Building Strategies into QBF Proofs

Beyersdorff, Olaf; Blinkhorn, Joshua; Mahajan, Meena

doi:10.1007/s10817-020-09560-1

Cited by 12 publications

(52 citation statements)

References 52 publications

(86 reference statements)

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“…MRes is a proof system for false QBFs introduced in [4]. We describe MRes briefly in this section, please refer to [4] for its formal definition.…”

Section: Mresmentioning

confidence: 99%

“…We define a family of proof systems MRes-R, inspired from the MRes proof system. In MRes ( [4]), strategies are built within the proof and are represented by merge maps. We observed that merge maps or any specific representations of strategies are not important for the soundness or completeness of the proof system.…”

Section: Completion Principle Formulas [17]mentioning

confidence: 99%

“…A new proof system Merge resolution (MRes) [4] has been developed recently. It follows a different QBF-solving approach.…”

Section: Introductionmentioning

confidence: 99%

“…It follows a different QBF-solving approach. In MRes, winning strategies for the universal player are explicitly represented within the proof in the form of deterministic branching programs, known as merge maps [4]. MRes builds partial strategies at each line of the proof such that the strategy at the last line (corresponding to the empty clause) forms the complete countermodel for the input QBF.…”

Section: Introductionmentioning

confidence: 99%

“…Introducing a new family of proof systems MRes-R: MRes [4] uses merge maps to store the countermodels within proofs. We observe that merge maps are not important for the soundness and completeness of the proof system.…”

mentioning

confidence: 99%

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Extending Merge Resolution to a Family of Proof Systems

Chede¹,

Shukla²

2021

Preprint

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Merge Resolution (MRes [4]) is a recently introduced proof system for false QBFs. Unlike other known QBF proof systems, it builds winning strategies for the universal player within the proofs. Every line of this proof system consists of existential clauses along with countermodels. MRes stores the countermodels as merge maps. Merge maps are deterministic branching programs in which isomorphism checking is efficient as a result MRes is a polynomial time verifiable proof system. In this paper, we introduce a family of proof systems MRes-R in which, the information of countermodels are stored in any pre-fixed complete representation R, instead of merge maps. Hence corresponding to each possible complete representation R, we have a sound and refutationally complete QBF-proof system in MRes-R. To handle arbitrary representations for the strategies, we introduce consistency checking rules in MRes-R instead of isomorphism checking in MRes. As a result these proof systems are not polynomial time verifiable. Consequently, the paper shows that using merge maps is too restrictive and can be replaced with arbitrary representations leading to several interesting proof systems. The paper also studies proof theoretic properties of the family of new proof systems MRes-R. We show that eFrege+∀red simulates all valid refutations from proof systems in MRes-R. Since proof systems in MRes-R may use arbitrary representations, in order to simulate them, we first represent the steps used by the proof systems as a new simple complete structure. As a consequence, the corresponding proof system belonging to MRes-R is able to simulate all proof systems in MRes-R. Finally, we simulate this proof system via eFrege+∀red using the ideas from [10]. On the lower bound side, we lift the lower bound result of regular MRes ([5]) for all regular proof systems in MRes-R. To be precise, we show that the completion principle formulas from [17] which are shown to be hard for regular MRes in [5], are also hard for any regular proof system in MRes-R. Thereby, the paper lifts the lower bound of regular MRes to an entire class of proof systems, which use some complete representation, including those undiscovered, instead of merge maps.

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“…MRes is a proof system for false QBFs introduced in [4]. We describe MRes briefly in this section, please refer to [4] for its formal definition.…”

Section: Mresmentioning

confidence: 99%

Section: Completion Principle Formulas [17]mentioning

confidence: 99%

“…A new proof system Merge resolution (MRes) [4] has been developed recently. It follows a different QBF-solving approach.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning