eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-08-23
81:1
81:18
10.4230/LIPIcs.MFCS.2024.81
article
Monoids of Upper Triangular Matrices over the Boolean Semiring
Ryzhikov, Andrew
1
https://orcid.org/0000-0002-2031-2488
Wolf, Petra
2
https://orcid.org/0000-0003-3097-3906
Department of Computer Science, University of Oxford, UK
LaBRI, CNRS, UniversitΓ© de Bordeaux, Bordeaux INP, France
Given a finite set π of square matrices and a square matrix B, all of the same dimension, the membership problem asks if B belongs to the monoid β³(π) generated by π. The rank one problem asks if there is a matrix of rank one in β³(π). We study the membership and the rank one problems in the case where all matrices are upper triangular matrices over the Boolean semiring. We characterize the computational complexity of these problems, and identify their PSPACE-complete and NP-complete special cases.
We then consider, for a set π of matrices from the same class, the problem of finding in β³(π) a matrix of minimum rank with no zero rows. We show that the minimum rank of such matrix can be computed in linear time.We also characterize the space complexity of this problem depending on the size of π, and apply all these results to the ergodicity problem asking if β³(π) contains a matrix with a column consisting of all ones. Finally, we show that our results give better upper bounds for the case where each row of every matrix in π contains at most one non-zero entry than for the general case.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol306-mfcs2024/LIPIcs.MFCS.2024.81/LIPIcs.MFCS.2024.81.pdf
matrix monoids
membership
rank
ergodicity
partially ordered automata