Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Buss, Samuel R.; Kołodziejczyk, Leszek Aleksander; Zdanowski, Konrad

doi:10.1090/s0002-9947-2015-06233-3

Cited by 46 publications

(15 citation statements)

References 39 publications

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“…Leszek Kołodziejczyk pursued an original path of research in the model theory of arithmetics in connection with complexity theory; let us mention his recent work (with co-authors) on proof complexity [32]. Mikołaj Bojańczyk initiated successful new research in Automata Theory, in connection with logic and database theory.…”

Section: The Limits Of Automata Theorymentioning

confidence: 99%

Contribution of Warsaw Logicians to Computational Logic

Niwiński

2016

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Abstract:The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt's dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal µ-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the µ-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations. Later, Bojanczyk opened a new chapter by introducing an unboundedness quantifier, which allowed for expressing some quantitative properties of programs. Yet another topic, linking the past with the future, is the subject of automata founded in the Fraenkel-Mostowski set theory. The studies on intuitionism found their continuation in the studies of Curry-Howard isomorphism. Łukasiewicz's landmark idea of many-valued logic found its continuation in various approaches to incompleteness and uncertainty.

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Section: The Limits Of Automata Theorymentioning

confidence: 99%

Contribution of Warsaw Logicians to Computational Logic

Niwiński

2016

Axioms

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“…Complexity of counting. In [5], Buss, Ko lodziejczyk and Zdanowski derived Toda's theorem in an extension of the theory AP C 2 .…”

Section: Cryptography Recently Dai Tri Man Lementioning

confidence: 99%

“…For a fixed prime p ≥ 2, they denote by C k p for k ∈ [p] quantifiers counting mod p. The intended meaning of C k p x ≤ tA(x) is that the number of values x ≤ t for which A is true is congruent to k mod p. See [5] for the explicit list of axioms defining C k p .…”

Section: Cryptography Recently Dai Tri Man Lementioning

confidence: 99%

“…Theory Theorem Reference P V 1 Cook-Levin's theorem Section 4 (n, d, λ)-graphs Section 6 the PCP theorem Section 7 P V 1 + W P HP (P V 1 ) PARITY / ∈ AC 0 [18] AP C 1 BPP, ZPP, AM,... [15] Goldreich-Levin's theorem [11] the exponential PCP theorem Section 5 HARD ǫ Impagliazzo-Wigderson's derandom. [ Toda's theorem [5] The theories are listed from the weakest to the strongest one.…”

Section: Introductionmentioning

confidence: 99%

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Logical strength of complexity theory and a formalization of the PCP theorem in bounded arithmetic

Pich¹

2015

Logical Methods in Computer Science

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We present several known formalizations of theorems from computational complexity in bounded arithmetic and formalize the PCP theorem in the theory P V1 (no formalization of this theorem was known). This includes a formalization of the existence and of some properties of the (n, d, λ)-graphs in P V1. 2012 ACM CCS: [Theory of computation]: Computational complexity and cryptography-Complexity theory and logic / Proof complexity.

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“…The lower bound problem for R(LIN/F 2 ) seems interesting also because the top proof system is logical. Note that Buss, Kolodziejczyk and Zdanowski [6] proved that, in fact, the AC 0 [p]-Frege system collapses (with a quasi-polynomial blow-up in proof size) to a proof system operating with clauses of conjunctions of low degree polynomials.…”

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confidence: 99%

Randomized feasible interpolation and monotone circuits with a local oracle

Krajíček

2018

J. Math. Log.

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The feasible interpolation theorem for semantic derivations from K. (1997) [16] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two NP sets U and V a small communication protocol (a general dag-like protocol in the sense of K. (1997) [16]) computing the Karchmer-Wigderson multi-function KW [U, V ] associated with the sets, and such a protocol further yields a small circuit separating U from V . When U is closed upwards the protocol computes the monotone Karchmer-Wigderson multi-function KW m [U, V ] and the resulting circuit is monotone. K. (1998) [18] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP).In this paper we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for KW m [U, V ] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of U, V , U closed upwards, yields a small randomized protocol for KW m [U, V ] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system R(LIN/F2) operating with clauses formed by linear Boolean functions over F2. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of K. (1998) [17] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [5]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.Consider a propositional proof system R(LIN/F 2 ) that operates with clauses of linear equations over F 2 and combines the rules of both resolution and linear equational calculus. A line C in a proof has the form {f 1 , . . . , f k }

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Collapsing modular counting in bounded arithmetic and constant depth propositional proofs

Cited by 46 publications

References 39 publications

Contribution of Warsaw Logicians to Computational Logic

Contribution of Warsaw Logicians to Computational Logic

Logical strength of complexity theory and a formalization of the PCP theorem in bounded arithmetic

Randomized feasible interpolation and monotone circuits with a local oracle

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