Abstraction-Based Algorithm for 2QBF

Abstract

Quantified Boolean Formulas (QBFs) enable standard representation of PSPACE problems. In particular, formulas with two quantifier levels (2QBFs) enable representing problems in the second level of the polynomial hierarchy (Π P 2 , Σ P 2 ). This paper proposes an algorithm for solving 2QBF satisfiability by counterexample guided abstraction refinement (CEGAR). This represents an alternative approach to 2QBF satisfiability and, by extension, to solving decision problems in the second level of polynomial hierarchy. In addition, the paper presents a comparison of a prototype implementing the presented algorithm to state of the art QBF solvers, showing that a larger set of instances is solved.

Solving QBF with Counterexample Guided Refinement

“…Previous work on QBF shows how CEGAR can be used to solve formulas with 2 levels of quantifiers [17]. Here we generalize this approach to an arbitrary number of quantifiers by recursion.…”

On Conflicts and Strategies in QBF

“…It uses DPLL-style clause learning [30,12], abstraction & refinement learning of RAReQS [15,14], and the flat architecture of qesto [16].…”

“…This type of refinement is inspired by the approach of RAReQS [15,14]. While in RAReQS, only the universal variables at the quantification level k + 1 are set, strategies enable setting multiple levels at the same time.…”

“…Other solvers included in the evaluation are the non-CNF-based solver GhostQ [19], CDCLstyle solver with variable dependency computation DepQBF [21], the CEGAR-based solvers RAReQS and AReQS [15,14]. Figure 4 shows cactus plots for two benchmark families.…”

“…This is mitigated by the CEGAR approaches of the solver AReQS and its recursive extension RAReQS, which expand the formula gradually [15,14]. More recently the solvers qesto [16], QELL [29], and CAQE [27] build on RAReQS with variations in refinement but also in flat architecture, while RAReQS builds a tree of abstractions.…”