Trading Determinism for Noncommutativity in Edmonds' Problem

Document Type

Conference Article

Publication Title

Proceedings Annual IEEE Symposium on Foundations of Computer Science Focs

Abstract

Let X=X1 X2 ∪ · ∪ Xk be a partitioned set of variables such that the variables in each part xi are noncommuting but for any i≠q j, the variables x Xi commute with the variables x′ xj. Given as input a square matrix t whose entries are linear forms over QX, we consider the problem of checking if t is invertible or not over the universal skew field of fractions of the partially commutative polynomial ring QX [1]. In this paper, we design a deterministic polynomial-time algorithm for this problem for constant k. The special case k=1 is the noncommutative Edmonds' problem (NSINGULAR) which has a deterministic polynomial-time algorithm by recent results [2]-[4]. En-route, we obtain the first deterministic polynomial-time algorithm for the equivalence testing problem of k-tape weighted automata (for constant k) resolving a longstanding open problem [5], [6]. Algebraically, the equivalence problem reduces to testing whether a partially commutative rational series over the partitioned set X is zero or not [6]. Decidability of this problem was established by Harju and Karhumäki [5]. Prior to this work, a randomized polynomial-time algorithm for this problem was given by Worrell [6] and, subsequently, a deterministic quasipolynomial-time algorithm was also developed [7].

First Page

539

Last Page

559

DOI

10.1109/FOCS61266.2024.00042

Publication Date

1-1-2024

Comments

Open Access; Green Open Access

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