Two SAT solvers for solving quantified Boolean formulas with an arbitrary number of quantifier alternations

Florian Lonsing,Roderick Bloem,Martina Seidl,Nicolas Braud-Santoni,Vedad Hadzic,Uwe Egly

doi:10.1007/s10703-021-00371-7

Abstract

In recent years, expansion-based techniques have been shown to be very powerful in theory and practice for solving quantified Boolean formulas (QBF), the extension of propositional formulas with existential and universal quantifiers over Boolean variables. Such approaches partially expand one type of variable (either existential or universal) for obtaining a propositional abstraction of the QBF. If this formula is false, the truth value of the QBF is decided, otherwise further refinement steps are necessary. Classically, expansion-based solvers process the given formula quantifier-block wise and use one SAT solver per quantifier block. In this paper, we present a novel algorithm for expansion-based QBF solving that deals with the whole quantifier prefix at once. Hence recursive applications of the expansion principle are avoided and only two incremental SAT solvers are required. While our algorithm is naturally based on the forall Exp+Res calculus that is the formal foundation of expansion-based solving, it is conceptually simpler than present recursive approaches. Experiments indicate that the performance of our simple approach is comparable with the state of the art of QBF solving, especially in combination with other solving techniques.

Highlights

Efficient tools for deciding the satisfiability of Boolean formulas (SAT solvers) are the core technology in many verification and synthesis approaches [49]
For quantified Boolean formulas (QBF) with one quantifier alternation (2QBF), a natural approach is to use two SAT solvers: one that deals with the existentially quantified variables and another one that deals with the universally quantified variables. For generalizing this SAT-based approach to QBFs with an arbitrary number of quantifier alternations, expansion is recursively applied per quantifier block, requiring multiple SAT solvers realizing a counter-example guided extraction approach (CEGAR) [17]
Inspired by Counterexample-Guided Inductive Synthesis (CEGIS), we present a novel solving algorithm based on non-recursive expansion for QBFs with arbitrary quantifier prefixes using only two SAT solvers

Summary

Introduction

Efficient tools for deciding the satisfiability of Boolean formulas (SAT solvers) are the core technology in many verification and synthesis approaches [49]. QBFs extend propositional formulas by universal and existential quantifiers over Boolean variables [34] resulting in a decision problem that is PSPACE-complete. For QBFs with one quantifier alternation (2QBF), a natural approach is to use two SAT solvers: one that deals with the existentially quantified variables and another one that deals with the universally quantified variables. For generalizing this SAT-based approach to QBFs with an arbitrary number of quantifier alternations, expansion is recursively applied per quantifier block, requiring multiple SAT solvers realizing a counter-example guided extraction approach (CEGAR) [17]. As noted by Rabe and Tentrup [42], these CEGAR-based approaches show poor performance for formulas with many quantifier alternations in general

Methods

Results

Conclusion