Proof Complexity

Krajíček, Jan

doi:10.1017/9781108242066

Cited by 41 publications

(66 citation statements)

References 267 publications

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“…In particular, formalizations of complexity-theoretic concepts, constructions and results started with Paris, Wilkie and Buss. We refer to [28,Section 22.5] for a robust list of references. These results involve circuit complexity but at that time there were no circuit lower bounds to consider.…”

Section: Related Workmentioning

confidence: 99%

“…We thus originally hoped that KPT interpolation could be used to derive lower bounds for some proof systems for which feasible interpolation fails. This would present an alternative to a similar weakening of feasible interpolation, the so called feasible disjunction property [28,Problem 17.9.1]. However, in light of the subsequent work of Krajíček [29] this possibility seems less likely.…”

Section: Kpt-interpolation Propertymentioning

confidence: 99%

“…Since S 1 2 is Σ b 1 conservative over PV 1 , our proof can be implemented already in PV 1 . The 'moreover' part is proved similarly by inspecting the proof of [28,Theorem 5.6.1] or [41,Theorem 2].…”

Section: Appendixmentioning

confidence: 99%

See 2 more Smart Citations

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Pich

Santhanam

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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Section: Related Workmentioning

confidence: 99%

Section: Kpt-interpolation Propertymentioning

confidence: 99%

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Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Pich

Santhanam

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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“…For basic definitions on propositional proof complexity, please refer to the comprehensive overview of Krajicek [Kra19].…”

Section: Iterated Lower Bound Formulasmentioning

confidence: 99%

Iterated lower bound formulas: a diagonalization-based approach to proof complexity

Santhanam

Tzameret

2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally.We give an explicit sequence of CNF formulas {φ n } such that VNP = VP iff there are no polynomial-size IPS proofs for the formulas φ n . This provides the first natural equivalence between proof complexity lower bounds and standard algebraic complexity lower bounds. Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas φ n themselves assert the non-existence of short IPS proofs for formulas encoding VNP = VP at a different input length. Our result also has meta-mathematical implications: it gives evidence for the difficulty of proving strong lower bounds for IPS within IPS.More generally, for any strong enough propositional proof system R we propose a new explicit hard candidate, the iterated R-lower bound formulas, which inductively asserts the non-existence of short R proofs for formulas encoding this same statement at a different input length. We show that these formulas are unconditionally hard for Resolution following recent results of Atserias and Müller and of Garlik. We further give evidence in favour of this hypothesis for other proof systems.

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“…The modus ponens is defined as: if A → B and A are present in the hypothesis then B can be logically implied by the hypothesis. For a detailed definition and explanation refer [20].…”

Section: Efrege+∀redmentioning

confidence: 99%

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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Proof Complexity

Cited by 41 publications

References 267 publications

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Strong co-nondeterministic lower bounds for NP cannot be proved feasibly

Iterated lower bound formulas: a diagonalization-based approach to proof complexity

Extending Merge Resolution to a Family of Proof Systems

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