2 Search Results for "Ghasemloo, Kaveh"


Document
Extended Abstract
Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract)

Authors: Yuval Filmus, Or Meir, and Avishay Tal

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)


Abstract
Håstad showed that any De Morgan formula (composed of AND, OR and NOT gates) shrinks by a factor of O(p²) under a random restriction that leaves each variable alive independently with probability p [SICOMP, 1998]. Using this result, he gave an Ω̃(n³) formula size lower bound for the Andreev function, which, up to lower order improvements, remains the state-of-the-art lower bound for any explicit function. In this work, we extend the shrinkage result of Håstad to hold under a far wider family of random restrictions and their generalization - random projections. Based on our shrinkage results, we obtain an Ω̃(n³) formula size lower bound for an explicit function computed in AC⁰. This improves upon the best known formula size lower bounds for AC⁰, that were only quadratic prior to our work. In addition, we prove that the KRW conjecture [Karchmer et al., Computational Complexity 5(3/4), 1995] holds for inner functions for which the unweighted quantum adversary bound is tight. In particular, this holds for inner functions with a tight Khrapchenko bound. Our random projections are tailor-made to the function’s structure so that the function maintains structure even under projection - using such projections is necessary, as standard random restrictions simplify AC⁰ circuits. In contrast, we show that any De Morgan formula shrinks by a quadratic factor under our random projections, allowing us to prove the cubic lower bound. Our proof techniques build on the proof of Håstad for the simpler case of balanced formulas. This allows for a significantly simpler proof at the cost of slightly worse parameters. As such, when specialized to the case of p-random restrictions, our proof can be used as an exposition of Håstad’s result.

Cite as

Yuval Filmus, Or Meir, and Avishay Tal. Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 89:1-89:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


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@InProceedings{filmus_et_al:LIPIcs.ITCS.2021.89,
  author =	{Filmus, Yuval and Meir, Or and Tal, Avishay},
  title =	{{Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{89:1--89:7},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.89},
  URN =		{urn:nbn:de:0030-drops-136281},
  doi =		{10.4230/LIPIcs.ITCS.2021.89},
  annote =	{Keywords: De Morgan formulas, KRW Conjecture, shrinkage, random restrictions, random projections, bounded depth circuits, constant depth circuits, formula complexity}
}
Document
Theories for Subexponential-size Bounded-depth Frege Proofs

Authors: Kaveh Ghasemloo and Stephen A. Cook

Published in: LIPIcs, Volume 23, Computer Science Logic 2013 (CSL 2013)


Abstract
This paper is a contribution to our understanding of the relationship between uniform and nonuniform proof complexity. The latter studies the lengths of proofs in various propositional proof systems such as Frege and bounded-depth Frege systems, and the former studies the strength of the corresponding logical theories such as VNC1 and V0 in [Cook/Nguyen, 2010]. A superpolynomial lower bound on the length of proofs in a propositional proof system for a family of tautologies expressing a result like the pigeonhole principle implies that the result is not provable in the theory associated with the propositional proof system. We define a new class of bounded arithmetic theories n^epsilon-ioV^\infinity for epsilon < 1 and show that they correspond to complexity classes AltTime(O(1),O(n^epsilon)), uniform classes of subexponential-size bounded-depth circuits DepthSize(O(1),2^O(n^epsilon)). To accomplish this we introduce the novel idea of using types to control the amount of composition in our bounded arithmetic theories. This allows our theories to capture complexity classes that have weaker closure properties and are not closed under composition. We show that the proofs of Sigma^B_0-theorems in our theories translate to subexponential-size bounded-depth Frege proofs. We use these theories to formalize the complexity theory result that problems in uniform NC1 circuits can be computed by uniform subexponential bounded-depth circuits in [Allender/Koucky, 2010]. We prove that our theories contain a variation of the theory VNC1 for the complexity class NC1. We formalize Buss's proof in [Buss, 1993] that the (unbalanced) Boolean Formula Evaluation problem is in NC1 and use it to prove the soundness of Frege systems. As a corollary, we obtain an alternative proof of [Filmus et al, ICALP, 2011] that polynomial-size Frege proofs can be simulated by subexponential-size bounded-depth Frege proofs. Our results can be extended to theories corresponding to other nice complexity classes inside NTimeSpace(n^O(1), n^o(1)) such as NL. This is achieved by essentially formalizing the containment NTimeSpace(n^O(1), n^o(1)) \subseteq AltTime(O(1), O(n^epsilon)) for all epsilon > 0.

Cite as

Kaveh Ghasemloo and Stephen A. Cook. Theories for Subexponential-size Bounded-depth Frege Proofs. In Computer Science Logic 2013 (CSL 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 23, pp. 296-315, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)


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@InProceedings{ghasemloo_et_al:LIPIcs.CSL.2013.296,
  author =	{Ghasemloo, Kaveh and Cook, Stephen A.},
  title =	{{Theories for Subexponential-size Bounded-depth Frege Proofs}},
  booktitle =	{Computer Science Logic 2013 (CSL 2013)},
  pages =	{296--315},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-60-6},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{23},
  editor =	{Ronchi Della Rocca, Simona},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2013.296},
  URN =		{urn:nbn:de:0030-drops-42044},
  doi =		{10.4230/LIPIcs.CSL.2013.296},
  annote =	{Keywords: Computational Complexity Theory, Proof Complexity, Bounded Arithmetic, NC1-Frege, AC0- Frege}
}
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