Theory of Computing
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Title : Non-Commutative Arithmetic Circuits with Division
Authors : Pavel Hrubes and Avi Wigderson
Volume : 11
Number : 14
Pages : 357-393
URL : https://theoryofcomputing.org/articles/v011a014
Abstract
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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.