Certifying polynomials for AC0[⊕] circuits, with applications

Swastik Kopparty, Srikanth Srinivasan

Research output: Chapter in Book/Report/Conference proceedingConference contribution

17 Scopus citations

Abstract

In this paper, we introduce and develop the method of certifying polynomials for proving AC0[⊕] circuit lower bounds. We use this method to show that Approximate Majority cannot be computed by AC 0[⊕] circuits of size n1+o(1). This implies a separation between the power of AC0[⊕] circuits of near-linear size and uniform AC0[⊕] (and even AC0) circuits of polynomial size. This also implies a separation between randomized AC 0[⊕] circuits of linear size and deterministic AC 0[⊕] circuits of near-linear size. Our proof using certifying polynomials extends the deterministic restrictions technique of Chaudhuri and Radhakrishnan, who showed that Approximate Majority cannot be computed by AC circuits of size n1+o(1). At the technical level, we show that for every AC0[⊕] circuit C of near-linear size, there is a low degree variety V over F2 such that the restriction of C to V is constant. We also prove other results exploring various aspects of the power of certifying polynomials. In the process, we show an essentially optimal lower bound of Ω(logΘ(d) s · log 1/ε ] on the degree of ε-approximating polynomials for AC0[⊕] circuits of size s.

Original languageEnglish (US)
Title of host publication32nd International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012
Pages36-47
Number of pages12
DOIs
StatePublished - 2012
Event32nd International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012 - Hyderabad, India
Duration: Dec 15 2012Dec 17 2012

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume18
ISSN (Print)1868-8969

Other

Other32nd International Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2012
CountryIndia
CityHyderabad
Period12/15/1212/17/12

All Science Journal Classification (ASJC) codes

  • Software

Keywords

  • Constant-depth Boolean circuits
  • Polynomials over finite fields
  • Size hierarchies

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