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×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×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.
Original language | English |
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Pages (from-to) | 357-393 |
Number of pages | 37 |
Journal | Theory of Computing |
Volume | 11 |
DOIs | |
State | Published - 20 Dec 2015 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2015 Pavel Hrubeš and Avi Wigderson.
Keywords
- Arithmetic circuits
- Non-commutative rational function
- Skew field