We describe a procedure that extracts existential strategies from verification proofs in the Long-Distance Consensus (i.e., Term Resolution) proof system when augmented with dependency schemes. We prove that when the standard dependency scheme 𝙳^std is used, the extracted strategies are winning strategies, thus establishing soundness of the proof system LDQ(D^std)-Consensus. We show through a counterexample that this approach fails to show soundness for LDQ(D^rrs)-Consensus.

Quantified Conflict Driven Clause Leaning (QCDCL) is one of the main approaches to solving Quantified Boolean Formulas (QBF). Cube-learning is employed in this approach to ensure that true formulas can be verified. Dependency Schemes help to detect spurious dependencies that are implied by the variable ordering in the quantifier prefix of QBFs but are not essential for constructing (counter)models. This detection can provably shorten refutations in specific proof systems, and is expected to speed up runs of QBF solvers. The simplest underlying proof system [BeyersdorffBöhm-LMCS2023], formalises the reasoning in the QCDCL approach on false formulas, when neither cube-learning nor dependency schemes is used. The work of [BöhmPeitlBeyersdorff-AI2024] further incorporates cube-learning. The work of [ChoudhuryMahajan-JAR2024] incorporates a limited use of dependency schemes, but without cube-learning. In this work, proof systems underlying the reasoning of QCDCL solvers which use cube learning, and which use dependency schemes at all stages, are formalised. Sufficient conditions for soundness and completeness are presented, and it is shown that using the standard and reflexive resolution path dependency schemes (𝙳^{std} and 𝙳^{rrs}) to relax the decision order provably shortens refutations. When the decisions are restricted to follow quantification order, but dependency schemes are used in propagation and learning, in conjunction with cube-learning, the resulting proof systems using the dependency schemes 𝙳^{std} and 𝙳^{rrs} are investigated in detail and their relative strengths are analysed.

Abstract In Quantified Boolean Formulas QBFs, dependency schemes help to detect spurious or superfluous dependencies that are implied by the variable ordering in the quantifier prefix but are not essential for constructing countermodels. This detection can provably shorten refutations in specific proof systems, and is expected to speed up runs of QBF solvers. The proof system $$\texttt{QCDCL}$$ QCDCL recently defined by Beyersdorff and Boehm (LMCS 2023) abstracts the reasoning employed by QBF solvers based on conflict-driven clause-learning (CDCL) techniques. We show how to incorporate the use of dependency schemes into this proof system, either in a preprocessing phase, or in the propagations and clause learning, or both. We then show that when the reflexive resolution path dependency scheme $$\texttt{D}^{\texttt{rrs}}$$ D rrs is used, a mixed picture emerges: the proof systems that add $$\texttt{D}^{\texttt{rrs}}$$ D rrs to $$\texttt{QCDCL}$$ QCDCL in these three ways are not only incomparable with each other, but are also incomparable with the basic $$\texttt{QCDCL}$$ QCDCL proof system that does not use $$\texttt{D}^{\texttt{rrs}}$$ D rrs at all, as well as with several other resolution-based QBF proof systems. A notable fact is that all our separations are achieved through QBFs with bounded quantifier alternation.