Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution

Beyersdorff, Olaf; Blinkhorn, Joshua; Mahajan, Meena

doi:10.1145/3373718.3394793

Cited by 12 publications

(14 citation statements)

References 36 publications

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“…In order to make the notion of 'propositional hardness' formal, we will use a strengthening of Q-resolution that allows oracle calls. This framework was introduced in [14] and tailored towards Q-resolution in [9]. The oracle allows to collapse arbitrary propositional sub-derivations into just one inference step.…”

Section: Definition 57 ([7]mentioning

confidence: 99%

“…In fact, the lower bounds for the equality and random formulas above hold in this stronger model. Theorem 5.15 ([7,9]). Equality n requires Q NP -resolution refutations of size 2 n .…”

Section: Definition 57 ([7]mentioning

confidence: 99%

“…where u is a universal literal that in the quantifier prefix is quantified right of all variables in C, i.e., none of the literals in C depends on u. For Q-resolution we have a number of lower bounds [3,7,11] as well as lower bound techniques, some of them lifted from propositional proof complexity [12,13], but more interestingly some of them genuine to the QBF domain [7,9] that unveil deep connections between proof size and circuit complexity [10], unparalleled in the propositional domain. Unlike in the relation between SAT and CDCL, it is has been open whether QCDCL runs can be efficiently translated into Q-resolution.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, the parity formulas also exhibit 'genuine' QBF hardness, as they are hard in the NP-oracle version of Q-resolution [9]. Since they are easy for QCDCL, this means that not all genuine Q-resolution lower bounds lift to QCDCL.…”

Section: Introductionmentioning

confidence: 99%

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Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

Beyersdorff¹,

Böhm²

2021

Preprint

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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.

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Section: Definition 57 ([7]mentioning

confidence: 99%

“…In fact, the lower bounds for the equality and random formulas above hold in this stronger model. Theorem 5.15 ([7,9]). Equality n requires Q NP -resolution refutations of size 2 n .…”

Section: Definition 57 ([7]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

Beyersdorff¹,

Böhm²

2021

Preprint

Self Cite

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“…It also aims to explain the success stories of, as well as the obstacles faced by, algorithmic approaches to hard problems such as satisfiability (SAT) and Quantified Boolean Formulas (QBF) [19,29]. While propositional proof complexity, the study of proofs of unsatisfiability of propositional formulas, has been around for decades [20,27], the area of QBF proof complexity is relatively new, with theoretical studies gaining traction only in the last decade or so [2,7,10,11]. While inheriting and using a wealth of techniques from propositional proof complexity [12,14,25], QBF proof complexity has also given several new perspectives specific to QBF [5,24,35], and these perspectives and their connections to QBF solving [32,39] as well as their practical applications [34] have driven the search for newer proof systems [1,11,22,28,30].…”

Section: Introductionmentioning

confidence: 99%

Hard QBFs for Merge Resolution

Beyersdorff,

Blinkhorn,

Mahajan

et al. 2020

Preprint

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We prove the first proof size lower bounds for the proof system Merge Resolution (MRes [6]), 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 [6], 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.

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Proof Complexity of Symbolic QBF Reasoning

Mengel

Slivovsky

2021

Theory and Applications of Satisfiability Testing – SAT 2021

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Hardness Characterisations and Size-Width Lower Bounds for QBF Resolution

Cited by 12 publications

References 36 publications

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

Hard QBFs for Merge Resolution

Proof Complexity of Symbolic QBF Reasoning

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