Nisan-Wigderson Generators in Proof Complexity: New Lower Bounds

Author Erfan Khaniki

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


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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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)


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].

Subject Classification

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


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