Abstract
In this article, we study the identity testing problem of arithmetic read-once formulas (ROFs) and some related models. An ROF is a formula (a circuit whose underlying graph is a tree) in which the operations are { +, × } and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for ROFs, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of Shpilka and Yolkovich [51].
Highlights
In this article, we study the problem of polynomial identity testing (PIT): given an arithmetic circuit C over a field F, with input variables x1, x2, . . . , xn, determine whether C computes the identically zero polynomial
A polynomial P (x) is a preprocessed read-once polynomial (PROP) if it can be computed by a preprocessed read-once formulas (PROFs)
These PROPs generalize the “sum-of-univariates” model. (See Section 3.2 for a formal definition.) We begin with our main result: polynomial-time black-box PIT algorithm for PROFs
Summary
We study the problem of polynomial identity testing (PIT): given an arithmetic circuit C over a field F, with input variables x1, x2, . . . , xn, determine whether C computes the identically zero polynomial. We study the problem of polynomial identity testing (PIT): given an arithmetic circuit C over a field F, with input variables x1, x2, . For several restricted classes of arithmetic circuits, efficient deterministic black-box PIT algorithms were found. In Shpilka and Volkovich [51], it was shown that one cannot achieve polynomial-time black-box PIT algorithms if |F| = o(n/ log n)
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