What is imp ortan
is that it depicts the same logical structure as that of the previous state
Figure illustrates the entire runtime structure after the two constraints pow x y z
and pow y x y are stored in this order Figure illustrates the structure after a
new input constraint makes x entailed
In summary a stack is used to store delayed constraints and their reduced forms
An access structure maps a nite numb er of guards to lists of delayed constraints
The constraint solver is assumed to identify those conditions whichareentailed
The cost of one primitive op eration on delayed constraints delaying a constraint
upgrading the degree of one delayed constraint including awakening the constraint
and undoing the delay upgrade of one constraint is b ounded by the xed size
...
pow(5,y,z) z=# y=# #<>0 #<>1 x=0 y=1 z=0 z=1 pow(y,5,y) #<>1
pow(y,x,y)
pow(x,y,z)
Figure The index structure after x isentailed
of the underlying wakeup system The total cost of an op eration delaying a new
constraint pro cessing an entailed constraint backtracking on delayed constraints
is prop ortional to the number of the delayed constraints a ected by the op eration
Abstract Machine
This section discusses some ma jor issues in the design of an abstract machine for
the execution of CLP programs The primary fo cus here will b e on the design of the
instruction set with emphasis on the interaction b etween their use and information
obtained from a p otential program analyzer Some elements of the runtime structure
will also b e mentioned
In general the essential features of the parts of an abstract machine dealing
with constraints will di er greatly over CLP languages using di erent constraint
domains This is exempli ed in the literature on CLP R CHIP and
CLP FD The following presentation though based on one work con
tains material that is relevant to abstract machines for many CLP languages
We b egin by arguing that an abstract machine is the right approach in the rst
place Abstract machines have b een used for implementing programming languages
for many reasons Portability is one only an implementation of the abstract ma
chine needs to b e made available on each platform Another is simply convenience
it is easier to write a native co de compiler if the task is rst reduced to compiling
for an abstract machine that is semantically closer to the source language The b est
abstract machines sit at just the rightpoin t on the sp ectrum b etween the concep
tual clarity of the high level source language and the details of the target machine
In doing so they can often b e used to express programs in exactly the right form for
tackling the e ciency issues of a source language For example the Warren Abstract
Machine revolutionized the execution of Prolog since translating programs
to the WAM exp osed many opp ortunities for optimization that were not apparent
at the source level The b ene t from designing an appropriate abstract machine for
agiven source language can b e so great that even executing the abstract instruc
tion co de byinterpretation can lead to surprisingly e cient implementations of a
language Many commercial Prolog systems compile to WAM like co de Certainly
more e ciency can b e obtained from native co de compilation but the step that
made Prolog usable was that of compiling to the WAM
While the WAM made Prolog practical global analysis shows the p otential of
making another ma jor leap For example and used fairly e cient ana
lyzers to generate high quality native co de Based on certain examples they showed
that the co de qualitywas comparable to that obtained from a C compiler In the
case of CLP the opp ortunities for obtaining valuable information from analysis are
even greater than in Prolog This is b ecause the constraint solving step is in general
far more involved than the uni cation step
Instructions Next we consider the design of an abstract machine instruc
tion set in addition to the basic instruction set of the WAM While the examples
presented will b e for CLP R the discussions are made for CLP systems in general
More details on this material can obtained from the theses
Our rst requirement is a basic instruction for invoking the constraintsolver
The format can b e of the form
xxx X X X solve
n
where xxx indicates the kind of constraint and the X denotes the arguments Typi
i
cally these arguments are as in the WAM stack lo cations or registers For example
in CLP R there are instructions of the form initpf n and addpf n X where n
isanumb er and X a solver variable The former initializes a parametric form to
contain just the number n The latter adds an entry of the form n pf X tothe
parametric form b eing stored in an accumulator where pf X is the parametric
form for X in the store Thus the accumulator in general stores an expression exp
of the form n n X n X Then the instruction solve eq tests for
k k
the consistency of exp in conjunction with the store If consistent the solver
adds the equation to the store otherwise backtracking o ccurs There are similar
instructions for inequalities
There are imp ortant sp ecial kinds of constraints that justify making sp ecialized
versions of this basic instruction While there are clearly many kinds of sp ecial cases
some sp eci c to certain constraint domains there are three cases which stand out
the constraint is to b e added to the store but no satis abilitycheck is needed
the constraint need not b e added but its satis ability in conjunction with
the store needs to b e checked
the constraint needs to b e added and satis ability needs to b e checked but
the constraint is never used later
To exemplify the sp ecial case consider adding the constraint X Y
to the store Supp ose that Y Z is already in the store and that X is a
new variable A direct compilation results in the following Note that the rightmost
column depicts the current state of the accumulator
initpf accumul ator
addpf X accumul ator X
addpf Y accumul ator X Z
solve eq sol v e X Z
A b etter compilation can b e obtained by using a sp ecialized instruction solve no fail eq
X which adds the equation X exp to the store where exp is the expression in
eq is that no satis ability the accumulator The main di erence here with solve
check is p erformed For the ab ove example wenowcanhave
initpf accumul ator
addpf Y accumul ator Z
solve no fail eq X add X Z
In summary for this sp ecial case for CLP systems in general we often encounter
constraints which can b e organized into a form such that its consistency with the
store is obvious This typically happ ens when a new variable app ears in an equa
tion for example and new variables are often created in CLP systems Thus the
instructions of the form solve no fail xxx are justi ed
Next consider the sp ecial case and the following example CLP R program
sum
sum N X
N
N N
X X N
sum N X
Of concern to us here are constraints that if added to the store can b e shown to
b ecome redundant as a result of future additions to the store This notion of future
redundancy was rst describ ed in Nowifwe execute the goal sum N X using
the second rule ab ove we obtain the subgoal
N N N X X N sum N X
Continuing the execution wenowhavetwochoices cho osing the rst rule we
obtain the new constraint N and cho osing the second rule we obtain the
constraint N among others In each case the original constraint N is
made redundant The main p oint of this example is that the constraint N
in the second rule should b e implemented simply as a test and not added to the
no add xxx constraintstore We hence de ne the new class of instructions solve
This example shows that future redundant constraints do o ccur in CLP systems
However one apparent di culty with this sp ecial case is the problem of detecting its
o ccurrence We will mention relevantwork on program analysis b elow Meanwhile
we remark that exp eriments using CLP R haveshown that this sp ecial case leads
to the most substantial e ciency gains compared to the other two kinds of sp ecial
cases discussed in this section
Finally consider sp ecial case Of concern here are constraints which are neither
entailed by the store as in case nor are eventually made redundant as in case
but which are required to b e added to the store and checked for consistency What
makes these constraints sp ecial is that after they have b een added to the store and
the store is recomputed into its new internal form their variables app ear in those
parts of the store that are never again referred to Consider the sum program once
again The following sequence of constraints arise from executing the goal sum
X
X X
X X
X X
Up on encountering the second equation X X and simplifying into note
that the variable X will never b e referred to in future Hence equation can b e
deleted Similarly up on encountering the third equation X X and simpli
fying into the variable X will never b e referred to in future and so can b e
deleted In short only one equation involving X need b e stored at any p ointinthe
and delete computation We hence add the class of instructions of the form add
X which informs the solver that after considering the constraint asso ciated to X it
may delete all structures asso ciated to X In CLP R the corresp onding instruc
and delete n X theobvious variant of the previously describ ed tion is addpf
instruction addpf n X Compiling this sum example gives
init pf
addpf X
solve no fail eq X
init pf
addpf and delete X
solve no fail eq X
init pf
addpf and delete X
solve no fail eq X
Note that a di erent set of instructions is required for the rst equation from that
required for the remaining equations Hence the rst iteration needs to b e unrolled
to pro duce the most e cient co de The main challenge for this sp ecial case is as in
sp ecial case the detection of the sp ecial constraints Wenow address this issue
Techniques for CLP Program Analysis The kinds of program analysis
required to utilize the sp ecialized instructions include those techniques develop ed
for Prolog most prominently detecting sp ecial cases of uni cation and determin
istic predicates Algorithms for such analysis have b ecome familiar see
for example See for example for a description of how to extend the general
techniques of abstract interpretation applicable in LP to CLP Our considerations
ab ove however require rather sp eci c kinds of analyses
Detecting redundantvariables and future redundant constraints can in fact b e
done without data ow analysis One simple metho d involves unfolding the predi
cate de nition and typically once is enough and then in the case of detecting
redundantvariables simply insp ecting where variables o ccur last in the unfolded
de nitions For detecting a future redundant constraint the essential step is de
termining whether the constraints in an unfolded predicate de nition imply the
constraint b eing analyzed
An early work describing these kinds of optimizations is and some further
discussion can also b e found in The latter rst describ ed the abstract ma
chine CLAM for CLP R and the former rst de ned and examined the problem
of our sp ecial case that of detecting and exploiting the existence of future redun
dant constraints in CLP R More recently rep orted new algorithms for the
problem of sp ecial case that of detecting redundantvariables in CLP R The
work describ es in a more general setting a collection of techniques entitled
re nement removal and reordering for optimization in CLP systems See also
for an overview of the status of CLP R optimization and for detailed
empirical results
Despite the p otential of optimization as rep orted in these works the lack of full
implementations leaves op en the practicality of using these and other sophisticated
optimization techniques for CLP systems in general
Runtime Structure A CLP abstract machine requires the same basic
runtime supp ort as the WAM Some data structures needed are a routine extension
of those for the WAM the usual register stack heap and trail organization
The main new structures p ertain to the solver Variables involved in constraints
typically haveasolver identi er which is used to refer to that variable s lo cation
in the solver data structures
The mo di cations to the basic WAM architecture typically would b e
Solver identi ers
It is often necessary to haveaway to index from a variable to the constraints
it is involved in Since the WAM structure provides stack lo cations for the
dynamically created variables it remains just to haveatagandvalue struc
ture to resp ectively a identify the variable as a solver variable and b
access the constraint s asso ciated with this variable Note that the basic
uni cation algorithm assuming functors are used in the constraint system
needs to b e augmented to deal with this new typ e
Taggedtrail
As mentioned in Section the trail in the WAM merely consists of a
stack of addresses to b e reset on backtracking In CLP systems in general
the trail is also used to store changes to constraints Hence a tagged value
trail is required The tags sp ecify what op eration is to b e reversed and the
value comp onent if present contains any old data to b e restored
Time stamped data structures
Time stamps have b een brie y discussed in Section The basic idea
here is that the data structure representing a constraintmay go through
several changes without there b eing a new choice p oint encountered during
this activity Clearly only one state of the structure need b e trailed for each
choice p oint
Constraint accumulator
A constraintistypically built up using a basic instruction rep eatedly for
example the addpf instruction in CLP R During this pro cess the par
tially constructed constraint is represented in an accumulator One of the
solve instructions then passes the constraint to the solver We can think of
this linear form accumulator as a generalization of the accumulator in classi
cal computer architectures accumulating a partially constructed constraint
instead of a number
Paral lel Implementations
We brie y outline the main works involving CLP and parallelism The opp ortunities
for parallelism in CLP languages are those that arise and have already b een ad
dressed in the logic programming context such as or parallelism and parallelism
stream parallelism and those that arise b ecause of the presence of a p otentially
computationally costly constraintsolver
The rst work in this area was an exp erimental implementation of an or
parallel for a CLP language with domain FD That approach has b een pursued
with the development of the ElipSys system which is the most develop ed of
the parallel implementations of CLP languages
Atay presents the or parallelization of LP a language that computes
with linear inequalities over reals and integers but in which rules do not have
lo cal variables Another work deals with the or parallel implementation of a CLP
language over FD on massively parallel SIMD computers However the basis
for the parallelism is not the nondeterministic choice of rules as in conventional
LP or parallelism but the nondeterministic choice of values for a variable
Work on and parallelism in logic programming dep ends heavily on notions of
indep endence of atoms in a goal addresses this notion in a CLP context
and identify notions of indep endence for constraint solvers whichmust hold if the
advantages of and parallelism in LP are to b e fully realized in CLP languages
However there has not to our knowledge b een an y attempt to pro duce an and
parallel implementation of a CLP language
Twoworks address b oth stream parallelism and parallelism in constraint solving
GDCC is a committed choice language that can b e crudely characterized as a
committed choice version of CAL It uses constraints over domains of nite trees
Bo oleans real numb ers and integers mainly discusses the parallelization of the
Gro ebner basis algorithms which are the core of the solversfortherealnumb er and
Bo olean constraint domains and a parallel branch and b ound metho d that is used
in the integer solver Leung addresses the incorp oration of constraint solving
in b oth a committed choice language and a language based on the Andorra mo del
of computation He presents distributed solvers for nite domains the Bo olean
domain and linear inequalities over the reals The nite domain solver is based on
the solver for the reals parallelizes the Simplex algorithm and the Bo olean
solver parallelizes the uni cation algorithm of
Finally rep orts the design and initial implementationofCLP R with
an execution mo del in which the inference engine and constraint solver compute
concurrently and asynchronously One of the issues addressed is backtracking which
is di cult when the engine and solver are so lo osely coupled
Wesaythatavariable in a rule is local if it app ears in the b o dy of the rule but not in the head
Part I I I
Programming and Applications
In this nal part we discuss the practical use of CLP languages The format here
is essentially a selected list of successful applications across a variety of problem
domains Each application is given an overview with emphasis on the particular
programming paradigm and CLP features used
It seems useful to classify CLP applications broadly into two classes In one class
the essential CLP technique is to use constraints and rules to obtain a transparent
representation of the relationships underlying the problem Here the constraints
also provideapowerful query language The other class caters for the many prob
lems which can b e solved byenumeration algorithms the combinatorial search
problems Here the LP asp ect of CLP is useful for providing the enumeration facil
ity while constraints servetokeep the search space manageable
Mo delling of Complex Problems
We consider here the use of CLP as a sp eci cation language constraints allow
the declarativeinterpretation of basic relationships and rules combine these for
complex relationships
Analysis and Synthesis of AnalogCircuits
This presentation is adapted from an early application of CLP R Brie y the
general metho dology for representing prop erties of circuits are that constraints at a
base level describ e the relationship b etween variables corresp onding to a subsystem
such as Ohm s law and constraints at a higher level describ e the interaction b etween
these subsystems such as Kircho s law
Consider the following program fragment de ning the pro cedure circuit N V
the p otential di erence and I which sp eci es that across an electrical network N
current are V and I resp ectively The network is sp eci ed in an obvious waybya
term containing the functors resistor series and parallel In this program
the rst rule states the required voltage current relationship for a resistor and the
remaining rules combine such relationships in a network of resistors
circuit resistor R V I V I R
circuit series N N V I
I I
I I
V V V
circuit N V I
circuit N V I
circuit parallel N N V I
V V V V
I I I
circuit N V I
circuit N V I
For example the query
circuit series series resistor R resistor R resistor R V
asks for the voltage value if a currentvalue of is owing through a network
containing just three identical resistors in series The answer is R V
Additional rules can b e added for other devices For example the piece wise linear
mo del of a dio de describ ed by the voltage current relationship
V if V
V if V
I
V if V
is captured by the rules
circuit diode V V V
circuit diode V V V V
circuit diode V V V
This basic idea can b e extended to mo del AC networks For example supp ose we
wish to reason ab out an RLC network in steady state First we disp ense with
bers by representing X iY as a CLP R term c X Y and use complex num
c equal c Re Im c Re Im
c add c Re Im c Re Im c Re Re Im Im
c mult c Re Im c Re Im c Re Im
Re Re Re Im Im
Im Re Im Re Im
to implement the basic complex arithmetic op erations of equality addition and
multiplication
Now consider the following pro cedure circuit N V I W whichislikeits
namesakeabove except that the voltage and currentvalues are now complex num
b ers and the new parameter W a real numb er is the angular frequency It is note
worthy that this signi cant extension of the previous program fragmentfor circuit
has b een obtained so easily
circuit resistor R V I W cmult V I c R
circuit inductor L V I W cmult V I c W L
circuit capacitor C V I W c mult V I c W C
circuit series N N V I W
c equal I I c equal I I
c add V V V
circuit N V I W
circuit N V I W
circuit parallel N N V I W
c equal V V c equal V V
add I I I c
V V V V
I I I
circuit N V I W
circuit N V I W
We close this example application bymentioning that the work in notonly
contains further explanation of the ab ovetechnique but also addresses other prob
lems suchasthesynthesis of networks and digital signal ow Not only do es the
CLP approach provide a concise framework for mo delling circuits previously done
in a more ad ho c manner but it also provides additionality functionality b ecause
relationships as opp osed to values are reasoned ab out Evidence that this approach
can b e practical was given for example the mo delling can b e executed at the rate
of ab out a hundred circuit comp onents p er second on a RS workstation
Options Trading Analysis
Options are contracts whose value is contingent up on the value of some underlying
asset The most common typ e of option are those on company shares A cal l option
gives the holder the righttobuya xednumb er of shares at a xed exercise price
until a certain maturity expiration date Conversely aput option gives the holder
the right to sell at a xed price The option itself may b e b ought or sold For
example consider a call option costing which gives the righttopurchase
shares at p er share within some p erio d of time This call option can b e sold at
the current market price or exercised at a cost of Nowifthepriceofthe
share is then the option may b e exercised to obtain a pro t of p er share
taking the price of the option into account the net gain is After the sp eci ed
p erio d the call option if not exercised b ecomes worthless Figure shows payo
diagrams which are a simple mo del of the relationship b etween the value of a call
option and the share price Sell options have similar diagrams Note that c denotes
the cost of the option and x the exercise price
Options can b e combined in arbitrary ways to form arti cial nancial instru
ments Thisallows one to tailor risk and return in exible ways For example the
butter y strategy in gure consists of buying two calls one at a lower strike price
x and one at a higher price z and selling two calls at the middle strike price y This
makes a pro t if the share stays around the middle strike price and limits the loss
if the movement is large
Payoff Payoff Payoff
c
x Stock Stock Stock Price x Price Price x y z
c
Buy a Call Sell a Call Butterfly
Figure Payo Diagrams
The following presentation is due to Yap based in his work using CLP R
This material app eared in and the subsequently implemented OTAS system
is describ ed in There are several main reasons why CLP and CLP R in
particular are suitable for reasoning ab out option trading there are complex trad
ing strategies used which are usually formulated as rules there is a combinatorial
asp ect to the problem as there are manyways of combining options a combination
of symb olic and numeric computation is involved there are well develop ed math
ematical valuation mo dels and constraints on the relationships involved in option
pricing and nally exible what if typ e analysis is required
A simple mathematical mo del of valuing options and other nancial instruments
suchasstocks and b onds is with linear piecewise functions Let the heavyside
function h and the ramp function r b e de ned as follows
if x y if x y
h x y and r x y
otherwise y x otherwise
The payo function for call and put options can now b e describ ed by the following
matrix pro duct which creates a linear piecewise function
h b s
h b s
h r r payo h
r b s
r b s
where s is the share price b is either the strike price or and h and r are
i i i
multipliers of the heavyside and ramp functions In the following program the
variables S X R resp ectively denote the sto ck price the exercise price and the
interest rate
h X Y Z Y X Z
h X Y Z Y X Z
r X Y Z Y X Z
r X Y Z Y X Z Y X
value Type Buy or Sell S C P R X B Payoff
sign Buy or Sell Sign
data Type S C P R X B B B H H R R
h B S T h B S T r B S T r B S T
Payoff Sign H T H T R T R T
The parameters for the piecewise functions can b e expressed symb olically in the
following tables implemented simply as CLP facts
sign buy
sign sell
data stock S C P R X B S R
data call S C P R X B X C R
data put S C P R X B X P R X
data bond S C P R X B B R
This program forms the basis for evaluating option combinations The following
direct query evaluates the sale of a call option that expires in the money
Call X R S
R X Payoff value call sell S Call
giving the answer Payoff More general queries make use of the ability
to reason with inequalities We can ask for what share price do es the value exceed
Payoff C X R S
value call sell S C R X Payoff
The answer constraints returned illustrates the piecewise nature of the mo del
Payoff S
Payoff S S S
More complex combinations can b e constructed by comp osing them out of the base
nancial instruments and linking them together with constraints For example the
That is when the strike price is less than the share price
We will use to separate di erent sets of answer constraints in the output
following is a combination of two calls and two puts
R Payoff Payoff Payoff Payoff Payoff
P K value put sell S P R K Payoff
P K value put buy S P R K Payoff
C K value call buy S C R K Payoff
C K value call sell S C R K Payoff
The answer obtained illustrates howcombinations of options can b e tailored to
pro duce a custom linear piecewise payo function
Payoff S
Payoff S S S
Payoff S S
Payoff S S S
Payoff S
The ab ove is just a brief overview of the core ideas b ehind the work in
Amongst the imp ortant asp ects that are omitted are consideration of option pricing
mo dels and details of implementing the decision supp ort system OTAS As in
the circuit mo delling application describ ed ab ove the advantages of using CLP R
here are that the program is concise and that the query language is expressive
Temporal Reasoning
It is natural and common to mo del time as an arithmetic domain and indeed we
do this in everyday life Dep ending up on the application a discrete representation
such as the integers or a continuous representation such as the reals maybe
appropriate and varying amounts of the arithmetic signature are needed for ex
ample we might use only the ordering or use only a successor function In this
brief discussion we assume that time is linearly ordered although this is not a
universally accepted choice
Temp oral logic is often used as a language for expressing time related con
cepts Temp oral logic adds to standard rst order logic such constructs as next
meaning roughly in the next time instant always meaning in every future
time instant and sometime meaning in some future time instant The lan
guage Templog was designed based on a Horn like subset of temp oral logic in
which the meaning of function symb ols do es not vary with time but the meaning
of predicate symb ols do es It was shown in that the op erational b ehavior of
Templog could b e mimicked by a CLP language via the following natural transla
tion every predicate receives another argument representing time Then at time
t next is represented by t t and the future for always and sometime is
represented by t t In later work Brzoska has presented a more p owerful
temp oral logic language which also can b e viewed as and implemented through a
CLP language
Often we wish to manipulate the time parameter more directly than is p ossible in
conventional temp oral logic For example wemay wish to express durations as well
as times We can do this if we include in the signature of our domain mo delling
We assume that time is mo delled by the integers
time This is used in applications to scheduling among others as discussed in
section
The use of simple constraint domains to mo del time has b een explored exten
sively in the context of temp oral databases In this situation an item of data might
incorp orate the time interval for whichitisvalid Simple domains have b een con
sidered b ecause of over riding requirements for quick and terminating execution of
queries as discussed in section Furthermore often the restriction is made that
only one or two arguments in a tuple are time valued with the other arguments
taking constantvalues surveys work in this area using an integer mo del of
time
Combinatorial Search Problems
CLP o ers an easy realization of enumeration algorithms for the solving of com
binatorial problems Given decision variables x x one uses a CLP program
n
schema of the form
solve X Xn
constraints X Xn
enumerate X Xn
to implement a constrain and generate enumeration strategy also called implicit
enumeration as opp osed to naiveenumerate and test strategy to curtail the search
space We refer to the basic text chapter for further intro ductory material
to this CLP approach
The ab oveschema is used to represent the set of all solutions to the constraints
Often one desires an optimal solution according to some criteria say the solu
tion a a to x x that minimizes some giv en function cost x x
n n n
The simplest strategy to obtain this solution is simply to obtain and check each
and every solution of solve An easy improvement is obtained by augmenting the
search with a branch and bound strategy Brie y the cost of the b est solution en
countered so far is stored and the continuing search is constrained to nd only
new solutions of b etter cost More concretely CLP systems typically provide predi
cates suchasminimize solve X Xn Cost BestCost and similarly
maximize where solve X Xn Cost serves to obtain one solution
as explained ab ove with cost Cost andBestCost is a numb er representing the
cost of the b est solution found so far Initially thisnumber can be any su ciently
large numb er It is assumed here that the pro cedure solve X Xn Cost
maintains a lower b ound for a variable Cost which is computed as the values of the
decision variables are determined The minimize pro cedure then essentially b ehaves
as a rep eated invo cation of the goal Cost BestCost solve X Xn
Cost In general the choice of a suitable cost function can b e di cult Finally we
refer the reader to the text section for more a detailed explanation of
howbranch and b ound is used in CLP systems
ypically nite since the enumer The constraint domain at hand is discrete and t
ation must cover all candidate values for the sequence x x and therefore
n
constraint solving is almost always NP hard This in turn restricts implementa
tions to the use of partial solvers that is not all constraints will b e considered
active Recall that partial solvers are however required to b e conservativeinthe
sense that whenever unsatis ability is rep orted the tested constraints are indeed
unsatis able
In general the primary e ciency issues are
How complete is the constraintsolver In general there is tradeo b etween
the larger cost of a more complete solver and the smaller search space that
suchasolver can give rise to
What constraints to use to mo del the problem A sp ecial case of this is
sue concerns the use of redundant constraints that is constraints that do
not change the meaning of the constraint store In general redundantcon
straints will slowdown a CLP system with a complete solver With partial
solvers however redundant constraints may b e useful to the solver in case
the equivalent information in the constraint store is not active
In which order do wecho ose the decision variables for enumeration And
should such order b e dynamically determined
In which order do weenumerate the values for a given decision variable
And should such order b e dynamically determined
In this section we will outline a numb er of CLP applications in sp eci c combina
torial problem areas In each subsection b elow unless otherwise sp eci ed weshall
assume that the underlying constraint system is based on the integers
Cutting Stock
The following describ es a two dimensional cutting sto ck problem p ertaining to fur
niture manufacturing an early application of CHIP We are givenasawing
ood into a numb er of di erent sized shelves The machine which cuts a b oard of w
machine is able to cut in several con gurations eachofwhich determines the num
ber of each kind of shelf and some amountofwood wasted Let there b e N di erent
kinds of shelves and M di erent con gurations Let S i M j N
i j
denote the numb er of shelves j cut in con guration i LetW i M denote
i
the wastage in con guration i Let R i N denote the number of shelves i
i
required The problem now can b e stated as nding the con gurations suchthat
the required numb er of shelves are obtained and the wastage minimized
In there were kinds of shelves con gurations and the number of
b oards to b e cut was xed at Two solutions were then presented whichwenow
paraphrase
Let X i denote the numb er of b oards cut according to con guration i
i
Thus X X The requirements on the numb er of shelves are expressed
via the constraints X S X S for j N The ob jective
j j
function to b e minimized is X W X W The straightforward program
representation of all this is given b elow The enumerate pro cedure has the range
f g Note that solve is run rep eatedly in the search for the solution of lowest
Cost
solve X X Cost
X X
S R X S X
X S X S R
X S X S R
Cost X W X W
enumerate X X
The second solution uses the sp ecial CHIP constraint element describ ed ab ovein
th
section Recall that element X List E expresses that the X elementof
List is E In this second approach to the problem the variables X i
i
denote the con gurations chosen Thus X Let T i j
i i j
denote the number of shelves j in con guration i Let Cost i denote
i
the wastage in con guration X Thus the required shelves are obtained by the
i
constraints T T R where j and the total cost is simply
j j j
Cost C ost
In program b elow the constraints X X X X serve to eliminate con
sideration of symmetrical solutions The following group of element constraints
serve to compute the T variables in terms of the given S values and the com
i j i j
puted X values The next group of element constraints computes the Cost
i i
variables in terms of the given W variables The enumerate pro cedure has the
i
range f g Once again solve is run rep eatedly here in the search for the
lowest Cost
solve X X Cost
X X X X X X
element X S S T j
j j j
element X S S T
j j j
element X S S T
j j j
element X S S T
j j j
element X W W Cost
element X W W Cost
element X W W Cost
element X W W Cost
T T T T R j
j j j j i
Cost Cost Cost Cost Cost
enumerate X X X X
The second program has advantages over the rst Apart from a smaller searchspace
in comparison with it was able to avoid encountering approximately
symmetrical solutions The timings given in showed that the second program
ran much faster This comparison exempli es the ab ovementioned fact that the way
a problem is mo delled can greatly a ect e ciency
DNA Sequencing
We consider a simpli ed version of the problem of restriction site mapping RSM
Brie y a DNA sequence is a nite string over the letters fA C G T g and a restric
tion enzyme partitions a DNA sequence into certain fragments The problem is then
to reconstruct the original DNA sequence from the fragments and other information
obtained through exp eriments In what follows we consider an abstraction of this
problem which deals only with the lengths of fragments instead of the fragments
themselves
Consider the use of two enzymes Let the rst enzyme partition the DNA se
quence into A A and the second into B B Now a simultaneous use of
N M
the two enzymes also pro duces a partition D D corresp onding to combining
K
the previous two partitions That is
i j A A D D and i j B B D D and conversely
i j i j
j i D D A A D D B B
j i j i
Let a denote the length of A similarly for b and d Let a denote the subsequence
i i i i i
a a i N Similarly de ne b and d The problem at hand nowcan
i i i
b e stated as given the multisets a fa a g b fb b g and d
N M
b b and fd d g construct the sequences a a a b
M K N N M
d d d
K K
Our basic algorithm generates d d in order and extends the partitions for
a and b using the following invariant prop ertywhich can b e obtained from the
problem de nition ab ove Either
d is aligned with a that is d d a a or
k i k i
d is aligned with b but not with a that is d d b b
k j i k j
In the program b elow the main pro cedure solve takes as input three lists repre
b and d in the rst three arguments and outputs in the remaining three sentinga
arguments Enumeration is done bycho osing at each recursive step of the rsm
pro cedure one of two cases mentioned ab ove Hence the two rules for rsm Note
that the three middle arguments of rsm maintains the length of the subsequences
found so far and in all calls either lenA lenD lenB or lenB lenD lenA
holds the pro cedure choose initial cho oses the rst fragment and makes the
rst call to rsm with this invariant holding Finally the pro cedure choose deletes
some element from the given list and returns the resultant list Note that one more
rule for rsm is needed in case the A and B fragments do align anywhere except at
the extreme ends wehave omitted this p ossibility for simplicity
solve A B D AFrag MapA BFrag MapB DFrag MapD
choose initial A B D AFrag BFrag DFrag A B D
rsm A B D AFrag BFrag DFrag MapA MapB MapD
rsm A B D LenA LenB LenD MapA MapB MapD
empty A empty B empty D
MapA MapB MapD
rsm A B D LenA LenB LenD Ai MapA MapB Dk MapD
LenA LenD LenA LenB
Dk LenB LenA Ai Dk
choose Dk D D
choose Ai A A
rsm A B D LenA Ai LenB LenD Dk MapA MapB MapD
rsm A B D LenA LenB LenD MapA Bj MapB Dk MapD
LenB LenD LenB LenA
For simplicitywe assume that wenever have all three partitions aligned except at the b eginning and at the end
Dk LenA LenB Bj Dk
choose Dk D D
choose Bj B B
rsm A B D LenA LenB Bj LenD Dk MapA MapB MapD
This application of CLP is due to Yap and it is imp ortant to note that the
ab ove program is a considerable simpli cation of Yap s program A ma jor omission
is the consideration of errors in the fragment lengths b ecause these lengths are
obtained from exp erimentation A ma jor p ointinYap s approachisthatitgives
a robust and uniform treatment of the exp erimental errors inherent in the data
as compared with many of the approaches in the literature Furthermore
shows how the simple two enzyme problem can b e extended to a numb er of other
problem variations Because a map solution is just a set of answer constraints
returned by the algorithm it is easy to combine this with other maps compare
maps verify maps etc This kind of exibility is imp ortant as the computational
problem of just computing a consistent map is intractable and hence when dealing
with any substantial amount of data would havetotakeinto accountdatafrom
manyvarieties of mapping exp eriments as well as other information sp eci c to the
molecule in question
Scheduling
In this class of problems we are given a numb er of tasks and for each task a
task duration Each task also requires other resources to b e p erformed and there
are constraints on precedences of task p erformance and on resource usage The
problem is to schedule the tasks so that the resources are most e ciently used for
example p erform the tasks so that all are done as so on as p ossible
Consider now a basic job shop scheduling problem in whichisgiven a number m
of machines j sequences of tasks the task durations and the machine assigned to
tsarethatthetasksineach sequence called each task The precedence constrain
a job are p erformed in the sequence order The resource constraints are that each
machine p erforms at most one task at anygiven time
In the program b elow precedences sets up the precedence constraints for one
job and is called with two equally long lists The rst contains the task variables
whose values are the start times The second list contains the durations of the tasks
Thus precedences is called once for each job The pro cedure resources is called
rep eatedly once for each pair of tasks T and T whichmust p erformed without
overlapping their durations are given by D and D
precedences T T Tail D D Tail
T D T
precedences Tail Tail
precedences
resources T D T D T D T
resources T D T D T D T
A simple wa y to pro ceed is to x an ordering of the tasks p erformed on eachma
chine This corresp onds to cho osing one of the two resources rules for eachpair
of tasks assigned to the same machine This forms the basis of the enumerate pro
cedure b elow Once an ordering of tasks is xed it is a simple matter to determine
the b est start times for each task
This can b e done in the manner indicated in the solve pro cedure b elow An
imp ortant e ciency p ointisthatbycho osing a precedence b etween two tasks the
new constraints created by the use of resources in conjunction with the precedence
constraints can reduce the numb er of p ossible choices for the remaining pairs
We assume that the pro cedure define cost de nes Cost in suchaway that in
conjunction with other constraints it provides a conservativelowerboundofthe
real cost of the schedule determined so far Its precise de nition omitted here can
b e obtained in a similar way as in the second program of the cutting sto ck example
ab ove
solve T T T Cost
n
precedences one per job
precedences
cost T T T Cost define
n
enumerate T T T
n
generate start times T T T
n
enumerate T T T
n
resources one per pair of tasks assigned to same machine
resources
Finally this solve pro cedure can b e rep eatedly run within a branch and b ound
framework with a sp ecial minimize predicate mechanism as explained ab ove to
obtain the b est solution over all p ossible orderings
In this presentation of the program wehavechosen to simply list all calls to
precedences in the pro cedure solve to fo cus on the imp ortant pro cedures in
the program A real program would use an auxilliary predicate to iterate over the
jobs and generate the calls to precedences Similarly enumerate would iterate
to generate calls to resources Thus the program would b e indep endentofthe
numb er of jobs or the pattern in which tasks are assigned to machines Similar
comments apply to other programs in this section
There are variations and sp ecializations of CLP approaches to this problem Sec
tion of and section of on which this presentation is based further
discuss the problem and how particular features of CHIP can b e useful Another
CHIP approach but this time to a sp eci c and practical scheduling problem is
rep orted in In the fo cus is on a new feature of CHIP and how it can b e
used to obtain an optimal solution to a particular jobs and machine problem
which remained op en until recently
Real scheduling problems can involve more kinds of constraints than just those
men tioned ab ove For example one could require that there is at most a certain time
elapsed b etween the completion of one task and the commencement of another See
for a more complete discussion of the CLP approach to the general scheduling problem
Chemical Hypothetical Reasoning
This Prolog I I I application describ ed in some detail in uses b oth arithmetic
and b o olean constraints The problem at hand is that of elucidating chemical
reaction pathways and we quote given an instantiation of the two reagent
reaction schema A B T P P determine the pathway that is the set
k
of constituent reaction steps aswell as other molecules or species formed during
the reaction
The reaction step considered in contains at most two reactant molecules
and at most two pro duct molecules and so can b e describ ed in the form R
R P P where R R P P are p ossibly empty molecular formulas The
problem then is to determine given an overall reaction a collection of basic steps or
pathway which explain the overall reaction For example given C H N CH
C H N H O the following is a pathwaywhich explains the reaction
C H N H O C H N
C H N C H N C H N
C H N C H N C H N C H N
Here C H N and C H N are the previously unidenti ed sp ecies
ts to express requirements for a chemical reaction The program imp oses constrain
and to exclude uninteresting reactions In addition to the constraints on the number
of molecules there are two other constraints on reaction steps for eachchemical
element the numb er of reactant atoms equals the numb er of pro duct atoms i e
the step is chemical ly balanced and no molecular formula app ears in b oth sides of
a step
There are also constraints on the pathway Let the reaction schema under con
sideration b e A B T P P Then
k
All pathway sp ecies must b e formable from the tworeagents A and B
Neither A nor B alone is su cient to form the target pro duct T Here b o olean
variables are used to express the dep endency relation can b e formed from
For each pathway step R R P P we state the b o olean constraint
a a a a where a a a a are b o olean variables asso ciated with
R R P P resp ectively The constraint expresses that b oth P and P
can b e formed if b oth R and R can b e formed Let B denote the b o olean
formulas thus constructed over all the steps in a pathway Then expressing
that sp ecies R do es not by itself pro duce sp ecies P is tantamounttothe
satis ability of the b o olean constraint B a a where a and a
R P R P
are the b o olean variables asso ciated with R and P resp ectively Since we
havetwo original reagents we will need two sets of b o olean variables and
oid anyinterference b etween the two sets of dep endency constraints to av
two conditions
There is a notion of pathway consistency which is de ned to b e the satis
ability of a certain arithmetic formula constructed from the o ccurrences of
sp ecies in the pathway Essentially this formula is a conjunction of formulas
n n n n for each pathwaystep R R P P where
n n are the arithmetic variables of R R P P resp ectively
Finally in the ultimate output of the program no two pathways are identical
nor b ecome identical under transformations suchaspermuting the reactants
or pro ducts within a step or switching the reactants and pro ducts in a step
The program representation of a molecular formula is as a list of numb ers each
of which sp eci es the numb er of atoms of a certain chemical element Weshall
assume that there are only four chemical elements of interest in our presentation
and hence a molecular formula is a tuple A sp ecies is also represented bya
tuple n a b f where n is an arithmetic variable to b e used in the formulation
of the arithmetic formula mentioned ab ove a and b are b o olean variables to b e
used in expressing the formation dep endencies and f is the sp ecies formula A
step R R P P is represented by a tuple r r p p containing the
identi ers of the representations of R R P and P
The listing b elow is simpli ed and translated version of the Prolog I I I program
in In the main pro cedure solve the rst argument is a list of xed size say n
in whicheach element is a sp ecies template The rst three templates are given and
these represent the two initial reagents R R and nal target T Similarly Steps
is a list of xed size say m inwhicheach element is a step template Thus n and
of obtains m are parameters to the program The unde ned pro cedure formula
the sp ecies formula from a sp ecies that is it pro jects onto the last elementofthe
given tuple Similarly arith var of bool var a of and bool var b of pro ject
onto the rst second and third arguments resp ectively
duplicates asserts constraints which prevent duplicate sp ecies The pro cedure no
and steps and it also prevents symmetrical solutions we omit the details Calls to
the pro cedure formation dependencies generate the formation dep endencies The
pro cedure both reagents needed imp oses two constraints one for each reagent
that in conjunction with the formation dep endencies assert that R resp ectively
R alone cannot pro duce T Finally enumerate species is self explanatory
solve R R T Species Steps
no duplicates
balanced step for each step in Steps
pathway step consistency for each step in Steps
formation dependencies for each step in Steps
both reagents needed R R T
enumerate species
balanced step R R P P
formula of R C H N O formula of R C H N O
of P C H N O formula of P C H N O formula
C C C C
H H H H
N N N N
O O O O
pathway step consistency R R P P
arith var of R N arith var of R N
arith var of P N arith var of P N
N N N N
formation dependencies R R P P
bool var a of R A bool var b of R B
var a of R A bool var b of R B bool
bool var a of P A bool var b of P B
bool var a of P A bool var b of P B
A A A A
B B B B
both reagents needed R R T
bool var a of R A bool var b of R B
bool var a of R A bool var b of R B
bool var a of T A bool var b of T B
R T
R T
Propositional Solver
As mentioned ab ove in the discussion ab out the b o olean constraint domain one
approach to solving b o olean equations is to use clp FD representing the input
formulas in a straightforward way using variables constrained to b e or See
section of and for example What follows is from
Assuming without losing generality that the input is a conjunction of equations
of the form Z X Y Z X Y or X Y the basic algorithm is simply to
representeach equation
Z X Y bythe FD constraints Z X Y
Z X Z Y Y
Z Y Z X X
X Y by Z X Y X Y Z
Z Y X Z
Z X X Z
X Y by X Y
Y X
The following is a clp FD program fragmentwhich realizes these representations
What is not shown is a pro cedure which takes the input equation and calls the
and or and not pro cedures appropriately andanenumeration pro cedure over the
values and for all variables In this program val X delays execution of an FD
constraint containing it until X is ground at which time val X denotes the value
ely the currentlower and of X The meaning of min X andmax X are resp ectiv
upp er b ounds on X maintained by the constraintsolver as discussed in Section
A constraint Xins texpresses that s and t are resp ectively lower and upp er
b ounds for X
and X Y Z
Z in min X min Y max X max Y
X in min Z max Z max Y min Y
Y in min Z max Z max X min X
or X Y Z
Z in min X min Y min X min Y max X max Y max X max Y
X in min Z max Y max Z
Y in min Z max X max Z
not X Y
X in val Y Y in val X
We conclude here bymentioning the authors claim that this approach has great
e ciency In particular it is several times faster than eachoftwo b o olean solvers
deployed in CHIP and some sp ecial purp ose stand alone solvers
Further Applications
The applications discussed in the previous two sections are but a sample of CLP ap
plications Here we brie y mention some others to indicate the breadth of problems
that have b een addressed using CLP languages and techniques
Wehave exempli ed the use of CLP to mo del analog circuits ab ove A con
siderable amountofwork has also b een done on digital circuits in particular on
veri cation diagnosis synthesis and test pattern
generation Much of these works used the CHIP system See also fora
description of a large application In civil engineering used CLP R for the
analysis and partial synthesis of truss structures As with electrical circuits the
constraints implementphysical mo delling and are used to verify truss and supp ort
comp onents as well as to generate spatial con gurations There is also work in me
chanical engineering used CLP R to design gear b oxes and combined
techniques from qualitativephysics and CLP R to design mechanical systems from
behaviour sp eci cations In general engineering applications such as these use CLP
to sp ecify a hierarchical comp osition of complex systems and for rule based reason
ing
Another imp ortant application area for CLP is nance Wementioned the OTAS
work ab ove Some further work is which also deals with option valuations and
which deal with nancial planning These nancial applications have
tended to take the form of exp ert systems involving sophisticated mathematical
mo dels
There havebeenvarious prop osals for including certainty measures and proba
bilities in logic programs to provide some built in evidential reasoning that can b e
useful when writing exp ert systems Original prop osals intended Prolog
as the underlying language but it is clear that CLP languages provide for more
exible execution of such exp ert systems
The Applause Pro ject has develop ed applications that use the ElipSys
system a parallel implemen tation of a CLP language for man ufacturing planning
tourist advice molecular biology and environment monitoring and control
Finally we mention work on applying CLP languages to music car se
quencing aircraft tra c control building visual language parsers
awarehousing problem safety analysis frequency assignment for cellu
lar telephones timetabling o or planning spacecraft attitude con
trol interop erability of bre optic communications equipment interest
rate risk management in banking failure mo de and e ect analysis of complex
systems development of digitally controlled analog systems testing of
telecommunication proto cols causal graph management factory sched
uling etc The Applause Pro ject has develop ed applications that use the
ElipSys system for manufacturing planning tourist advice molecular biology and
environment monitoring and control
Acknow ledgeme nts
Wewould like to thank the following p eople for their comments on drafts of this
pap er and or help in other ways M Bruyno oghe N Heintze P van Hentenryck
A Herold J L Lassez S Michaylov C Palamidessi K Shuerman P Stuckey M
Wallace R Yap We also thank the anonymous referees for their careful reading
and helpful comments
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