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DOI: 10.4230/LIPIcs.STACS.2012.519
URN: urn:nbn:de:0030-drops-34307
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2012/3430/
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Jansen, Maurice ; Santhanam, Rahul

Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes

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Abstract

We associate to each Boolean language complexity class C the algebraic class a.C consisting of families of polynomials {f_n} for which the evaluation problem over the integers is in C. We prove the following lower bound and randomness-to-hardness results:

1. If polynomial identity testing (PIT) is in NSUBEXP then a.NEXP does not have poly size constant-free arithmetic circuits.
2. a.NEXP^RP does not have poly size constant-free arithmetic circuits.
3. For every fixed k, a.MA does not have arithmetic circuits of size n^k.

Items 1 and 2 strengthen two results due to (Kabanets and Impagliazzo, 2004). The third item improves a lower bound due to (Santhanam, 2009).

We consider the special case low-PIT of identity testing for (constant-free) arithmetic circuits with low formal degree, and give improved hardness-to-randomness trade-offs that apply to this case.

Combining our results for both directions of the hardness-randomness connection, we demonstrate a case where derandomization of PIT and proving lower bounds are equivalent. Namely, we show that low-PIT is in i.o-NTIME[2^{n^{o(1)}}]/n^{o(1)} if and only if there exists a family of multilinear polynomials in a.NE/lin that requires constant-free arithmetic circuits of super-polynomial size and formal degree.

BibTeX - Entry

@InProceedings{jansen_et_al:LIPIcs:2012:3430,
  author =	{Maurice Jansen and Rahul Santhanam},
  title =	{{Stronger Lower Bounds and Randomness-Hardness Trade-Offs Using Associated Algebraic Complexity Classes}},
  booktitle =	{29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)},
  pages =	{519--530},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-35-4},
  ISSN =	{1868-8969},
  year =	{2012},
  volume =	{14},
  editor =	{Christoph D{\"u}rr and Thomas Wilke},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2012/3430},
  URN =		{urn:nbn:de:0030-drops-34307},
  doi =		{10.4230/LIPIcs.STACS.2012.519},
  annote =	{Keywords: Computational Complexity, Circuit Lower Bounds, Polynomial Identity Testing, Derandomization}
}

Keywords: Computational Complexity, Circuit Lower Bounds, Polynomial Identity Testing, Derandomization
Collection: 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)
Issue Date: 2012
Date of publication: 24.02.2012

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