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Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees

Authors Ramprasad Saptharishi, Anamay Tengse



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Ramprasad Saptharishi
  • Tata Institute of Fundamental Research, Mumbai, India
Anamay Tengse
  • Tata Institute of Fundamental Research, Mumbai, India

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Ramprasad Saptharishi and Anamay Tengse. Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 6:1-6:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.FSTTCS.2018.6

Abstract

We study the class of non-commutative Unambiguous circuits or Unique-Parse-Tree (UPT) circuits, and a related model of Few-Parse-Trees (FewPT) circuits (which were recently introduced by Lagarde, Malod and Perifel [Guillaume Lagarde et al., 2016] and Lagarde, Limaye and Srinivasan [Guillaume Lagarde et al., 2017]) and give the following constructions: - An explicit hitting set of quasipolynomial size for UPT circuits, - An explicit hitting set of quasipolynomial size for FewPT circuits (circuits with constantly many parse tree shapes), - An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is bounded by a constant. The above three results are extensions of the results of [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic branching programs (ROABPs) to UPT-set-multilinear circuits. The main idea is to study shufflings of non-commutative polynomials, which can then be used to prove suitable depth reduction results for UPT circuits and thereby allow a careful translation of the ideas in [Manindra Agrawal et al., 2015], [Rohit Gurjar et al., 2015] and [Rohit Gurjar et al., 2016].

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Complexity classes
Keywords
  • Unambiguous Circuits
  • Read-once Oblivious ABPs
  • Polynomial Identity Testing
  • Lower Bounds
  • Algebraic Circuit Complexity

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