eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-03-11
27:1
27:16
10.4230/LIPIcs.STACS.2024.27
article
Nonnegativity Problems for Matrix Semigroups
D'Costa, Julian
1
https://orcid.org/0000-0003-2610-5241
Ouaknine, Joël
2
https://orcid.org/0000-0003-0031-9356
Worrell, James
1
https://orcid.org/0000-0001-8151-2443
Department of Computer Science, University of Oxford, UK
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
The matrix semigroup membership problem asks, given square matrices M,M₁,…,M_k of the same dimension, whether M lies in the semigroup generated by M₁,…,M_k. It is classical that this problem is undecidable in general, but decidable in case M₁,…,M_k commute. In this paper we consider the problem of whether, given M₁,…,M_k, the semigroup generated by M₁,…,M_k contains a non-negative matrix. We show that in case M₁,…,M_k commute, this problem is decidable subject to Schanuel’s Conjecture. We show also that the problem is undecidable if the commutativity assumption is dropped. A key lemma in our decidability proof is a procedure to determine, given a matrix M, whether the sequence of matrices (Mⁿ)_{n = 0}^∞ is ultimately nonnegative. This answers a problem posed by S. Akshay [S. Akshay et al., 2022]. The latter result is in stark contrast to the notorious fact that it is not known how to determine, for any specific matrix index (i,j), whether the sequence (Mⁿ)_{i,j} is ultimately nonnegative. Indeed the latter is equivalent to the Ultimate Positivity Problem for linear recurrence sequences, a longstanding open problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol289-stacs2024/LIPIcs.STACS.2024.27/LIPIcs.STACS.2024.27.pdf
Decidability
Linear Recurrence Sequences
Schanuel’s Conjecture