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Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds

Authors: Erfan Khaniki

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

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

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)

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  author =	{Khaniki, Erfan},
  title =	{{Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.17},
  URN =		{urn:nbn:de:0030-drops-165799},
  doi =		{10.4230/LIPIcs.CCC.2022.17},
  annote =	{Keywords: Proof complexity, Bounded arithmetic, Bounded depth Frege, Nisan-Wigderson generators, Meta-complexity, Lower bounds}
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