Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T06:53:41.463Z Has data issue: false hasContentIssue false
  • English
  • Français

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; [email protected]
Juhani Karhumäki
Affiliation:
Dept. of Mathematics and TUCS, University of Turku, 20014 Turku, Finland; [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J. Berstel, Transductions and context-free languages. B.G. Teubner (1979).
Cassaigne, J., Harju, T. and Karhumäki, J., On the undecidability of freeness of matrix semigroups. Internat. J. Algebra Comput. 9 (1999) 295–305. CrossRef
Choffrut, C., A remark on the representation of trace monoids. Semigroup Forum 40 (1990) 143–152. CrossRef
M. Chrobak and W. Rytter, Unique decipherability for partially commutative alphabets. Fund. Inform. X (1987) 323–336.
S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press (1974).
Harju, T., Decision questions on integer matrices. Lect. Notes Comp. Sci. 2295 (2002) 57–68. CrossRef
T. Harju and J. Karhumäki, Morphisms, in Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa. Springer-Verlag 1 (1997) 439–510.
Jacob, G., La finitude des représentations linéaires de semigroupes est décidable. J. Algebra 52 (1978) 437–459. CrossRef
J. Karhumäki, Some opem problems in combinatorics of words and related areas, in Proc. of RIMS Symposium on Algebraic Systems, Formal Languages and Computation. RIMS Institute 1166 (2000) 118–130.
Klarner, D.A., Birget, J.-C. and Satterfield, W., On the undecidability of the freeness of integer matrix semigroups monoids. Internat. J. Algebra Comput. 1 (1991) 223–226. CrossRef
R. Lyndon and P. Schupp, Combinatorial Group Theory, of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag 89 (1977).
Magnus, W., The use of 2 by 2 matrices in combinatorial group theory. Resultate der Mathematik 4 (1981) 171–192. CrossRef
Mandel, A. and Simon, I., On finite semigroups of matrices. Theoret. Comput. Sci. 5 (1978) 101–112. CrossRef
Markov, A.A., On certain insoluble problems concerning matrices (russian). Doklady Akad. Nauk SSSR (N.S.) 57 (1947) 539–542.
Open problems in group theory: http://zebra.sci.ccny.edu/cgi-bin/LINK.CGI?/www/web/problems/oproblems.html
Paterson, M.S., Unsolvability in 3 x 3 matrices. Stud. Appl. Math. 49 (1970) 105–107. CrossRef
J.J. Rotman, An introduction to the Theory of Groups. Ally and Bacon Inc. (1965).