A Duality-Aware Calculus for Quantified Boolean Formulas

Wir präsentieren ein formales Rahmenwerk, das es ermöglicht das Verhalten von QBF-Beweisen zu beschreiben.Learning and backjumping are essential features in search-based decision procedures for Quantified Boolean Formulas (QBF). To obtain a better understanding of such procedures, we present a formal framework, which allows to simultaneously reason on prenex conjunctive and disjunctive normal form. It captures both satisfying and falsifying search states in a symmetric way. This symmetry simplifies the framework and offers potential for further variants.W1255-N23S11408-N23(VLID)193237

DepQBF 6.0: A Search-Based QBF Solver Beyond Traditional QCDCL

We present the latest major release version 6.0 of the quantified Boolean formula (QBF) solver DepQBF, which is based on QCDCL. QCDCL is an extension of the conflict-driven clause learning (CDCL) paradigm implemented in state of the art propositional satisfiability (SAT) solvers. The Q-resolution calculus (QRES) is a QBF proof system which underlies QCDCL. QCDCL solvers can produce QRES proofs of QBFs in prenex conjunctive normal form (PCNF) as a byproduct of the solving process. In contrast to traditional QCDCL based on QRES, DepQBF 6.0 implements a variant of QCDCL which is based on a generalization of QRES. This generalization is due to a set of additional axioms and leaves the original Q-resolution rules unchanged. The generalization of QRES enables QCDCL to potentially produce exponentially shorter proofs than the traditional variant. We present an overview of the features implemented in DepQBF and report on experimental results which demonstrate the effectiveness of generalized QRES in QCDCL.Comment: 12 pages + appendix; to appear in the proceedings of CADE-26, LNCS, Springer, 201

Hard QBFs for Merge Resolution

We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [Olaf Beyersdorff et al., 2020]), a refutational proof system for prenex quantified Boolean formulas (QBF) with a CNF matrix. Unlike most QBF resolution systems in the literature, proofs in MRes consist of resolution steps together with information on countermodels, which are syntactically stored in the proofs as merge maps. As demonstrated in [Olaf Beyersdorff et al., 2020], this makes MRes quite powerful: it has strategy extraction by design and allows short proofs for formulas which are hard for classical QBF resolution systems. Here we show the first exponential lower bounds for MRes, thereby uncovering limitations of MRes. Technically, the results are either transferred from bounds from circuit complexity (for restricted versions of MRes) or directly obtained by combinatorial arguments (for full MRes). Our results imply that the MRes approach is largely orthogonal to other QBF resolution models such as the QCDCL resolution systems QRes and QURes and the expansion systems ?Exp+Res and IR

QCDCL vs QBF Resolution: Further Insights

We continue the investigation on the relations of QCDCL and QBF resolution systems. In particular, we introduce QCDCL versions that tightly characterise QU-Resolution and (a slight variant of) long-distance Q-Resolution. We show that most QCDCL variants - parameterised by different policies for decisions, unit propagations and reductions - lead to incomparable systems for almost all choices of these policies

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies

A game characterisation of tree-like Q-Resolution size

We provide a characterisation for the size of proofs in tree-like Q-Resolution and tree-like QU-Resolution by a Prover–Delayer game, which is inspired by a similar characterisation for the proof size in classical tree-like Resolution. This gives one of the first successful transfers of one of the lower bound techniques for classical proof systems to QBF proof systems. We apply our technique to show the hardness of three classes of formulas for tree-like Q-Resolution. In particular, we give a proof of the hardness of the parity formulas from Beyersdorff et al. (2015) for tree-like Q-Resolution and of the formulas of Kleine Büning et al. (1995) for tree-like QU-Resolution

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies

Quantified Boolean Formulas: Proof Complexity and Models of Solving

Quantified Boolean formulas (QBF), which form the canonical PSPACE-complete decision problem, are a decidable fragment of first-order logic. Any problem that can be solved within a polynomial-size space can be encoded succinctly as a QBF, including many concrete problems in computer science from domains such as verification, synthesis and planning. Automated solvers for QBF are now reaching the point of industrial applicability. In this thesis, we focus on dependency awareness, a dedicated solving paradigm for QBF. We show that dependency schemes can be envisaged in terms of dependency quantified Boolean formulas (DQBF), exposing strong connections between these two previously disparate entities. By introducing new lower-bound techniques for QBF proof systems, we study the relative strengths of models of dependency-aware solving, including the proposal of new, stronger models. Proof Complexity: Using the strategy extraction paradigm, we introduce new lower-bound techniques that apply to resolution-based QBF proof systems. In particular, we use the technique to prove exponential lower bounds for a new family of QBFs called the equality formulas. Our technique also affords considerably simpler, more intuitive proofs of some existing QBF proof-size lower bounds. Models of Solving: We apply our lower bound techniques to show new separations for QBF proof systems parametrised by dependency schemes. We also propose new models of dynamic dependency-aware solving and prove that they are exponentially stronger than the existing static models. Finally, we introduce Merge Resolution, a proof system modelling CDCL-style solving for DQBF, which is the first of its kind