On TC0, AC0, and arithmetic circuits

M. Agrawal, E. Allender, S. Datta

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

10 Scopus citations


Continuing a line of investigation that has studied the function classes P, we study the class of functions AC0. One way to define AC0 is as the class of functions computed by constant-depth polynomial-size arithmetic circuits of unbounded fanin addition and multiplication gates. In contrast to the preceding function classes, for which we know no nontrivial lower bounds, lower bounds for AC0 follow easily from established circuit lower bounds. One of our main results is a characterization of TC0 in terms of AC0: A language A is in TC0 if and only if there is a AC0 function f and a number k such that x∈AhArr/f(x)=2/sup |x|k/. Using the naming conventions, this yields: TC0=PAC0=C=AC0. Another restatement of this characterization is that TC0 can be simulated by constant-depth arithmetic circuits, with a single threshold gate. We hope that perhaps this characterization of TC0 in terms of AC0 circuits might provide a new avenue of attack for proving lower bounds. Our characterization differs markedly from earlier characterizations of TC0 in terms of arithmetic circuits over finite fields. Using our model of arithmetic circuits, computation over finite fields yields ACC0. We also prove a number of closure properties and normal forms for AC0.

Original languageEnglish (US)
Title of host publicationProceedings - 12th Annual IEEE Conference on Computational Complexity, CCC 1997 (Formerly
Subtitle of host publicationStructure in Complexity Theory Conference)
PublisherIEEE Computer Society
Number of pages15
ISBN (Electronic)0818679077
StatePublished - 1997
Event12th Annual IEEE Conference on Computational Complexity, CCC 1997 - Ulm, Germany
Duration: Jun 24 1997Jun 27 1997

Publication series

NameProceedings of the Annual IEEE Conference on Computational Complexity
ISSN (Print)1093-0159


Other12th Annual IEEE Conference on Computational Complexity, CCC 1997

All Science Journal Classification (ASJC) codes

  • Software
  • Theoretical Computer Science
  • Computational Mathematics


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