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

Authors:
Meena Mahajan;B.V. Raghavendra Rao;Karteek Sreenivasaiah
Affiliations:
The Institute of Mathematical Sciences, Chennai, India;Indian Institute of Technology Madras, Chennai, India;The Institute of Mathematical Sciences, Chennai, India
Venue:
Theoretical Computer Science
Year:
2014

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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.