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Towards Blackbox Identity Testing of Log-Variate Circuits

Authors Michael A. Forbes, Sumanta Ghosh, Nitin Saxena

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Michael A. Forbes
  • University of Illinois at Urbana-Champaign, USA
Sumanta Ghosh
  • Department of Computer Science, IIT Kanpur, India
Nitin Saxena
  • Department of Computer Science, IIT Kanpur, India

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Michael A. Forbes, Sumanta Ghosh, and Nitin Saxena. Towards Blackbox Identity Testing of Log-Variate Circuits. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 54:1-54:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ICALP.2018.54

Abstract

Derandomization of blackbox identity testing reduces to extremely special circuit models. After a line of work, it is known that focusing on circuits with constant-depth and constantly many variables is enough (Agrawal,Ghosh,Saxena, STOC'18) to get to general hitting-sets and circuit lower bounds. This inspires us to study circuits with few variables, eg. logarithmic in the size s.
We give the first poly(s)-time blackbox identity test for n=O(log s) variate size-s circuits that have poly(s)-dimensional partial derivative space; eg. depth-3 diagonal circuits (or Sigma wedge Sigma^n). The former model is well-studied (Nisan,Wigderson, FOCS'95) but no poly(s2^n)-time identity test was known before us. We introduce the concept of cone-closed basis isolation and prove its usefulness in studying log-variate circuits. It subsumes the previous notions of rank-concentration studied extensively in the context of ROABP models.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Pseudorandomness and derandomization
  • Computing methodologies → Algebraic algorithms
  • Mathematics of computing → Combinatoric problems
Keywords
  • hitting-set
  • depth-3
  • diagonal
  • derandomization
  • polynomial identity testing
  • log-variate
  • concentration
  • cone closed
  • basis isolation

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