Generic complexity of the membership problem for semigroups of integer matrices

Author:

Rybalov A. N.,

Abstract

The membership problem for finitely generated subgroups (subsemigroups) of groups (semigroups) is a classical algorithmic problem, actively studied for many decades. Already for sufficiently simple groups and semigroups, this problem becomes undecidable. For example, K. A. Mikhailova in 1966 proved the undecidability of the membership problem for finitely generated subgroups (hence and for subsemigroups) of a direct product F2×F2 of two free groups of rank 2. Since, by the well-known Sanov theorem, the group F2 has an exact representation by integer matrices of order 2, the group F2×F2 is a subgroup of the group GL4(ℤ) of integer matrices of order 4. It easily implies the undecidability of this problem for the group GLk(ℤ) for k ≥ 4. Undecidability of the membership problem for finitely generated subsemigroups of semigroups of integer matrices of order ≥ 3 follows from Paterson’s result proved in 1970. In this paper, we propose a strongly generic algorithm deciding the membership problem for semigroups of integer matrices of arbitrary order for inputs from a subset whose sequence of frequencies exponentially fast converges to 1 with increasing size.

Publisher

Tomsk State University

Subject

Applied Mathematics,Computational Theory and Mathematics,Discrete Mathematics and Combinatorics,Signal Processing,Theoretical Computer Science

Cited by 2 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Effective Lower Bounds on the Matrix Rank and Their Applications;Programming and Computer Software;2023-10

2. Effective Lower Bounds on the Matrix Rank and Their Applications;Программирование;2023-09-01

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