Theoretical study of symmetries in propositional calculus and applications

Benhamou, Belaïd; Saïs, Lakhdar

doi:10.1007/3-540-55602-8_172

Cited by 32 publications

(26 citation statements)

References 8 publications

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“…Aguirre [1] and Crawford, Ginsberg, Luks and Roy [7] each define symmetry similarly: if S is a set of clauses in CNF, then a permutation % of the variables in those clauses is a symmetry of S if %ðSÞ ¼ S. The expression %ðSÞ denotes the result of applying the permutation % to the clauses in S. If this permutation simply reorders the literals in individual clauses, and reorders the clauses, then it leaves S effectively unchanged, and so in this case %ðSÞ ¼ S and % is a symmetry. Benhamou and Sais [4] use a slightly more general definition, in which a symmetry is a permutation defined on the set of literals that preserves the set of clauses. For example, given two variables x and y, x may be mapped to :y.…”

Section: Definitionmentioning

confidence: 99%

Symmetry Definitions for Constraint Satisfaction Problems

et al. 2006

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We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.

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Section: Definitionmentioning

confidence: 99%

Symmetry Definitions for Constraint Satisfaction Problems

et al. 2006

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“…Aguirre [1] and Crawford, Ginsberg, Luks & Roy [6] each define symmetry similarly: if S is a set of clauses in CNF, then a permutation π of the variables in those clauses is a symmetry of S if π(S) = S. The expression π(S) denotes the result of applying the permutation π to the clauses in S. If this permutation simply re-orders the literals in individual clauses, and reorders the clauses, then it leaves S effectively unchanged, and so in this case π(S) = S and π is a symmetry. Benhamou and Sais [4] use a slightly more general definition, in which a symmetry is a permutation defined on the set of literals that preserves the set of clauses. For example, given two variables x and y, x may be mapped to ¬y.…”

Section: V K } ⊆ V and A I ∈ D For All I Such That 1 ≤ I ≤ K Notmentioning

confidence: 99%

Symmetry Definitions for Constraint Satisfaction Problems

Cohen

Jeavons

Jefferson

et al. 2005

Principles and Practice of Constraint Programming - CP 2005

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Abstract. We review the many different definitions of symmetry for constraint satisfaction problems (CSPs) that have appeared in the literature, and show that a symmetry can be defined in two fundamentally different ways: as an operation preserving the solutions of a CSP instance, or else as an operation preserving the constraints. We refer to these as solution symmetries and constraint symmetries. We define a constraint symmetry more precisely as an automorphism of a hypergraph associated with a CSP instance, the microstructure complement. We show that the solution symmetries of a CSP instance can also be obtained as the automorphisms of a related hypergraph, the k-ary nogood hypergraph and give examples to show that some instances have many more solution symmetries than constraint symmetries. Finally, we discuss the practical implications of these different notions of symmetry.

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“…Symmetries can be classified into semantic or syntactic [5]. Semantic symmetries are permutations of the formula that preserves its set of the models (or solutions) and, therefore, can be regarded as properties of the underlying Boolean function, independent of the particular syntactic representation.…”

Section: Symmetries In Automated Theorem Provingmentioning

confidence: 99%

“…Since then, many articles discuss how to detect and exploit syntactic symmetries in SAT solving [7,5,8,9,10,11,12,13,14]. Symmetries have been also extensively investigated and successfully exploited in other domains besides SAT like Constraint Satisfaction Problem [15,16], Integer Programming [17,18], Planning [19,20], Model Checking [21,22,23,24], Quantified Boolean Formulas (QBF) [25,26,27], and Satisfiability Modulo Theories (SMT) [28,29,30].…”

Section: Symmetries In Automated Theorem Provingmentioning

confidence: 99%

Symmetries in Modal Logics

Areces

Hoffmann

Orbe

2013

Electron. Proc. Theor. Comput. Sci.

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In this paper we develop the theoretical foundations to exploit symmetries in modal logics. We generalize the notion of symmetries of propositional formulas in conjunctive normal form to modal formulas using the framework provided by coinductive modal models introduced in [1]. Hence, the results apply to a wide class of modal logics including, for example, hybrid logics. We present two graph constructions that enable the reduction of symmetry detection in modal formulas to the graph automorphism detection problem, and we evaluate the graph constructions on modal benchmarks.

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Theoretical study of symmetries in propositional calculus and applications

Cited by 32 publications

References 8 publications

Symmetry Definitions for Constraint Satisfaction Problems

Symmetry Definitions for Constraint Satisfaction Problems

Symmetry Definitions for Constraint Satisfaction Problems

Symmetries in Modal Logics

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