Hostname: page-component-788cddb947-pt5lt Total loading time: 0 Render date: 2024-10-12T12:25:09.588Z Has data issue: false hasContentIssue false

On the decidability of semigroup freeness

Published online by Cambridge University Press:  29 May 2012

Julien Cassaigne
Affiliation:
Institut de mathématiques de Luminy, case 907, 163 avenue de Luminy, 13288 Marseille Cedex 9, France. cassaigne@iml.univ-mrs.fr
Francois Nicolas
Affiliation:
Lehrstuhl für Bioinformatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743 Jena, Germany; francois.nicolas@uni-jena.de
Get access

Abstract

This paper deals with the decidability of semigroup freeness. More precisely, the freeness problem over a semigroup S is defined as: given a finite subset X ⊆ S, decide whether each element of S has at most one factorization over X. To date, the decidabilities of the following two freeness problems have been closely examined. In 1953, Sardinas and Patterson proposed a now famous algorithm for the freeness problem over the free monoids. In 1991, Klarner, Birget and Satterfield proved the undecidability of the freeness problem over three-by-three integer matrices. Both results led to the publication of many subsequent papers. The aim of the present paper is (i) to present general results about freeness problems, (ii) to study the decidability of freeness problems over various particular semigroups (special attention is devoted to multiplicative matrix semigroups), and (iii) to propose precise, challenging open questions in order to promote the study of the topic.

Type
Research Article
Copyright
© EDP Sciences 2012

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

Beardon, A.F., Pell’s equation and two generator free Möbius groups. Bull. London Math. Soc. 25 (1993) 527532. Google Scholar
Bell, P. and Potapov, I., Reachability problems in quaternion matrix and rotation semigroups. Inform. Comput. 206 (2008) 13531361. Google Scholar
Benois, M., Parties rationnelles du groupe libre. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A 269 (1969) 11881190. Google Scholar
J. Berstel, D. Perrin and C. Reutenauer, Codes and Automata, in Encycl. Math. Appl. 129 (2009).
Blondel, V.D. and Canterini, V., Undecidable problems for probabilistic automata of fixed dimension. Theor. Comput. Syst. 36 (2003) 231245. Google Scholar
Blondel, V.D. and Tsitsiklis, J.N., When is a pair of matrices mortal? Inf. Proc. Lett. 63 (1997) 283286. Google Scholar
Blondel, V.D. and Tsitsiklis, J.N., The boundedness of all products of a pair of matrices is undecidable. Syst. Control Lett. 41 (2000) 135140. Google Scholar
V.D. Blondel, J. Cassaigne and J. Karhumäki, Freeness of multiplicative matrix semigroups, in Unsolved problems in mathematical systems and control theory, edited by V.D. Blondel and A. Megretski. Princeton University Press (2004) 309–314.
Bournez, O. and Branicky, M., The mortality problem for matrices of low dimensions. Theor. Comput. Syst. 35 (2002) 433448. Google Scholar
Brenner, J.L. and Charnow, A., Free semigroups of 2 × 2 matrices. Pacific J. Math. 77 (1978) 5769. Google Scholar
Cassaigne, J., Harju, T. and Karhumäki, J., On the undecidability of freeness of matrix semigroups. Int. J. Algebra Comput. 9 (1999) 295305. Google Scholar
Choffrut, C. and Karhumäki, J., Some decision problems on integer matrices. RAIRO – Theor. Inf. Appl. 39 (2005) 125131. Google Scholar
Claus, V., Some remarks on PCP(k) and related problems. Bull. Eur. Assoc. Theoret. Comput. Sci. 12 (1980) 5461. Google Scholar
Ehrenfeucht, A., Karhumäki, J. and Rozenberg, G., The (generalized) Post correspondence problem with lists consisting of two words is decidable. Theoret. Comput. Sci. 21 (1982) 119144. Google Scholar
Gawrychowski, P., Gutan, M. and Kisielewic, A., On the problem of freeness of multiplicative matrix semigroups. Theoret. Comput. Sci. 411 (2010) 11151120. Google Scholar
R.H. Gilman, Computations with rational subsets of confluent groups, in Proc. of the 3rd International Symposium on Symbolic and Algebraic Computation (EUROSAM 84), edited by J. Fitch. Lect. Notes Comput. Sci. 174 (1984) 207–212.
Z. Grunschlag, Algorithms in Geometric Group Theory. Ph.D. thesis, Graduate division of the University of California at Berkeley (1999).
Grytczuk, A. and Wójtowicz, M., Beardon’s diophantine equations and non-free Möbius groups. Bull. Lond. Math. Soc. 32 (2000) 305310. Google Scholar
L. Guyot, Private Communication (2008).
Halava, V. and Harju, T., Mortality in matrix semigroups. Am. Math. Mon. 108 (2001) 649653. Google Scholar
Halava, V., Harju, T. and Hirvensalo, M., Binary (generalized) Post correspondence problem. Theoret. Comput. Sci. 276 (2002) 183204. Google Scholar
Halava, V., Harju, T. and Hirvensalo, M., Undecidability bounds for integer matrices using Claus instances. Int. J. Found. Comput. Sci. 18 (2007) 931948. Google Scholar
T. Harju and J. Karhumäki, Morphisms, in Handbook of formal languages, edited by G. Rozenberg and A. Salomaa 1 (1997) 439–510.
J.E. Hopcroft, R. Motwani and J.D. Ullman, Introduction to automata theory, languages, and computation, 2nd edition. Addison-Wesley (2001).
R.A. Horn and C.R. Johnson, Matrix analysis. Corrected reprint of the 1985 original. Cambridge University Press (1990).
Jacob, G., Un algorithme calculant le cardinal, fini ou infini, des demi-groupes de matrices. Theoret. Comput. Sci. 5 (1977) 183204. Google Scholar
G.J. Janusz, Algebraic number fields, in Pure Appl. Math. 55 (1973).
Kambites, M., Silva, P.V. and Steinberg, B., On the rational subset problem for groups. J. Algebra 309 (2007) 622639. Google Scholar
Kannan, R. and Lipton, R.J., Polynomial-time algorithm for the orbit problem. J. Assoc. Comput. Mach. 33 (1986) 808821. Google Scholar
Klarner, D.A., Birget, J.-C. and Satterfield, W., On the undecidability of the freeness of integer matrix semigroups. Int. J. Algebra Comput. 1 (1991) 223226. Google Scholar
Krom, M. and Krom, M., More on mortality. Am. Math. Mon. 97 (1990) 3738. Google Scholar
M. Lothaire, Combinatorics on words, 2nd edition. Cambridge Mathematical Library, Cambridge University Press (1997).
M. Lothaire, Algebraic combinatorics on words, in Encycl. Math. Appl. 90 (2002).
R.C. Lyndon and P.E. Schupp, Combinatorial group theory, in Ergebnisse der Mathematik und ihrer Grenzgebiete 89 (1977).
Mandel, A. and Simon, I., On finite semigroups of matrices. Theoret. Comput. Sci. 5 (1977) 101111. Google Scholar
Matiyasevich, Yu. and Sénizergues, G., Decision problems for semi-Thue systems with a few rules. Theoret. Comput. Sci. 330 (2005) 145169. Google Scholar
F. Mazoit, Autour de quelques problèmes de décidabilité sur des semigroupes de matrices. Rapport de stage MIM 2. École normale supérieure de Lyon (1998).
Miller, M.A., Mortality for sets of 2 × 2 matrices. Math. Mag. 67 (1994) 210213. Google Scholar
F. Nicolas, (Generalized) Post correspondence problem and semi-Thue systems. Available at http://arxiv.org/abs/0802.0726 (2008).
Paterson, M.S., Unsolvability in 3 × 3 matrices. Stud. Appl. Math. 49 (1970) 105107. Google Scholar
Post, E.L., A variant of a recursively unsolvable problem. Bull. Am. Math. Soc. 52 (1946) 264268. Google Scholar
Post, E.L., Recursive unsolvability of a problem of Thue. J. Symbolic Logic 12 (1947) 111. Google Scholar
G. Richomme, Private Communication (2007).
J. Sakarovitch, Éléments de théorie des automates. Vuibert (2003).
Schultz, P., Mortality of 2 × 2 matrices. Am. Math. Mon. 84 (1977) 463464. Google Scholar
A. Tarski, A decision method for elementary algebra and geometry, 2nd edition. University of California Press (1951).