A SAT Based Approach for Solving Formulas over Boolean and Linear Mathematical Propositions

Audemard, Gilles; Bertoli, Piergiorgio; Cimatti, Alessandro; Korniłowicz, Artur; Sebastiani, Roberto

doi:10.1007/3-540-45620-1_17

Cited by 84 publications

(99 citation statements)

References 10 publications

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“…Our approach has the potential of resulting in fast solvers for testing unsatisfiability of nonlinear constraints. This is especially significant in the context of satisfiability testing tools [21,9,1] that are being increasingly used for program analysis. There is also much recent progress in computational aspects of real algebraic geometry and computational tools for building a sumsof-squares representation using semi-definite programming [16,15,4], which indicates that our work will be actively refined and developed further in the future.…”

Section: Resultsmentioning

confidence: 99%

“…The same is also needed in the lazy approach of extending constraint solvers to handle boolean combination of constraints. Tools such as ICS [9], CVC [21], and MathSat [1], which are used in bounded model-checking of discrete systems, also implement some form of incomplete nonlinear constraint solving. Fast and sound, but potentially incomplete, implementations that can solve large problem instances are useful in several "incompleteness-tolerant" applications such as the process of creating abstractions, where incompleteness only causes creation of a coarser abstraction.…”

Section: Introductionmentioning

confidence: 99%

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An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints

Tiwari

2005

Computer Science Logic

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Abstract. We describe a simple algebraic semi-decision procedure for detecting unsatisfiability of a (quantifier-free) conjunction of nonlinear equalities and inequalities. The procedure consists of Gröbner basis computation plus extension rules that introduce new definitions, and hence it can be described as a critical-pair completion-based logical procedure. This procedure is shown to be sound and refutationally complete. When projected onto the linear case, our procedure reduces to the Simplex method for solving linear constraints. If only finitely many new definitions are introduced, then the procedure is also terminating. Such terminating, but potentially incomplete, procedures are used in "incompleteness-tolerant" applications.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints

Tiwari

2005

Computer Science Logic

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“…Unlike with SAT, in SMT there is very-limited tradition in testing on random problems (e.g., [2,3]). However, for a matter of scientific curiosity and/or to leverage to SMT a popular test for SLS SAT procedures, here we present also a brief comparison of WALKSMT vs. MATHSAT4 on randomly-generated, unstructured 3-CNF LA(Q)-formulas.…”

Section: Walksmt On Random Instancesmentioning

confidence: 99%

Stochastic Local Search for SMT: Combining Theory Solvers with WalkSAT

Griggio¹,

Phan

Sebastiani

et al. 2011

Frontiers of Combining Systems

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Abstract.A dominant approach to Satisfiability Modulo Theories (SMT) relies on the integration of a Conflict-Driven-Clause-Learning (CDCL) SAT solver and of a decision procedure able to handle sets of atomic constraints in the underlying theory T (T -solver). In pure SAT, however, Stochastic Local-Search (SLS) procedures sometimes are competitive with CDCL SAT solvers on satisfiable instances. Thus, it is a natural research question to wonder whether SLS can be exploited successfully also inside SMT tools. In this paper we investigate this issue. We first introduce a general procedure for integrating a SLS solver of the WalkSAT family with a T -solver. Then we present a group of techniques aimed at improving the synergy between these two components. Finally we implement all these techniques into a novel SLSbased SMT solver for the theory of linear arithmetic over the rationals, combining UBCSAT/UBCSAT++ and MathSAT, and perform an empirical evaluation on satisfiable instances. The results confirm the potential of the approach.

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“…(2) in Section 4) whenever a theory does not admit infinite models. Last, in Section 5.4, we describe how to combine rewrite-based procedures [1,2] with Satisfiability Modulo Theory (SMT) tools, such as [10,3,11,12], in order to obtain automatic methods to solve constraint satisfiability problems involving theories admitting only finite models (e.g., enumerated datatypes).…”

Section: Decidabilitymentioning

confidence: 99%

“…, then there must exists an integer k ≥ 0 such that C ∈ S k (recall Definition 5.3). Second, when a cardinality constraint clause C is derived from T ∪ Γ, a bound on the cardinality of the domains of any model can be immediately obtained by the cardinal associated to C. It is possible to use such a bound to build a set of clauses which is equisatisfiable to T ∪ Γ (see Figure 1) and pass it to an efficient decision procedure for the pure theory of equality, such as those provided by many SMT tools (see, e.g., [10,3,11,12]). The observations above motivate the following relaxation of the notion of a ∃-superposition-decidable theory.…”

Section: Combining Superposition Modules and Smt Proceduresmentioning

confidence: 99%

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

Bonacina

Ghilardi

Nicolini

et al. 2006

Automated Reasoning

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In the context of combinations of theories with disjoint signatures, we classify the component theories according to the decidability of constraint satisfiability problems in finite and infinite models, respectively. We exhibit a theory T 1 such that satisfiability is decidable, but satisfiability in infinite models is undecidable. It follows that satisfiability in T 1 ∪ T 2 is undecidable, whenever T2 has only infinite models, even if signatures are disjoint and satisfiability in T2 is decidable.In the second part of the paper we strengthen the Nelson-Oppen decidability transfer result, by showing that it applies to theories over disjoint signatures, whose satisfiability problem, in either finite or infinite models, is decidable. We show that this result covers decision procedures based on rewriting, generalizing recent work on combination of theories in the rewrite-based approach to satisfiability.

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A SAT Based Approach for Solving Formulas over Boolean and Linear Mathematical Propositions

Cited by 84 publications

References 10 publications

An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints

An Algebraic Approach for the Unsatisfiability of Nonlinear Constraints

Stochastic Local Search for SMT: Combining Theory Solvers with WalkSAT

Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures

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