QBF Proof Complexity

Quantified Boolean Formulas (QBF) and their proof complexity are not as well understood as propositional formulas, yet remain an area of interest due to their relation to QBF solving. Proof systems for QBF provide a theoretical underpinning for the performance of these solvers. We define a novel calculus IR-calc, which enables unification of the principal existing resolution-based QBF calculi and applies to the more powerful Dependency QBF (DQBF). We completely reveal the relative power of important QBF resolution systems, settling in particular the relationship between the two different types of resolution-based QBF calculi. The most challenging part of this comparison is to exhibit hard formulas that underlie the exponential separations of the proof systems. In contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. To this end we exhibit a new proof technique for showing lower bounds in QBF proof systems based on strategy extraction. We also find that the classical lower bound techniques of the prover-delayer game and feasible interpolation can be lifted to a QBF setting and provide new lower bounds. We investigate more powerful proof systems such as extended resolution and Frege systems. We define and investigate new QBF proof systems that mix propositional rules with a reduction rule, we find the strategy extraction technique also works and directly lifts lower bounds from circuit complexity. Such a direct transfer from circuit to proof complexity lower bounds has often been postulated, but had not been formally established for propositional proof systems prior to this work. This leads to strong lower bounds for restricted versions of QBF Frege, in particular an exponential lower bound for QBF Frege systems operating with AC0[p] circuits. In contrast, any non-trivial lower bound for propositional AC0[p]-Frege constitutes a major open problem

Proof complexity of resolution-based QBF calculi

Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of important QBF solvers. However, the proof complexity of these proof systems is currently not well understood and in particular lower bound techniques are missing. In this paper we exhibit a new and elegant proof technique for showing lower bounds in QBF proof systems based on strategy extraction. This technique provides a direct transfer of circuit lower bounds to lengths of proofs lower bounds. We use our method to show the hardness of a natural class of parity formulas for Q-resolution and universal Q-resolution. Variants of the formulas are hard for even stronger systems as long-distance Q-resolution and extensions. With a completely different lower bound argument we show the hardness of the prominent formulas of Kleine Büning et al. [34] for the strong expansion-based calculus IR-calc. Our lower bounds imply new exponential separations between two different types of resolution-based QBF calculi: proof systems for CDCLbased solvers (Q-resolution, long-distance Q-resolution) and proof systems for expansion-based solvers (∀Exp+Res and its generalizations IR-calc and IRM-calc). The relations between proof systems from the two different classes were not known before

Building Strategies into QBF Proofs

Strategy extraction is of great importance for quantified Boolean formulas (QBF), both in solving and proof complexity. So far in the QBF literature, strategy extraction has been algorithmically performed from proofs. Here we devise the first QBF system where (partial) strategies are built into the proof and are piecewise constructed by simple operations along with the derivation. This has several advantages: (1) lines of our calculus have a clear semantic meaning as they are accompanied by semantic objects; (2) partial strategies are represented succinctly (in contrast to some previous approaches); (3) our calculus has strategy extraction by design; and (4) the partial strategies allow new sound inference steps which are disallowed in previous central QBF calculi such as Q-Resolution and long-distance Q-Resolution. The last item (4) allows us to show an exponential separation between our new system and the previously studied reductionless long-distance resolution calculus. Our approach also naturally lifts to dependency QBFs (DQBF), where it yields the first sound and complete CDCL-style calculus for DQBF, thus opening future avenues into CDCL-based DQBF solving

Lifting QBF Resolution Calculi to DQBF

We examine the existing resolution systems for quantified Boolean formulas (QBF) and answer the question which of these calculi can be lifted to the more powerful Dependency QBFs (DQBF). An interesting picture emerges: While for QBF we have the strict chain of proof systems Q-Res < IR-calc < IRM-calc, the situation is quite different in DQBF. Q-Res and likewise universal resolution are too weak: they are not complete. IR-calc has the right strength: it is sound and complete. IRM-calc is too strong: it is not sound any more, and the same applies to long-distance resolution. Conceptually, we use the relation of DQBF to EPR and explain our new DQBF calculus based on IR-calc as a subsystem of first-order resolutio

On unification of QBF resolution-based calculi

Several calculi for quantified Boolean formulas (QBFs) exist, but relations between them are not yet fully understood. This paper defines a novel calculus, which is resolution-based and enables unification of the principal existing resolution-based QBF calculi, namely Q-resolution, long-distance Q-resolution and the expansion-based calculus Exp+Res. All these calculi play an important role in QBF solving. This paper shows simulation results for the new calculus and some of its variants. Further, we demonstrate how to obtain winning strategies for the universal player from proofs in the calculus. We believe that this new proof system provides an underpinning necessary for formal analysis of modern QBF solvers. © 2014 Springer-Verlag Berlin Heidelberg

Feasible Interpolation for QBF Resolution Calculi

In sharp contrast to classical proof complexity we are currently short of lower bound techniques for QBF proof systems. In this paper we establish the feasible interpolation technique for all resolution-based QBF systems, whether modelling CDCL or expansion-based solving. This both provides the first general lower bound method for QBF proof systems as well as largely extends the scope of classical feasible interpolation. We apply our technique to obtain new exponential lower bounds to all resolution-based QBF systems for a new class of QBF formulas based on the clique problem. Finally, we show how feasible interpolation relates to the recently established lower bound method based on strategy extraction