Fuzzy Constraint Satisfaction

Zs. Ruttkay

AI Group, Dept. of Mathematics and Computer Science, Vrije Universiteit AmsterdamDe Boelelaan 1081a, 1081HV Amsterdam, The Netherlands

e-mail: zsofi@cs.vu.nl

Abstract In this paper the issue of softconstraint satisfaction is discussed from a fuzzyset theoretical point of view. A fuzzy constraint isconsidered as a fuzzy relation. Different possibledefinitions for the degree of joint satisfaction of aset of fuzzy constraints are given, covering specificother soft constraint satisfaction problem (CSP)types such a partial and hierarchical CSP. It isshown that the classical CSP solving heuristicsbased on variable and value evaluations can begeneralised and used to guide the solutionconstruction process for solving fuzzy CSPs, andthat the heuristic search can be replaced bybranch-and-bound search. The solution process isillustrated with an example from the CSPliterature. Finally, research issues are discussed.

1. Introduction

In the recent years there has been a growing interestin soft constraint satisfaction. In general, in a softCSP not all the given constraints need to be satisfied either because all of them cannot be met,theoretically, or in practice it would require too muchtime to find a solution, or simply, the real problemto be modelled is such that soft constraints areadequate and should be introduced, e.g. to expresspossible alternative requirements. Different kinds ofsoft CSPs and solution techniques have beendiscussed in the CSP literature: partial constraintsatisfaction [8], hierarchical constraint satisfaction[1, 7], structured constraint satisfaction [16], softconstraint relaxation [9] and constrained heuristicsearch [6]. In all these approaches constraints (orsets of constraints) are compared, e.g. on the basis ofweights (e.g. indicating importance) assigned to thegiven crisp constraints.

Another approach to deal with soft constraints isto generalise the notion of crisp constraint. A crispconstraint is characterised by the set of tuples forwhich the constraint holds. Hence, it is a naturalextension that a fuzzy set characterises a fuzzy

constraint. This possibility has recently receivedattention [3, 4, 11, 12, 17]. In the case of a fuzzyconstraint, different tuples satisfy the givenconstraint to a different degree. We find the fuzzy settheoretical approach to deal with soft constraintsappealing for two reasons. From a technical point ofview, the existing concepts and techniques of fuzzyset theory can be adapted for defining and solvingfuzzy CSPs. From a conceptual point of view, bycomparing tuples the preferences are expressed in anexplicit way, which may facilitate the knowledgeacquisition and model building process.

In this paper we discuss possible definitions forfuzzy solutions of fuzzy CSPs. Then we show howthe basic solution technique for solving CSPs constructive heuristic search can be adapted forsolving fuzzy CSPs. We discuss appropriategeneralisations of variable and value evaluationswhich can be used to guide the search, and theapplicability of the branch and bound technique toprune the search space. Finally, we formulate someresearch issues.

2. Basic definitions

The fuzzy relation, cardinality, domain of a fuzzyrelation and fuzzy operators common in fuzzy settheory [2] can be used to generalise essentialconcepts of crisp CSPs for fuzzy CSPs.

2.1 Fuzzy constraint

A fuzzy constraint is a mapping form aD=D1x...xDk domain (the direct product of the finitedomain of the variables referred by the constraint) tothe [0,1] interval. For a fuzzy constraint c the numberc(v1,...vk) denotes 'how well' the tuple (v1,...vk)satisfies the constraint. A fuzzy constraintcorresponds to the membership function of a fuzzyset. For a (v1,...vk) we call c(v1,...vk) the degree ofsatisfaction of the constraint c by the instantiation(v1,...vk). If c(v1,...vk)=1 then we say that (v1,...vk)

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fully satisfies c, while if c(v1,...vk)=0 then we saythat (v1,...vk) fully violates c. The (v1,...vk) elementsfor which c (v1,...vk)>0 form the so-called supportset, denoted by S. One may be interested in elementswhich satisfy a constraint better than a giventhreshold , where 1>0. The -support set,denoted by S, is the set of all (v1,...vk)D1x...xDkfor which

c(v1,...vk).

The cardinality of c is

|| c || = v1...vk

c(v1,...vk).

2.2 Fuzzy CSP

A fuzzy CSP (FCSP) is given as a list of variables(x1,...xK), a list of finite domains of possible valuesfor the variables (D1,...DK), and a list of fuzzyconstraints (c1,...cN), each of them referring to someof the given variables. We will consider binaryFCSPs only. We will also assume that there is atmost one constraint referring to any two variables.These restrictions are made only to simplify thediscussion. All that will be introduced below can beeasily generalised for FCSPs with non-binaryconstraints, and with possible multiple constraintsreferring to the same variables.

We will use the notation xi for some (xi1, xi2

),that is the pair of variables referred to by theconstraint ci , and vi for some (vi1

, vi2)Di1xDi 2.

We will also use the notation cij for a constraintreferring to the variables xi and xj , x ij for (x i, xj),and vij for (vi , vj )DixDj . If a vD is given, then itsatisfies the individual c1,...cN constraints to thedegree c1(v1),...cN(vN), respectively. The domainof a constraint cij is a fuzzy set, with the membershipfunction domij , where

domij (v)=max {cij (v, w) | wDj } vDi .

2.3 Fuzzy solution

The solution of an FCSP is also fuzzy: for anyinstantiation the degree of satisfaction indicates howwell the constraints are satisfied jointly. There areseveral ways to define the degree of joint satisfactionin terms of the degree of satisfaction of theindividual constraints. We will introduce threeplausible and simple definitions.As an individual constraint can be considered as themembership function of a fuzzy set, the problem of

defining the degree of satisfaction of severalconstraints can be considered as the problem ofdefining the membership function for intersection offuzzy sets. Adapting two common fuzzy settheoretical intersection definitions [2], we get twopossible ways to define the degree of jointsatisfaction of a set of fuzzy constraints, and hence,of an FCSP.

Definition 1: Based on the conjunctive combinationprinciple, the degree of joint satisfaction of theconstraints c1,...cN by the instantiation v D is theminimum of the satisfaction of the individualconstraints. That is,

Cmin((c1,...cN), v) = min {ci(vi ) | i=1,...N}.

Definition 2: Based on the productive combinationprinciple, the degree of joint satisfaction of theconstraints c1,...cN by the instantiation v D is theproduct of the satisfaction of the individualconstraints. That is,

Cpro((c1,...cN), v) =i=1

N ci (vi ).

However, form a CSP perspective it also makessense to consider the joint satisfaction as thecumulated satisfaction of the individual constraints,related to the fuzzy-set theoretical concept of unionof fuzzy sets.

Definition 3: Based on the average combinationprinciple, the degree of joint satisfaction of theconstraints c1,...cN by the instantiation v D is theaverage of the satisfaction of the individualconstraints. That is,

Cave((c1,...cN), v) = 1N

i=1

Nci (vi ).

As it is discussed in [4], the first definition is toorough, not discriminating instantiations which areequally 'bad' concerning the least satisfied constraint,but differ much in the degree of satisfaction of therest of the constraints. In [4], two refinements of theconjunctive combination principle are discussedbased on inclusion and lexicographical ordering.

Our last two definitions are also such that theabove mentioned 'drowning effect' does not occur.The second definition does not differentiate betweeninstantiations which fully violate at least oneconstraint, while the third definition does. All thedefinitions above can be modified in such a way that

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importance of the individual constraints (relative orabsolute) is taken into account. It also can be shownthat partial constraint satisfaction corresponds to thecase when the constraints are crisp, but the notion ofjoint satisfaction is defined on the averagecombination principle.

Without respect to the specific definition of thedegree of joint satisfaction further on denoted byC, in general , the concepts of perfect and bestsolution can be defined (the same concepts weredefined for the specific partial CSP case in [8]). An

instantiation v * D is a perfect solution if all theindividual constraints are satisfied in the bestpossible way, that is

cij (v*ij )=max { cij (v

*i , vj ) | vj Dj } for all cij .

An instantiation v * D is a best solution if thedegree of joint satisfaction of all the constraints is themaximal possible, that is

C((c1,...cN) v * )=max { C((c1,...cN), v) | vD}.

When an FCSP is given, one can be interested infinding a best solution, or in a globally good enoughsolution a solution for which the joint satisfactionexceeds a given minimum , or in a locally goodenough solution an instantiation for which certainindividual constraints are satisfied better than a givenminimum level. The threshold can be given in anabsolute way, or as a percentage of the degree ofsatisfaction of the (theoretical) perfect solution.

3. Constructing a best solution

The solution mechanism for finding a best or goodenough solution to be discussed below can be appliedfor FCSPs with any of the above definitions of fuzzysolutions. More precisely, we will exploit only twocharacteristics of the definition of the degree of jointsatisfaction:

(I) The degree of joint satisfaction can be computedon the basis of the satisfaction of the individualconstraints, independent of each other.

(II ) The joint satisfaction is monotone with respectto the satisfaction of the individual constraints,that is, if v'D and v''D are such that

ci (v'i)=ci (v'' i) for i=1,...N, ij, andcj (v'j)cj (v'' j), then C((c1,...cN), v')C((c1,...cN), v'').

3.1 Heuristic search

The most common way to solve a CSP is to constructa solution by extending a partial solution, that isinstantiating variables one after the other in such away that all the constraints which can be evaluatedare satisfied. If a dead-end situation occurs (for thevariable on turn no appropriate value can be found),backtracking has to be performed. As extensivelydiscussed in the CSP literature [13], the order inwhich the variables are instantiated and the order inwhich the possible values are tried have an essentialeffect on the number of steps required to construct asolution. The critical selections (of a variable to beinstantiated and of a value to be assigned) are oftenperformed on the basis of the general idea of'instantiating the most constrained variables first',and 'assigning the least constraining values first'. Forpossible measures for these concepts, see [13, 14].

The same principles can be used when solvingFCSPs. There can be different measures used forguiding the variable and value selection whenexploring the search space in the case of a FCSPsolving as well. Below we discuss the moststraightforward possibilities.

The appropriateness of a value vDi for avariable xi is evaluated on the basis of the degree ofthe best possible joint satisfaction of the constraintsreferring to xi. That is,

ai (v) = max { C((cil 1,...cil i

), v ) |

v Dil 1x...xDil k-1x{v}xD il k+1x...xDil i}.

Because of (I ), ai(v) can be computed on the basis ofdomil 1

(v),...domil i(v), that is the best possible

satisfaction of the individual constraints referring toxi .

The difficulty of a variable xi is evaluated onthe basis of the evaluation of the possible values forthe variable. That is,

di = vDi

ai (v) .

The search for a best solution of a given FCSP, usingthe above evaluations for variable and valueselection, considers variables first for which fewgood values exist, and selects one of the best valuesfor the variable. Note, however, that in general thealgorithm does not guarantee that the constructedsolution is a best one. Another problem is thatbacktracking takes place only if it is known that anyextension of the current partial solution fully violatesthe given constraints. This can be proven for a partialsolution if the definition of the joint satisfaction is

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such that the full violation of a subset of theconstraints entails the full violation of the entire setof constraints. For the joint satisfactions Cmin andCpro this is true, but for Cave not.

3.2 Pruning not good enough solutions

The above mentioned anomalies can be cured byextending the solution scheme with pruning thesearch space, on the basis of the common discreteoptimisation technique branch-and-bound [5]. Whenexploring the search space, an upper bound for thedegree of satisfaction of the best possible extensionof a given partial solution is computed. As aconsequence of (II ), the degree of satisfaction of apartial solution provides an upper bound for thedegree of satisfaction of any extension of the givenpartial solution.

All partial instantiations (not yet explored leavesof the search tree) for which the computed upperbound is lower than the degree of satisfaction of thebest solution found so far are excluded from furtherconsiderations. Hence, as soon as a solution isfound, some (ideally, all) other possible alternativesmay be pruned. The upper limit can be used also toguide the search, not only to prune the search space.Then the partial instantiations with the highestestimate are explored first.

constraints satis-faction

f t s

c1 1 S D

0.4 S B0.2 S G0.8 C G0.5 C B

c2 1 S L

0.7 S W1 C W

0.1 C Lc3 1 D W

0.7 D L

1 B W

0.4 B L

1 G L

0.6 G W

Fig.1The fuzzy CSP model of the robot-dressing problem

(Not indicated pairs fully violate the appropriateconstraint.)

4 An example

The above described solution mechanism will beillustrated on an somewhat revised example takenfrom [8]. The problem a robot is facing is to getdressed in such a way that the pieces he puts onmatch. The robot has a choice of sneakers orCordovans for foot-wear, a choice of denim, blue andgrey trousers, and a choice of white or light-greyshirts. The problem can be modelled as an FCSP,with variables f, t and s with domains {S, C}, {D,B, G} and {W, L}, respectively. The matchingdegree of the possible pairs is expressed as degree ofsatisfaction of the appropriate binary fuzzyconstraints (given in Fig. 1). The definition of jointsatisfaction is based on the 'production combination'principle.

It can be seen that this problem has noperfect solution, and the only best solution is f=S,t=D, s=W with a joint satisfaction degree 0.7. Thereare 12 possible instantiations of the variables, andonly two of them violate fully the prescribedconstraints. In general, there are 72 different ways toconstruct a full instantiation (each completeinstantiation can be constructed in 6 different ways,depending on the order of instantiating the variables).By applying the heuristics described above, the bestsolution is constructed at first, in 3 steps.

0.7W

0.8

f

ts

D B G

s

L

S1.0

C

1.0 0.4 0.2

1.82.3

0.49

Fig. 2.The construction of the first solution, based on theevaluation of possible values and uninstantiated

variables

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The solution process is illustrated in Fig .2. In orderto decide which variable to instantiate first, thepossible values for each variable are evaluated. It canbe seen that

af(S)=1, af(C)=0.8, and thus df=1.8;

at(D)=1, at(B)=0.5, at(G)=0.8, and thus dt=2.3;

as(L)=1, as(W)=1, and thus ds=2.

Hence, the most critical variable, f is instantiated firstto the best possible value S. For selecting the secondvariable to be instantiated, the evaluations are asfollows (note that some of them have decreased dueto the fact that f=S):

at(D)=1, at(B)=0.4, at(G)=0.2, and thus dt=1.6;

as(L)=1, as(W)=0.7, and thus ds=1.7.

As a result, t is the next variable to be instantiated,and the best possible value, D is assigned to it.Finally, for the remaining variable s the best value Wis chosen. The degree of satisfaction of the solutionis 0.7.

Though the found solution is quite good, it has tobe checked if it is a best one. At each value selectionstep, the value evaluations multiplied by thedegree of satisfaction of those constraints which canbe evaluated due to earlier instantiations serve asan upper bound for the degree of satisfaction of thebest possible extension, assuming the value inquestion.

f

ts

D B G

s

0.7LW

S1.0

C0.8

1.0 0.4 0.2

1.82.3

0.49

t1.3

s1.1

LW1.1 0.1

GB0.5 0.48

t

Fig. 3.Pruning poor possible values ( symbol), and

exploring the remaining choices (- - - line).

As the estimates for alternative values for t and s areall lower than the degree of satisfaction of the foundsolution, these alternatives need not to be explored,the search space can be pruned (see Fig. 3).

The alternative value for the variable f has to beconsidered, as af(C)=0.8. Exploring this assignment,it can be seen that the next variable to be instantiatedis s, and the best assignment is s=W. The otheralternative for s, L is pruned as as(L)=0.1, hence nobetter solution than the already found one exists forwhich s=L. However, calculating the estimates forthe possible values for the remaining, not yetinstantiated variable t, it is obvious that for any of thepossible values the degree of satisfaction is below0.7. We need not explore alternatives correspondingto different variable orders, as for both possiblevalues for f we have shown that there is at most onecandidate for a better solution than the alreadyconstructed one.

5. Discussion

We have shown that fuzzy set theory provides a goodand general mathematical framework to define fuzzyconstraints, FCSPs and the degree of satisfaction ofsolutions, and that the classical constructive solutionmethod can be adapted for the FCSP case. Theproposed heuristics can be used with any definitionof the degree of solution for which the independentcomputation (I) and the monotonicity (II ) criteriahold.

Space did not allow us to discuss in this paperfurther generalisations of classical CSP concepts andsolution principles. For a-priory analysis of the FCSPand filtering of the domains, local consistency can begeneralised on the basis of the -support sets, andcan be used to estimate the degree of satisfaction ofthe best solution. However, while in the classicalcase local consistency is necessary, but not sufficientcondition for the existence of a solution, in the caseof a FCSP the necessity depends on the definition ofthe degree of joint satisfaction. In our definitions,only for the conjunctive combination is local -consistency necessary for the existence of a solutionwith a degree of satisfaction not smaller than .

The topology of the constraint graph can play asimilar role as in the case of crisp CSP incharacterising the difficulty of the problem. It isinteresting to investigate if the number and length ofcycles in a fuzzy graph representing an FCSP can beused to characterise the difficulty of the problem,similarly to the case of crisp CSPs.

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Further fuzzy set theorethical concepts, such the ascardinality of a constraint or the different definitionsto characterise the fuzziness of the support sets mayalso be used to estimate the difficulty of an FCSP.The outlined branch-and-bound search could befurther improved by introducing heuristics for theorder of checking the constraints.

Domains for variables with a given, naturalmetric are special, as the degree of satisfaction fortuples for a fuzzy constraint can be defined on thebasis of the distance of the tuple from the set oftuples which fully satisfy a given crisp constraint.Hence, the fuzzy relaxation of a crisp constraintcould be generated automatically, even in aninterwoven way with the solution process. Thedistance of fuzzy sets [15] could be used to indicatethe distance of the relaxed problem an FCSP from the original CSP.

A challenging and from a modelling point ofview very useful further generalisation of the crispCSP would be to add one more factor of fuzziness,by considering fuzzy constraints over fuzzy domains.

Acknowledgement

Thanks for Paul ten Hagen and Wojtek Kowalczykfor his constructive comments on this paper. Thepaper was written in the framework of the projectSKBS A3.

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