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 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 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.
Highlights
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
Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division
We show that X−1 can be computed by a polynomial size circuit, but on the other hand, every formula computing an entry of X−1 must have an exponential size
Summary
Arithmetic circuit complexity studies the computation of polynomials and rational functions using the basic operations addition, multiplication, and division. We prove the following completeness result: if a rational function f can be computed by a formula of size s f can be expressed as an entry of A−1, where A is a 2s × 2s-matrix whose entries are variables or field elements. This is an analog of Valiant’s [49] theorem on completeness of determinant in the commutative, division-free, setting. In Appendix A, we discuss a connection between the open problems and invariant theory
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