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Arithmetic circuits: the chasm at depth four gets wider

Abstract : In their paper on the ``chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n*n matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of polynomial size. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials.
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Preprints, Working Papers, ...
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Contributor : Pascal Koiran <>
Submitted on : Wednesday, June 23, 2010 - 10:33:41 PM
Last modification on : Friday, September 10, 2021 - 2:34:03 PM
Long-term archiving on: : Friday, September 24, 2010 - 5:35:55 PM


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  • HAL Id : ensl-00494642, version 1
  • ARXIV : 1006.4700



Pascal Koiran. Arithmetic circuits: the chasm at depth four gets wider. 2010. ⟨ensl-00494642v1⟩



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