Abstract
Recently, Forbes, Kumar and Saptharishi [CCC, 2016] proved that there exists an explicit d O(1) - variate and degree d polynomial Pd ∈ VNP such that if any depth four circuit C of bounded formal degree d which computes a polynomial of bounded individual degree O(1), that is functionally equivalent to Pd, then C must have size 2 Ω(√ d log d) . The motivation for their work comes from Boolean Circuit Complexity. Based on a characterization for ACC0 circuits by Yao [FOCS, 1985] and Beigel and Tarui [CC, 1994], Forbes, Kumar and Saptharishi [CCC, 2016] observed that functions in ACC0 can also be computed by algebraic Σ∧ΣΠ circuits (i.e., circuits of the form – sums of powers of polynomials) of 2 logO(1) n size. Thus they argued that a 2 ω(poly log n) “functional” lower bound for an explicit polynomial Q against Σ∧ΣΠ circuits would imply a lower bound for the “corresponding Boolean function” of Q against non-uniform ACC0 . In their work, they ask if their lower bound be extended to Σ∧ΣΠ circuits. In this paper, for large integers n and d such that ω(log2 n) ≤ d ≤ n 0.01, we show that any Σ∧ΣΠ circuit of bounded individual degree at most O d k2 that functionally computes Iterated Matrix Multiplication polynomial IMMn,d (∈ VP) over {0, 1} n 2d must have size n Ω(k) . Since Iterated Matrix Multiplication IMMn,d over {0, 1} n 2d is functionally in GapL, improvement of the afore mentioned lower bound to hold for quasipolynomially large values of individual degree would imply a fine-grained separation of ACC0 from GapL. For the sake of completeness, we also show a syntactic size lower bound against any Σ∧ΣΠ circuit computing IMMn,d (for the same regime of d) which is tight over large fields. Like Forbes, Kumar and Saptharishi [CCC, 2016], we too prove lower bounds against circuits of bounded formal degree which functionally compute IMMn,d, for a slightly larger range of individual degree. 2012 ACM Subject Classification Theory of computation → Circuit complexity; Theory of computation → Algebraic complexity theory Keywords and phrases Functional Lower