Read-once polynomial identity testing

Amir Shpilka, Ilya Volkovich

Research output: Contribution to journalArticlepeer-review

Abstract

An arithmetic read-once formula (ROF for short) is a formula (a circuit whose underlying graph is a tree) in which the operations are (Formula presented.) and such that every input variable labels at most one leaf. A preprocessed ROF (PROF for short) is a ROF in which we are allowed to replace each variable xi with a univariate polynomial Ti(xi). In this paper, we study the problems of designing deterministic identity testing algorithms for models related to preprocessed ROFs. Our main result gives PIT algorithms for the sum of k preprocessed ROFs, of individual degrees at most d (i.e., each Ti(xi) is of degree at most d), that run in time (Formula presented.) in the white-box setting and in time (Formula presented.) in the black-box setting. We also obtain better algorithms when the formulas have a small depth that lead to an improvement in the best PIT algorithm for multilinear depth-3 (Formula presented.) circuits. Our main technique is to prove a hardness of representation result, namely a theorem showing a relatively mild lower bound on the sum of k PROFs. We then use this lower bound in order to design our PIT algorithm.

Original languageEnglish
Pages (from-to)477-532
Number of pages56
JournalComputational Complexity
Volume24
Issue number3
DOIs
StatePublished - 21 Sep 2015

Keywords

  • Derandomization
  • arithmetic circuits
  • bounded-depth circuits
  • identity testing
  • read-once formulas

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