Why are Proof Complexity Lower Bounds Hard?

Pich, Ján; Santhanam, Rahul

doi:10.1109/focs.2019.00080

Cited by 10 publications

(16 citation statements)

References 36 publications

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“…Then the sequence of formulas expressing that VNP = VP is a fixed point for lb 2 R in the sense that it preserves truth when R is IPS: if VNP = VP holds then lb R (lb R (VNP = VP, m ω(1) ), m ω(1) ) holds. Indeed, our diagonalisation approach is inspired partly by Atserias and Müller, and Garlik [AM20, Gar19], who showed implicitly that every sequence of formulas is such a fixed point for lb 2 R when R is Resolution, and by [PS19] who showed implicitly that for every strong enough (nonuniform) propositional proof system R (simulating Extended Frege), the distribution of random truth table formulas is a fixed point for lb 2 R . Here we explore the idea that iterating lb R provides an explicit hard sequence of formulas for R. Assume that R is not polynomially bounded, and let φ be a fixed formula that does not have |φ| c size R-proofs, for some constant c.…”

Section: Iterated Lower Bounds Formulasmentioning

confidence: 99%

“…We are also inspired by recent work of [AM20,PS19]. Atserias and Muller [AM20] settle the long-standing open problem of whether Resolution is automatable (assuming P = NP) by giving a reduction from SAT to proof complexity lower bounds for Resolution via variants of the proof complexity lower bound formulas.…”

Section: Related Workmentioning

confidence: 99%

“…Their reduction relies on the hardness of the Pigeonhole Principle, which only holds for weak proof systems such as Resolution, while we are interested here in strong proof systems such as IPS. Pich and Santhanam [PS19] unconditionally establish a version of the Natural Proofs barrier [RR97] in proof complexity, but they do so for a proof system that is defined non-constructively and moreover for instances which are randomly generated. In contrast, we are interested here in the meta-mathematics of proof complexity for well-studied and concrete proof systems such as IPS and for explicit instances.…”

Section: Related Workmentioning

confidence: 99%

“…Grochow and Pitassi [GP18] show that super-polynomial lower bounds for CNFs in the algebraic proof system IPS imply VNP = VP; thus if we believe VNP = VP is hard to show, then we must believe the same for IPS lower bounds. In a recent work, [PS19] take a different approach, formulating an analogue of the natural proofs barrier for proof complexity. They show unconditionally that for some (non-uniform) propositional proof system R, super-polynomial lower bounds on R-proofs for random truth table formulas cannot be shown efficiently in any non-uniform propositional proof system.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Iterated Lower Bounds Formulasmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Self Cite

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“…This question can be made precise by asking about the computational complexity of MCSP. (See also [55] for a different approach. )…”

Section: Meta-complexity Mcsp and Kolmogorov Complexitymentioning

confidence: 99%

Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

Allender

2021

NZ J Math

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Information in Propositional Proofs and Algorithmic Proof Search

Krajíček¹

2021

J. symb. log.

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We study from the proof complexity perspective the (informal) proof search problem (cf. [17, Sections 1.5 and 21.5]): • Is there an optimal way to search for propositional proofs? We note that, as a consequence of Levin’s universal search, for any fixed proof system there exists a time-optimal proof search algorithm. Using classical proof complexity results about reflection principles we prove that a time-optimal proof search algorithm exists without restricting proof systems iff a p-optimal proof system exists. To characterize precisely the time proof search algorithms need for individual formulas we introduce a new proof complexity measure based on algorithmic information concepts. In particular, to a proof system P we attach information-efficiency function $i_P(\tau )$ assigning to a tautology a natural number, and we show that: • $i_P(\tau )$ characterizes time any P-proof search algorithm has to use on $\tau $ , • for a fixed P there is such an information-optimal algorithm (informally: it finds proofs of minimal information content), • a proof system is information-efficiency optimal (its information-efficiency function is minimal up to a multiplicative constant) iff it is p-optimal, • for non-automatizable systems P there are formulas $\tau $ with short proofs but having large information measure $i_P(\tau )$ . We isolate and motivate the problem to establish unconditional super-logarithmic lower bounds for $i_P(\tau )$ where no super-polynomial size lower bounds are known. We also point out connections of the new measure with some topics in proof complexity other than proof search.

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Why are Proof Complexity Lower Bounds Hard?

Cited by 10 publications

References 36 publications

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

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

Vaughan Jones, Kolmogorov Complexity, and the New Complexity Landscape around Circuit Minimization

Information in Propositional Proofs and Algorithmic Proof Search

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