Theory of Computing ------------------- Title : Non-Commutative Arithmetic Circuits with Division Authors : Pavel Hrubes and Avi Wigderson Volume : 11 Number : 14 Pages : 357-393 URL : http://www.theoryofcomputing.org/articles/v011a014 Abstract -------- We initiate the study of the complexity of arithmetic circuits with division gates over non-commuting variables. Such circuits and formulae 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 an $n\times n$ matrix consisting of $n^2$ distinct mutually non-commuting variables, we show that: 1. $X^{-1}$ can be computed by a circuit of polynomial size. 2. Every formula computing some entry of $X^{-1}$ must have size at least $2^{\Omega(n)}$. We also show that matrix inverse is complete in the following sense: 1. 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 $A^{-1}$. 2. 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.