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Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds

Author Erfan Khaniki

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LIPIcs.CCC.2022.17.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Theory of computation → Complexity theory and logic
Keywords
  • Proof complexity
  • Bounded arithmetic
  • Bounded depth Frege
  • Nisan-Wigderson generators
  • Meta-complexity
  • Lower bounds

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Abstract

A map g:{0,1}ⁿ → {0,1}^m (m > n) is a hard proof complexity generator for a proof system P iff for every string b ∈ {0,1}^m ⧵ Rng(g), formula τ_b(g) naturally expressing b ∉ Rng(g) requires superpolynomial size P-proofs. One of the well-studied maps in the theory of proof complexity generators is Nisan-Wigderson generator. Razborov [A. A. {Razborov}, 2015] conjectured that if A is a suitable matrix and f is a NP∩CoNP function hard-on-average for 𝖯/poly, then NW_{f, A} is a hard proof complexity generator for Extended Frege. In this paper, we prove a form of Razborov’s conjecture for AC⁰-Frege. We show that for any symmetric NP∩CoNP function f that is exponentially hard for depth two AC⁰ circuits, NW_{f,A} is a hard proof complexity generator for AC⁰-Frege in a natural setting. As direct applications of this theorem, we show that:  
1) For any f with the specified properties, τ_b(NW_{f,A}) (for a natural formalization) based on a random b and a random matrix A with probability 1-o(1) is a tautology and requires superpolynomial (or even exponential) AC⁰-Frege proofs. 
2) Certain formalizations of the principle f_n ∉ (NP∩CoNP)/poly requires superpolynomial AC⁰-Frege proofs.  These applications relate to two questions that were asked by Krajíček [J. {Krajíček}, 2019].

Cite As Get BibTex

Erfan Khaniki. Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.CCC.2022.17

Author Details

Erfan Khaniki
  • Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
  • Institute of Mathematics of the Czech Academy of Sciences, Prague, Czech Republic

Funding

This work was supported by the GACR grant 19-27871X, the institute grant RVO: 67985840, and the Specific university research project SVV-2020-260589. Part of this work was done while the author was participating in the program Satisfiability: Theory, Practice, and Beyond at the Simons Institute for the Theory of Computing.

Acknowledgements

We are grateful to Jan Bydžovský, Susanna de Rezende, Emil Jeř{á}bek, Jan Krajíček, Jan Pich and Pavel Pudlák for their different forms of help in different stages of this work. We are also indebted to anonymous referees for their helpful suggestions, which led to a better presentation of the paper.

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