Rational series and their languages
Rational series and their languages
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
The Parallel Evaluation of General Arithmetic Expressions
Journal of the ACM (JACM)
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Deterministic polynomial identity testing in non-commutative models
Computational Complexity
More on Noncommutative Polynomial Identity Testing
CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Relationless Completeness and Separations
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
On P vs. NP and geometric complexity theory: Dedicated to Sri Ramakrishna
Journal of the ACM (JACM)
Algebraic Complexity Theory
Noncommutative rational functions, their difference-differential calculus and realizations
Multidimensional Systems and Signal Processing
The GCT program toward the P vs. NP problem
Communications of the ACM
FOCS '12 Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
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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.