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Scalar ambiguity and freeness in matrix semigroups over bounded languages

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conference contribution
posted on 2016-01-20, 13:47 authored by Paul Bell, Shang Chen, Lisa JacksonLisa Jackson
There has been much research into freeness properties of finitely generated matrix semigroups under various constraints, mainly related to the dimensions of the generator matrices and the semiring over which the matrices are defined. A recent paper has also investigated freeness properties of matrices within a bounded language of matrices, which are of the form M1M2 · · · Mk ⊆ F n×n for some semiring F [9]. Most freeness problems have been shown to be undecidable starting from dimension three, even for upper-triangular matrices over the natural numbers. There are many open problems still remaining in dimension two. We introduce a notion of freeness and ambiguity for scalar reachability problems in matrix semigroups and bounded languages of matrices. Scalar reachability concerns the set {ρ TMτ |M ∈ S}, where ρ, τ ∈ F n are vectors and S is a finitely generated matrix semigroup. Ambiguity and freeness problems are defined in terms of uniqueness of factorizations leading to each scalar. We show various undecidability results.

History

School

  • Science

Department

  • Computer Science

Published in

Language and Automata Theory and Applications (LATA 2016)

Citation

BELL, P.C., CHEN, S. and JACKSON, L.M., 2016. Scalar ambiguity and freeness in matrix semigroups over bounded languages. Lecture Notes in Computer Science, 9618, pp.493-505.

Publisher

© Springer Verlag

Version

  • AM (Accepted Manuscript)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2016

Notes

This paper was presented at LATA 2016: 10th International Conference on Language and Automata Theory and Applications http://grammars.grlmc.com/lata2016/. This is a pre‐copyedited version of a Lecture Notes in Computer Science published by Springer. The definitive authenticated version is available online via http://dx.doi.org/10.1007/978-3-319-30000-9_38

ISSN

0302-9743

Language

  • en

Location

Prague, Czech Republic