Non-commutative arithmetic circuits with division

Abstract

We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulas compute non-commutative rational functions, which, despite their name, can no longer be expressed as ratios of polynomials. We prove some lower and upper bounds, completeness and simulation results, as follows. If X is n x n matrix consisting of n2 distinct mutually non-commuting variables, we show that: (i). X-1 can be computed by a circuit of polynomial size, (ii). every formula computing some entry of X-1 must have size at least 2Ω(n). We also show that matrix inverse is complete in the following sense: (i). Assume that a non-commutative rational function f can be computed by a formula of size s. Then there exists an invertible 2s x 2s-matrix A whose entries are variables or field elements such that f is an entry of A-1. (ii). If f is a non-commutative polynomial computed by a formula without inverse gates then A can be taken as an upper triangular matrix with field elements on the diagonal. We show how divisions can be eliminated from non-commutative circuits and formulae which compute polynomials, and we address the non-commutative version of the "rational function identity testing" problem. As it happens, the complexity of both of these procedures depends on a single open problem in invariant theory.