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DOI: 10.4230/LIPIcs.MFCS.2018.78
URN: urn:nbn:de:0030-drops-96607
URL: http://drops.dagstuhl.de/opus/volltexte/2018/9660/
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Rajgopal, Ninad ; Santhanam, Rahul ; Srinivasan, Srikanth

Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds

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Abstract

We give a deterministic algorithm for counting the number of satisfying assignments of any AC^0[oplus] circuit C of size s and depth d over n variables in time 2^(n-f(n,s,d)), where f(n,s,d) = n/O(log(s))^(d-1), whenever s = 2^o(n^(1/d)). As a consequence, we get that for each d, there is a language in E^{NP} that does not have AC^0[oplus] circuits of size 2^o(n^(1/(d+1))). This is the first lower bound in E^{NP} against AC^0[oplus] circuits that beats the lower bound of 2^Omega(n^(1/2(d-1))) due to Razborov and Smolensky for large d. Both our algorithm and our lower bounds extend to AC^0[p] circuits for any prime p.

BibTeX - Entry

@InProceedings{rajgopal_et_al:LIPIcs:2018:9660,
  author =	{Ninad Rajgopal and Rahul Santhanam and Srikanth Srinivasan},
  title =	{{Deterministically Counting Satisfying Assignments for Constant-Depth Circuits with Parity Gates, with Implications for Lower Bounds}},
  booktitle =	{43rd International Symposium on Mathematical Foundations  of Computer Science (MFCS 2018)},
  pages =	{78:1--78:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-086-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{117},
  editor =	{Igor Potapov and Paul Spirakis and James Worrell},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9660},
  URN =		{urn:nbn:de:0030-drops-96607},
  doi =		{10.4230/LIPIcs.MFCS.2018.78},
  annote =	{Keywords: circuit satisfiability, circuit lower bounds, polynomial method, derandomization}
}

Keywords: circuit satisfiability, circuit lower bounds, polynomial method, derandomization
Seminar: 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)
Issue Date: 2018
Date of publication: 20.08.2018


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