License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.15
URN: urn:nbn:de:0030-drops-165770
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16577/
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Sokolov, Dmitry

Pseudorandom Generators, Resolution and Heavy Width

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Abstract

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^?(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.

BibTeX - Entry

@InProceedings{sokolov:LIPIcs.CCC.2022.15,
  author =	{Sokolov, Dmitry},
  title =	{{Pseudorandom Generators, Resolution and Heavy Width}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{15:1--15:22},
  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/opus/volltexte/2022/16577},
  URN =		{urn:nbn:de:0030-drops-165770},
  doi =		{10.4230/LIPIcs.CCC.2022.15},
  annote =	{Keywords: proof complexity, pseudorandom generators, resolution, lower bounds}
}

Keywords: proof complexity, pseudorandom generators, resolution, lower bounds
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022

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