Solving Quantified Boolean Formulas with Few Existential Variables

TL;DR

This paper introduces a fixed-parameter tractable algorithm for solving QBF problems with a limited number of existential variables, addressing a key challenge in making QBF more practical for complex AI problems.

Contribution

The paper develops a novel FPT algorithm for QBF with few existential variables and provides hardness results for cases with unbounded clause length.

Findings

FPT algorithm applicable to QBF with bounded clause length

W[1]-hardness result for unbounded clause length QBF

Sharper lower bounds under exponential-time hypothesis

Abstract

The quantified Boolean formula (QBF) problem is an important decision problem generally viewed as the archetype for PSPACE-completeness. Many problems of central interest in AI are in general not included in NP, e.g., planning, model checking, and non-monotonic reasoning, and for such problems QBF has successfully been used as a modelling tool. However, solvers for QBF are not as advanced as state of the art SAT solvers, which has prevented QBF from becoming a universal modelling language for PSPACE-complete problems. A theoretical explanation is that QBF (as well as many other PSPACE-complete problems) lacks natural parameters} guaranteeing fixed-parameter tractability (FPT). In this paper we tackle this problem and consider a simple but overlooked parameter: the number of existentially quantified variables. This natural parameter is virtually unexplored in the literature which one might find surprising given the general scarcity of FPT algorithms for QBF. Via this parameterization we then develop a novel FPT algorithm applicable to QBF instances in conjunctive normal form (CNF) of bounded clause length. We complement this by a W[1]-hardness result for QBF in CNF of unbounded clause length as well as sharper lower bounds for the bounded arity case under the (strong) exponential-time hypothesis.