Answer Set Programming based on Propositional Satisfiability

Answer set programming (ASP) emerged in the late 1990s as a new logic programming paradigm that has been successfully applied in various application domains. Also motivated by the availability of efficient solvers for propositional satisfiability (SAT), various reductions from logic programs to SAT were introduced. All these reductions, however, are limited to a subclass of logic programs or introduce new variables or may produce exponentially bigger propositional formulas. In this paper, we present a SAT-based procedure, called ASPSAT, that (1) deals with any (nondisjunctive) logic program, (2) works on a propositional formula without additional variables (except for those possibly introduced by the clause form transformation), and (3) is guaranteed to work in polynomial space. From a theoretical perspective, we prove soundness and completeness of ASPSAT. From a practical perspective, we have (1) implemented ASPSAT in Cmodels, (2) extended the basic procedures in order to incorporate the most popular SAT reasoning strategies, and (3) conducted an extensive comparative analysis involving other state-of-the-art answer set solvers. The experimental analysis shows that our solver is competitive with the other solvers we considered and that the reasoning strategies that work best on ‘small but hard’ problems are ineffective on ‘big but easy’ problems and vice versa

Linear Encodings of Bounded LTL Model Checking

We consider the problem of bounded model checking (BMC) for linear temporal logic (LTL). We present several efficient encodings that have size linear in the bound. Furthermore, we show how the encodings can be extended to LTL with past operators (PLTL). The generalised encoding is still of linear size, but cannot detect minimal length counterexamples. By using the virtual unrolling technique minimal length counterexamples can be captured, however, the size of the encoding is quadratic in the specification. We also extend virtual unrolling to Buchi automata, enabling them to accept minimal length counterexamples. Our BMC encodings can be made incremental in order to benefit from incremental SAT technology. With fairly small modifications the incremental encoding can be further enhanced with a termination check, allowing us to prove properties with BMC. Experiments clearly show that our new encodings improve performance of BMC considerably, particularly in the case of the incremental encoding, and that they are very competitive for finding bugs. An analysis of the liveness-to-safety transformation reveals many similarities to the BMC encodings in this paper. Using the liveness-to-safety translation with BDD-based invariant checking results in an efficient method to find shortest counterexamples that complements the BMC-based approach.Comment: Final version for Logical Methods in Computer Science CAV 2005 special issu

The optimality of a fast CNF conversion and its use with SAT

Abstract. Despite the widespread use and study of Boolean satisfiability for a diverse range of problem domains, encoding of problems is usually given to general propositional logic with little or no discussion of the conversion to clause form that will be necessary. In this paper we present a fast and easy to implement conversion to equisatisfiable clause form for Boolean circuits. Since the conversion is equivalent to that of Boy de la Tour it is optimal in the number of clauses produced. We present experimental results comparing this and other conversion procedures on BMC problems and conclude that the CNF conversion plays a large part in reducing the overall solving time.

The Optimality of a Fast CNF Conversion and its Use with SAT

Despite the widespread use and study of Boolean satisfiability for a diverse range of problem domains, encoding of problems is usually given to general propositional logic with little or no discussion of the conversion to clause form that will be necessary. In this paper we present a fast and easy to implement conversion to equisatisfiable clause form for Boolean circuits, a popular representation of propositional logic formulae. We show that the conversion is equivalent to that of Boy de la Tour and is hence optimal in the number of clauses produced for linear input formulae (formul excluding #), and we discuss the optimality for other input formulae

Temporal Logic Encodings for SAT-based Bounded Model Checking

Since its introduction in 1999, bounded model checking (BMC) has quickly become a serious and indispensable tool for the formal verification of hardware designs and, more recently, software. By leveraging propositional satisfiability (SAT) solvers, BMC overcomes some of the shortcomings of more conventional model checking methods. In model checking we automatically verify whether a state transition system (STS) describing a design has some property, commonly expressed in linear temporal logic (LTL). BMC is the restriction to only checking the looping and non-looping runs of the system that have bounded descriptions. The conventional BMC approach is to translate the STS runs and LTL formulae into propositional logic and then conjunctive normal form (CNF). This CNF expression is then checked by a SAT solver. In this thesis we study the effect on the performance of BMC of changing the translation to propositional logic. One novelty is to use a normal form for LTL which originates in resolution theorem provers. We introduce the normal form conversion early on in the encoding process and examine the simplifications that it brings to the generation of propositional logic. We further enhance the encoding by specialising the normal form to take advantage of the types of runs peculiar to BMC. We also improve the conversion from propositional logic to CNF. We investigate the behaviour of the new encodings by a series of detailed experimental comparisons using both hand-crafted and industrial benchmarks from a variety of sources. These reveal that the new normal form based encodings can reduce the solving time by a half in most cases, and up to an order of magnitude in some cases, the size of the improvement corresponding to the complexity of the LTL expression. We also compare our method to the popular automata-based methods for model checking and BMC