Pseudorandom Generators, Resolution and Heavy Width

Author Dmitry Sokolov

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Dmitry Sokolov
  • St. Petersburg State University, Russia
  • St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences, Russia


I would like to thank Anastasia Sofronova, Edward Hirsch for fruitful discussions and attempts to fix my writing; anonymous reviewers for improving the text; Alexander Razborov and anonymous reviewers for pointing out incorrect operations with "Canonical Form" in an earlier draft of the paper. The work was supported by the Theoretical Physics and Mathematics Advancement Foundation "BASIS".

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Dmitry Sokolov. Pseudorandom Generators, Resolution and Heavy Width. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


Following the paper of Alekhnovich, Ben-Sasson, Razborov, Wigderson [Michael Alekhnovich et al., 2004] we call a pseudorandom generator ℱ:{0, 1}ⁿ → {0, 1}^m hard for a propositional proof system P if P cannot efficiently prove the (properly encoded) statement b ∉ Im(ℱ) for any string b ∈ {0, 1}^m. In [Michael Alekhnovich et al., 2004] the authors suggested the "functional encoding" of the considered statement for Nisan-Wigderson generator that allows the introduction of "local" extension variables. These extension variables may potentially significantly increase the power of the proof system. In [Michael Alekhnovich et al., 2004] authors gave a lower bound of exp[Ω(n²/{m⋅2^{2^Δ}})] on the length of Resolution proofs where Δ is the degree of the dependency graph of the generator. This lower bound meets the barrier for the restriction technique. In this paper, we introduce a "heavy width" measure for Resolution that allows us to show a lower bound of exp[n²/{m 2^𝒪(εΔ)}] on the length of Resolution proofs of the considered statement for the Nisan-Wigderson generator. This gives an exponential lower bound up to Δ := log^{2 - δ} n (the bigger degree the more extension variables we can use). In [Michael Alekhnovich et al., 2004] authors left an open problem to get rid of scaling factor 2^{2^Δ}, it is a solution to this open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • proof complexity
  • pseudorandom generators
  • resolution
  • lower bounds


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