QBF Modeling: Exploiting Player Symmetry for Simplicity and Efficiency

Abstract

Quantified Boolean Formulas (QBFs) present the next big challenge for automated propositional reasoning. Not surprisingly, most of the present day QBF solvers are extensions of successful propositional satisfiability algorithms (SAT solvers). They directly integrate the lessons learned from SAT research, thus avoiding re-inventing the wheel. In particular, they use the standard conjunctive normal form (CNF) augmented with layers of variable quantification for modeling tasks as QBF. We argue that while CNF is well suited to “existential reasoning” as demonstrated by the success of modern SAT solvers, it is far from ideal for “universal reasoning” needed by QBF. The CNF restriction imposes an inherent asymmetry in QBF and artificially creates issues that have led to complex solutions, which, in retrospect, were unnecessary and sub-optimal. We take a step back and propose a new approach to QBF modeling based on a game-theoretic view of problems and on a dual CNF-DNF (disjunctive normal form) representation that treats the existential and universal parts of a problem symmetrically. It has several advantages: (1) it is generic, compact, and simpler, (2) unlike fully non-clausal encodings, it preserves the benefits of pure CNF and leverages the support for DNF already present in many QBF solvers, (3) it doesn’t use the so-called indicator variables for conversion into CNF, thus circumventing the associated illegal search space issue, and (4) our QBF solver based on the dual encoding (Duaffle) consistently outperforms the best solvers by two orders of magnitude on a hard class of benchmarks, even without using standard learning techniques.