Read-once polynomial identity testing

Abstract

An arithmetic read-once formula (ROF for short) is a formula (a circuit in which the fanout of every gate is at most 1) in which the operations are {+, ×} and such that every input variable labels at most one leaf. In this paper we study the problems of identity testing and reconstruction of read-once formulas. the following are some of the results that we obtain.1. Given k ROFs in n variables, over a field F, we give a deterministic (non black-box) algorithm that checks whether they sum to zero or not. The running time of the algorithm is n2. We give an n O(d+k 2 ) time deterministic algorithm for checking whether a black box holding the sum of k depth d ROFs in n variables computes the zero polynomial. In other words, we provide a hitting set of size n O(d+k 2 ) for the sum of k depth d ROFs. If |F| is too small then we make queries from a polynomial size extension field. This implies a deterministic algorithm that runs in time n O(d) for the reconstruction of depth d ROFs.3. We give a hitting set of size exp(Õ( √ n + k 2 )) for the sum of k ROFs (without depth restrictions). In particular this implies a sub-exponential time deterministic algorithm for black-box identity testing and reconstructing of ROFs. As before, if |F| is too small then we make queries from a polynomial size extension field.To the best of our knowledge our results give the first sub-exponential time black-box identity testing algorithm for the sum of (a constant number of) ROFs and the first polynomial time identity testing algorithm for the sum of (a constant number of) ROFs, in the non black-box setting.Another question that we study is the following generalization of the polynomial identity testing problem. Given an arithmetic circuit computing a polynomial P (x), decide whether there is a ROF computing P (x). If there is such a formula then output it. Otherwise output "No". We call this question the read-once testing problem (ROT for short). Previous (randomized) algorithms for reconstruction of ROFs imply that there exists a randomized algorithm for the read-once testing problem. In this work we show that most previous algorithms for polynomial identity testing can be strengthen to yield algorithms for the read-once testing problem. In particular we give ROT algorithms for the following circuit classes: Depth-2 circuits (circuits computing sparse polynomials), Depth-3 circuits with bounded top fan-in (both in the black-box and non black-box settings, where the running time depends on the model), non-commutative formulas and sum of k ROFs. The running time of the ROT algorithm is essentially the same running time as the corresponding PIT algorithm for the class.The main tool in most of our results is a new connection between polynomial identity testing and reconstruction of read-once formulas. Namely, we show that in any model that is closed under partial derivatives (that is, a partial derivative of a polynomial computed by a circuit in the model, can also be computed by a circuit in the model) and that has an efficient det...

On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

“…This can be done by replacing, for each i and j, the j-th occurrence of x i with a new variable x i,j . Now, using PIT algorithm for read-once formulas [SV08,SV09], check whether this formula is zero or not. If it is zero then the original formulas was also zero and we are done.…”

“…Thus, we somehow have to find a way of verifying whether a linear function is a factor of a multilinear formula. Notice that as we start with a read-once formula for which PIT is known [SV08,SV09], we can assume that we know many inputs on which the formula does not vanish. One may hope that before replacing x i,j with x i we somehow managed to obtain inputs that will enable us to verify whether x i − x i,j is a factor of the formula or not.…”

“…For example the second approach does not work for the case of (sums of) read-once formulas (see [SV08] for a definition).…”

“…In [SV08,SV09] justifying assignments were used in conjunction with new PIT algorithms in order to obtain deterministic quasi-polynomial time interpolation algorithms for read-once formulas. We rely on the ideas from [SV09] for obtaining justifying assignments from PIT algorithms.…”

“…Our approach uses ideas from [SV08], that showed that the polynomial identity testing problem for a circuit class C is essentially equivalent to the problem of deciding whether a circuit from C computes a polynomial that has a read-once arithmetic formula.…”