QBF-Based Formal Verification: Experience and Perspectives

Abstract

The language of Quantified Boolean Formulas (QBF) has a lot of potential applications to Formal Verification (FV) tasks, as it captures many of these tasks in a natural and compact way. Practical experience has been disappointing though. When compared with contending approaches such as SAT, QBF-based FV has invariably yielded unfavorable experimental results. This paper makes two contributions. We first provide an account of the status quo in QBF-based FV. We examine commonly adopted formalizations and the relative strengths of different decision procedures. In the second part of this paper, we investigate for the first time the relevance of some advanced QBF techniques to FV tasks. In particular, we describe the use and the benefits of restricted quantifiers, QBF certificates, alternative encodings for classical model checking problems, and encodings with free variables. These promising research perspectives seem to reverse the negative standing of QBF applied to FV, as confirmed by the experimental evidence we discuss. Experiments are conducted by extending the publicly available solver sKizzo in several ways, and they include the first case studies where QBF compares favorably to SAT, its traditional competitor. QBF turns out to be an order of magnitude faster than SAT in some tasks (e.g., automated design debugging of large circuits). Moreover, as the size of the problems grows, the SAT encodings result in excessive memory requirements leading to out-of-memory conditions, while the more compact QBF encodings continue to be manageable and solvable.

Incrementally Computing Minimal Unsatisfiable Cores of QBFs via a Clause Group Solver API

“…The clauses in these groups amount to an unsatisfiable core of the PCNF. That core is obtained by internally collecting all selector variables relevant for unsatisfiability 4 Table 1. Statistics for unsatisfiable instances from the QBF Gallery 2014 where MUCs were successfully computed.…”

Q-Resolution with Generalized Axioms

“…The fact that many important practical reasoning, verification, and synthesis problems fall into the latter complexity class (cf. [3] for an overview) strongly motivates the quest for efficient QBF solvers.…”

Efficient extraction of Skolem functions from QRAT proofs

“…A natural way to encode synthesis problems is offered by quantified Boolean formulas (QBFs) [1], [2], which extend propositional logic by quantifiers over propositional variables [3]. The QBF formalism provides a convenient framework for modeling finite two-player games [4] which is reflected by the popular game-based view on the semantics of QBFs.…”