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 n O(k 2 ) .2. 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 eff...

Deterministic polynomial identity tests for multilinear bounded-read formulae

“…Shpilka and Volkovich considered a special type of bounded-read formulae, namely formulae that are the sum of k read-once formulae. For such formulae and constant k, they established a deterministic polynomial-time non-blackbox identity test as well as a deterministic blackbox algorithm that runs in quasi-polynomial time, more precisely in time s O(log s) , on formulae of size s (Shpilka & Volkovich 2008, 2009). These results have been extended to sums of k read-once algebraic branching programs (Jansen et al 2009).…”

“…For example, they have been used to exhibit lower bounds (Baur & Strassen 1983;Mignon & Ressayre 2004;Nisan & Wigderson 1996;Shpilka & Wigderson 2001), learn arithmetic circuits (Klivans & Shpilka 2006), and produce polynomial identity tests (Karnin et al 2013;Shpilka & Volkovich 2008, 2009). In our setting, partial derivatives give us a handle on the structure of constant-read formulae, which we in turn exploit to develop our identity tests.…”

“…We exhibit an explicit multilinear read-twice formula with n variables that requires Ω(n) terms when written as a sum of read-once formulae. In order to do so, we follow an approach similar to that which Shpilka & Volkovich (2008, 2009 use to show "hardness of representation" results for sums of read-once formulae.…”

“…We require a property of the SV-generator that is implicit in Shpilka & Volkovich (2008, 2009. Informally, it states that if a class of polynomials is disjoint from D and is closed under zerosubstitution, then the SV-generator hits this class of polynomials.…”

“…For restricted classes of circuits, one may hope to exploit their structure and come up with other types of witnesses. The prior deterministic identity tests we mentioned (Karnin et al 2013;Karnin & Shpilka 2008;Kayal & Saraf 2009;Saraf & Volkovich 2011;Saxena & Seshadhri 2009;Shpilka & Volkovich 2008, 2009) follow the latter general outline. More specifically, they exhibit a measure for the complexity of the restricted circuit that can be efficiently computed when given the circuit as input and prove that (i) restricted circuits that are zero have low complexity and (ii) restricted circuits of low complexity are easy to test.…”