Lower Bounds for Non-Commutative Skew Circuits

Nisan (STOC 1991) exhibited a polynomial which is computable by linear-size non-commutative circuits but requires exponential-size non-commutative algebraic branching programs. Nisan’s hard polynomial is in fact computable by linear-size “skew circuits.” Skew circuits are circuits where every multiplication gate has the property that all but one of its children is an input variable or a scalar. Such multiplication gates are called skew gates. We prove that any non-commutative skew circuit which computes the square of the polynomial defined by Nisan must have exponential size. A simple extension of this result then yields an exponential lower bound on the size of non-commutative circuits where each multiplication gate has an argument of degree at most one-fifth of the total degree. We also extend our techniques to prove an exponential lower bound for a class of circuits which is a restriction of general non-commutative circuits and a generalization of noncommutative skew circuits. We define the non-skew depth of a circuit to be the maximum number of non-skew gates on any path from an input gate to the output gate. We prove lower bounds for non-commutative circuits of small non-skew depth. More precisely, we show that for any k < d, there is an explicit polynomial of degree d over n variables that has non-commutative circuits of polynomial size but such that any circuit with non-skew depth k must have size at least nΩ(d/k). It is not hard to see that any Research funded by IFCPAR/CEFIPRA Project No 4702-1(A) and by the ANR project CompA (ANR-13-BS02-0001-01). ACM Classification: F.1.3 AMS Classification: 68Q15, 68Q17