A finite state intersection approach to propositional satisfiability

José M Castaño,Rodrigo Castaño

doi:10.1016/j.tcs.2012.04.030

Abstract

We use a finite state (FSA) construction approach to address the problem of propositional satisfiability (SAT). We present a very simple translation from formulas in conjunctive normal form (CNF) to regular expressions and use regular expressions to construct an FSA. As a consequence of the FSA construction, we obtain an ALL-SAT solver and model counter. This automata construction can be considered essentially a finite state intersection grammar (FSIG). We also show how an FSIG approach can be encoded. Several variable ordering (state ordering) heuristics are compared in terms of the running time of the FSA and FSIG construction. We also present a strategy for clause ordering (automata composition). Running times of state-of-the-art model counters and BDD based SAT solvers are compared and we show that both the FSA and FSIG approaches obtain an state-of-the-art performance on some hard unsatisfiable benchmarks. It is also shown that clause learning techniques can help improve performance. This work brings up many questions on the possible use of automata and grammar models to address SAT.

Highlights

There is a long tradition that analyzed transformations of logic formulas and automata formally [9,15,39,38]
We present an approach to SAT solving as a parsing problem using Finite State Intersection Grammars (FSIGs) [42]
We show that it is possible to address SAT problems using automata construction, and that this approach is capable of reaching a reasonable performance compared to other equivalent approaches, in particular we compare them with binary decision diagrams (BDD) and NNF approaches [12], and it will be more efficient in unsatisfiable hard cases with state-of-the-art DPLL SAT solvers

Summary

Introduction

There is a long tradition that analyzed transformations of logic formulas and automata formally [9,15,39,38]. Studies by Büchi [9], Elgot [15] and Trakhtenbrot analyzed transformations from formulas to automata and vice versa in the context of the relation between FSA and Monadic Second Order logic (MSO). Sometimes referenced as the Büchi–Elgot–Trakhtenbrot Theorem, it was established that FSA and MSO have the same expressive power [38] This was the basis of the approach that uses Büchi automata to decide satisfiability of modal logic formulas (LTL) [39]. This is the only reference to an automata based approach to satisfiability found in [7]. Schützenberger, McNaughton, Papert and Kamp established the equivalence between star-free regular expressions, counter-free finite state automata, first-order logic and temporal logic, see [32]

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Results

Conclusion