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Local Search and Restart Strategies for Satisfiability Solving in Fuzzy

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Local Search and Restart Strategies for Satisfiability Solving in Fuzzy Local Search and Restart Strategies for Satisfiability Solving in Fuzzy Logics Tim Brys∗, Madalina M. Drugan∗, Peter A.N. Bosmany, Martine De Cockx and Ann Nowe´∗ ∗Artificial Intelligence Lab, VUB Pleinlaan 2, 1050 Brussels, Belgium timbrys, mdrugan, anowe @vub.ac.be f g yCentrum Wiskunde & Informatica (CWI) P.O. Box 94079, 1090 GB Amsterdam, The Netherlands [email protected] xDept. of Applied Math. and Comp. Sc. UGent Krijgslaan 281 (S9), 9000 Gent, Belgium [email protected] Abstract—Satisfiability solving in fuzzy logics is a subject that as not suffering from the complexity issues associated with an- has not been researched much, certainly compared to satisfiability alytical solvers, as our results showed. The disadvantage of this in propositional logics. Yet, fuzzy logics are a powerful tool approach is that only given an optimization algorithm proven for modelling complex problems. Recently, we proposed an optimization approach to solving satisfiability in fuzzy logics and to converge to the global optimum can it be a complete solver, compared the standard Covariance Matrix Adaptation Evolution i.e. deciding both SAT and UNSAT . The optimization 1 1 Strategy algorithm (CMA-ES) with an analytical solver on a set of algorithm used in [12] was the Covariance Matrix Adaptation benchmark problems. Especially on more finegrained problems Evolution Strategy (CMA-ES)[13], which is considered to did CMA-ES compare favourably to the analytical approach. be state-of-the-art in black-box optimization on a continuous In this paper, we evaluate two types of hillclimber in addition to CMA-ES, as well as restart strategies for these algorithms. domain. In this paper we explore some alternatives to this Our results show that a population-based hillclimber outperforms algorithm as the core for this SAT solver. Our findings are 1 CMA-ES on the harder problem class. somewhat surprising in that CMA-ES is outperformed by a hillclimber on one problem class. I. INTRODUCTION The rest of the paper is structured as follows: in Section The problem of SAT in propositional logic [1] is well known II, we give a brief description of fuzzy logics and formally and is of interest to researchers from various domains [2], [3], define SAT as an optimization problem. Section III contains 1 as many problems can be reformulated as a SAT problem and the description of a hillclimber algorithm (III-A), a population- subsequently solved by a state-of-the-art SAT solver. based version of that algorithm (III-B) and CMA-ES (III-C), as In fuzzy logics – logics with an infinite number of truth well as restart strategies for these algorithms. These algorithms degrees – the same principle of satisfiability exists, SAT , are then evaluated and compared on a set of benchmark 1 and it is, like its classical counterpart, useful for solving problems in Section IV. a variety of problems. Indeed, many fuzzy reasoning tasks can be reduced to SAT , including reasoning about vague II. FUZZY LOGIC AND SAT 1 1 concepts in the context of the semantic web [4], fuzzy spatial In fuzzy logics [14], truth is expressed as a real number reasoning [5] and fuzzy answer set programming [6], which in taken from the unit interval [0; 1]. Essentially, there are an itself is an important framework for non-monotonic reasoning infinite number of truth degrees possible. A formula in a fuzzy over continuous domains (see e.g. [7], [8], [9]). logic is built from a set of variables , constants taken from In contrast to SAT, SAT has received much less attention V 1 [0; 1] and n-ary connectives for n N. An interpretation is 2 from the various research communities; there are only two a mapping : [0; 1] that maps every variable to a truth I V! types of analytical solvers to be found in the literature, one degree. We can extend this fuzzy interpretation to formulas I using mixed integer programming [10], and one using a con- as follows: straint satisfaction approach [11]. Both methods suffer from For each constant c in [0; 1], [c] = c. exponential complexity issues associated respectively with • I For each variable v in , [v] = (v). mixed integer programming itself and an iteratively refining • V I I Each n-ary connective f is interpreted by a function f : discretization. • [0; 1]n [0; 1]. Furthermore we define Recently, we proposed a new approach to solving SAT ! 1 [12] that models satisfiability as an optimization problem [f(α1; : : : ; αn)] = f([α1] ;:::; [αn] ) over a continuous domain, and has the advantage of being I I I independent of the underlying logic and its operators, as well for formulas α with 1 i n. i ≤ ≤ [0, 1]n [0, 1]. Furthermore we define 1 ! [0, 1]n [0, 1]. Furthermore we define ! 1 [f(↵1,...,↵n)] = f([↵1] ,...,[↵n] ) ) I I I i α f ( [f(↵1,...,↵n)] = ([↵1] ,...,[↵n] ) I ) f I I I i α ( I for formulas ↵i with 1 i n. f for formulas↵ with 1 i n. The connectives in fuzzy logicsi typically correspond to con- The connectives in fuzzy logics typically correspond to con- 0 l u 1 nectives from classical logic, such as conjunction, disjunction, [α ] n 0 l i I u 1 nectives from classical logic, such as conjunction, disjunction, [α ] implication[0 and, 1] negation,[0, 1] which. Furthermore are interpreted we define respectively i I implication! and negation, which are interpreted respectively 1 by a t-norm, a t-conorm,n an implicator and a negator. A Fig. 1. f (↵i) for a formula ↵i with lower bound l, upper bound u and by a[0 t-norm,, 1][f(↵1,..., a[0 t-conorm,,↵1]n)] = anf implicator([↵1] ,..., and[↵n a] negator.) A I Fig. 1. f (↵i) for a formula ↵i with lower bound l, upper bound u and . Furthermore we define ) triangular norm or t-norm T is anI increasing,I associativeI degree of satisfaction [↵Ii] .i 1 α degree of satisfactionI( [↵ ] . triangular norm! or t-norm T is an increasing, associative I i 2 f and commutative [0, 1] [0, 1] 2mapping that satisfies the I forand formulas commutative↵i with[0, 1]1 i[0, 1]n.mapping that satisfies the ![f(↵1,...,↵n)] = f([↵1] ,...,[↵n] ) boundary condition T(1,x)=x for! all x in [0, 1]. Simi- ) Theboundary connectives condition inT fuzzy(1,x)= logicsx Ifor typically all x in correspond[0I , 1]. Simi- toI con- i α The connectives in fuzzy logics typically correspond to con- ( formula. Given these n formulas ↵ , boundsI (u ,l ), and an larly, a triangular conorm or t-conorm S is an increasing, i f i i nectiveslarly, a from triangular classical conorm logic, or such t-conorm as conjunction,S is an increasing, disjunction,formula. Given0 these1 nl formulas ↵i, boundsu (ui,li), and1 an nectives fromfor formulasclassical logic,↵i2 with such as1 conjunction,2 i n. disjunction,interpretation , we define the objective function[α ] f as follows: associative and commutative [0, 1] [0[0,,1] mapping[0, 1] that interpretation , we define the objectivei I function f as follows: implicationassociative and and negation,commutative! which are interpretedmapping respectively that I I satisfiesimplication the boundary and condition negation,S(0 which,x)= arex. interpreted An! implicator respectively bysatisfiesThe a t-norm, connectives the boundary a t-conorm, in condition fuzzy anS logics implicator(0,x)= typicallyx. Anand implicator a correspond negator. A to con- n f (↵ n,l ,u ) by a2 t-norm, a2 t-conorm, an implicator and a negator. A i=1 ii=1i f i(↵i,li,ui) I is a [0, 1]I is a[0[0, 1], 1]mapping[0, 1] thatmapping is decreasing that is decreasing in its first in its first Fig. 1.f(f)=(↵i) ffor( a)= formulaI ↵i0Iwith lowerl bound (2)l, upper bound(2) u u and 1 nectives! from classical logic,T such as conjunction, disjunction,I [α ] triangulartriangular norm norm or! t-norm or t-normT is anis increasing, an increasing, associative associativedegree ofI satisfactionI[↵i]n. n i I argument, increasingargument, in increasing its second in argument2 its second and argument that satisfies and that satisfies P IP the propertiesandandimplication commutativetheI commutative(0 properties, 0) = I(0 andI[0(0,,1)1], [00)2 negation,=; =1]I(1I(0[0, 1),,1)1] =[0 = 1whichmapping;Iand1](1,mapping1)I(1 = are, 0) that1 and = interpreted satisfies0 thatI.(1,and,0) satisfies = the 0. respectivelyand, the ! ! boundaryboundarybyNA a negator condition t-norm, conditionN isT a(1 decreasing[0 t-conorm,,xT, (11])=; x)x[0 =[0,for,1]1] anx allfor implicator[0 x all, 1] in xmapping[0,in1] and.[0 Simi-; that1] a. Simi- negator. A A negator is a decreasing mapping that Fig.0 1. f (↵i) for a formula ↵i with lower bound l,1 upper bound u and ! ! 1 if li [↵i] ui satisfies N(0) = 1 and N(1) = 0. S formula. Given1 these nifIformulasli [↵i] ↵iu,i bounds (ui,li), and an satisfieslarly,Nlarly,(0)triangular a = triangular a 1 and triangularN norm(1) conorm = 0 conorm. or or t-norm t-conorm or t-conormT isS is an an increasing,is increasing, an increasing, associative degree of[↵ satisfaction] [↵i] . I [↵i] i I 2 2 f (↵i,li,ui)= I if [↵i] I<li (3) As an exampleAs an of example a particularly of a particularly popular2 [0; fuzzy1] popular logic,[0 fuzzy; in1] logic, in f (interpretation↵i,li,ui)= , weI define8 ifli[↵ thei] <l objectivei function(3) f as follows: associativeassociative and and commutative commutative[0, 1] [0, 1] mappingmapping that I that I 8 li I and commutative [0, 1] [0, 1] mapping that satisfies the 1I [↵ ] > 1 [↵i] I Łukasiewicz logic, negation , conjunction! ! , disjunction > i < − I if [↵i] >ui Łukasiewiczsatisfiessatisfies logic, the the boundarynegation boundary, condition conjunction conditionS!(0,x,S disjunction)=(0; xx).
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