Some decision problems on integer matrices
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 125-131.

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.

DOI : 10.1051/ita:2005007
Classification : 20M05, 68R15 
@article{ITA_2005__39_1_125_0,
     author = {Choffrut, Christian and Karhum\"aki, Juhani},
     title = {Some decision problems on integer matrices},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {125--131},
     publisher = {EDP-Sciences},
     volume = {39},
     number = {1},
     year = {2005},
     doi = {10.1051/ita:2005007},
     mrnumber = {2132582},
     zbl = {1081.20066},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2005007/}
}
TY  - JOUR
AU  - Choffrut, Christian
AU  - Karhumäki, Juhani
TI  - Some decision problems on integer matrices
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2005
SP  - 125
EP  - 131
VL  - 39
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita:2005007/
DO  - 10.1051/ita:2005007
LA  - en
ID  - ITA_2005__39_1_125_0
ER  - 
%0 Journal Article
%A Choffrut, Christian
%A Karhumäki, Juhani
%T Some decision problems on integer matrices
%J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
%D 2005
%P 125-131
%V 39
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ita:2005007/
%R 10.1051/ita:2005007
%G en
%F ITA_2005__39_1_125_0
Choffrut, Christian; Karhumäki, Juhani. Some decision problems on integer matrices. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 39 (2005) no. 1, pp. 125-131. doi : 10.1051/ita:2005007. http://www.numdam.org/articles/10.1051/ita:2005007/

[1] J. Berstel, Transductions and context-free languages. B.G. Teubner (1979). | MR | Zbl

[2] J. Cassaigne, T. Harju and J. Karhumäki, On the undecidability of freeness of matrix semigroups. Internat. J. Algebra Comput. 9 (1999) 295-305. | Zbl

[3] C. Choffrut, A remark on the representation of trace monoids. Semigroup Forum 40 (1990) 143-152. | Zbl

[4] M. Chrobak and W. Rytter, Unique decipherability for partially commutative alphabets. Fund. Inform. X (1987) 323-336. | Zbl

[5] S. Eilenberg, Automata, Languages and Machines, Vol. A. Academic Press (1974). | MR | Zbl

[6] T. Harju, Decision questions on integer matrices. Lect. Notes Comp. Sci. 2295 (2002) 57-68. | Zbl

[7] T. Harju and J. Karhumäki, Morphisms, in Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa. Springer-Verlag 1 (1997) 439-510. | Zbl

[8] G. Jacob, La finitude des représentations linéaires de semigroupes est décidable. J. Algebra 52 (1978) 437-459. | Zbl

[9] 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. | Zbl

[10] D.A. Klarner, J.-C. Birget and W. Satterfield, On the undecidability of the freeness of integer matrix semigroups monoids. Internat. J. Algebra Comput. 1 (1991) 223-226. | Zbl

[11] R. Lyndon and P. Schupp, Combinatorial Group Theory, of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag 89 (1977). | MR | Zbl

[12] W. Magnus, The use of 2 by 2 matrices in combinatorial group theory. Resultate der Mathematik 4 (1981) 171-192. | Zbl

[13] A. Mandel and I. Simon, On finite semigroups of matrices. Theoret. Comput. Sci. 5 (1978) 101-112. | Zbl

[14] A.A. Markov, On certain insoluble problems concerning matrices (russian). Doklady Akad. Nauk SSSR (N.S.) 57 (1947) 539-542. | Zbl

[15] Open problems in group theory: http://zebra.sci.ccny.edu/cgi-bin/LINK.CGI?/www/web/problems/oproblems.html

[16] M.S. Paterson, Unsolvability in 3×3 matrices. Stud. Appl. Math. 49 (1970) 105-107. | Zbl

[17] J.J. Rotman, An introduction to the Theory of Groups. Ally and Bacon Inc. (1965). | MR | Zbl

Cité par Sources :