Abstract. 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 Qresolution, 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.

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.

The question whether a set of formulae Γ implies a formula ϕ is fundamental. The present paper studies the complexity of the above implication problem for propositional formulae that are built from a systematically restricted set of Boolean connectives. We give a complete complexity-theoretic classification for all sets of Boolean functions in the meaning of Post's lattice and show that the implication problem is efficiently solvable only if the connectives are definable using the constants {0, 1} and only one of {∧, ∨, ⊕}. The problem remains coNP-complete in all other cases. We also consider the restriction of Γ to singletons which makes the problem strictly easier in some cases.

A general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [2007]. There the authors show important results on tree-like Parameterized Resolution-a parameterized version of classical Resolution-and their gap complexity theorem implies lower bounds for that system.The main result of the present paper significantly improves upon this by showing optimal lower bounds for a parameterized version of bounded-depth Frege. More precisely, we prove that the pigeonhole principle requires proofs of size n Ω(k) in parameterized bounded-depth Frege, and, as a special case, in dag-like Parameterized Resolution. This answers an open question posed in [Dantchev et al. 2007]. In the opposite direction, we interpret a well-known technique for FPT algorithms as a DPLL procedure for Parameterized Resolution. Its generalization leads to a proof search algorithm for Parameterized Resolution that in particular shows that tree-like Parameterized Resolution allows short refutations of all parameterized contradictions given as bounded-width CNFs.

Modern QBF solvers typically use two different paradigms, conflict-driven clause learning (CDCL) solving or expansion solving. Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of these solvers, with Q-Resolution and its extensions relating to CDCL solving and ∀Exp+Res relating to expansion solving. This article defines two novel calculi, which are resolution-based and enable unification of some of the principal existing resolution-based QBF calculi, namely Q-resolution, long-distance Q-resolution and the expansion-based calculus ∀Exp+Res. However, the proof complexity of the QBF resolution proof systems is currently not well understood. In this article, we completely determine the relative power of the main QBF resolution systems, settling in particular the relationship between the two different types of resolution-based QBF calculi: proof systems for CDCL-based solvers (Q-resolution, universal, and long-distance Q-resolution) and proof systems for expansion-based solvers (∀Exp+Res and its generalizations IR-calc and IRM-calc defined here). The most challenging part of this comparison is to exhibit hard formulas that underlie the exponential separations of the aforementioned proof systems. To this end, 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 such as long-distance Q-resolution and extensions. With a completely different and novel counting argument, we show the hardness of the prominent formulas of Kleine Büning et al. [51] for the strong expansion-based calculus IR-calc.