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Some decision problems on integer matrices

Published online by Cambridge University Press:  15 March 2005

Christian Choffrut  and
Juhani Karhumäki
Christian Choffrut
Affiliation:
L.I.A.F.A, Université Paris VII, Tour 55-56, 1 étage, 2 pl. Jussieu, 75 251 Paris Cedex, France; Christian.Choffrut@liafa.jussieu.fr
Juhani Karhumäki
Affiliation:
Dept. of Mathematics and TUCS, University of Turku, 20014 Turku, Finland; Juhani.Karhumaki@cs.utu.fi
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Abstract

Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs to a given finitely generated semigroup.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 125 - 131
Copyright
© EDP Sciences, 2005

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