Nisan-Wigderson generators in proof complexity: New lower bounds

0559510 - MÚ 2023 RIV DE eng C - Konferenční příspěvek (zahraniční konf.)
Khaniki, Erfan
Nisan-Wigderson generators in proof complexity: New lower bounds.
37th Computational Complexity Conference (CCC 2022). Dagstuhl: Schloss Dagstuhl, Leibniz-Zentrum für Informatik, 2022 - (Lovett, S.), s. 1-15, č. článku 17. Leibniz International Proceedings in Informatics, 234. ISBN 978-3-95977-241-9. ISSN 1868-8969.
[37th Computational Complexity Conference (CCC 2022). Philadelphia (US), 20.07.2022-23.07.2022]
Grant CEP: GA ČR(CZ) GX19-27871X
Institucionální podpora: RVO:67985840
Klíčová slova: proof complexity * bounded arithmetic * bounded depth Frege * Nisan-Wigderson generators
Obor OECD: Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
https://dx.doi.org/10.4230/LIPIcs.CCC.2022.17

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].
Trvalý link: https://hdl.handle.net/11104/0332787