Monomials, multilinearity and identity testing in simple read-restricted circuits

Abstract

We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero. We give a deterministic polynomial time algorithm for this problem when the inputs are read-twice or read-thrice formulas. In the process, these algorithms also test if the input circuit is computing a multilinear polynomial. We further study three related computational problems on arithmetic circuits. Given an arithmetic circuit C,(1) ZMC: test if a given monomial in C has zero coefficient or not,(2) MonCount: compute the number of monomials in C, and (3) MLIN: test if C computes a multilinear polynomial or not. These problems were introduced by Fournier, Malod and Mengel (2012)[11], and shown to characterise various levels of the counting hierarchy (CH). We address the above problems on read-restricted arithmetic circuits and branching programs. We prove several complexity characterisations for the above problems on these restricted classes of arithmetic circuits.