On the decidability of semigroup freeness∗

Julien Cassaigne; Francois Nicolas

RAIRO - Theoretical Informatics and Applications (2012)

  • Volume: 46, Issue: 3, page 355-399
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Cassaigne, Julien, and Nicolas, Francois. "On the decidability of semigroup freeness∗." RAIRO - Theoretical Informatics and Applications 46.3 (2012): 355-399. <http://eudml.org/doc/277830>.

@article{Cassaigne2012,
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.},
author = {Cassaigne, Julien, Nicolas, Francois},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Decidability; semigroup freeness; matrix semigroups; Post correspondence problem; decidability; free monoids; freeness problem; undecidability},
language = {eng},
month = {8},
number = {3},
pages = {355-399},
publisher = {EDP Sciences},
title = {On the decidability of semigroup freeness∗},
url = {http://eudml.org/doc/277830},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Cassaigne, Julien
AU - Nicolas, Francois
TI - On the decidability of semigroup freeness∗
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/8//
PB - EDP Sciences
VL - 46
IS - 3
SP - 355
EP - 399
AB - 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.
LA - eng
KW - Decidability; semigroup freeness; matrix semigroups; Post correspondence problem; decidability; free monoids; freeness problem; undecidability
UR - http://eudml.org/doc/277830
ER -

References

top
  1. A.F. Beardon, Pell’s equation and two generator free Möbius groups. Bull. London Math. Soc.25 (1993) 527–532.  
  2. P. Bell and I. Potapov, Reachability problems in quaternion matrix and rotation semigroups. Inform. Comput.206 (2008) 1353–1361.  
  3. M. Benois, Parties rationnelles du groupe libre. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A269 (1969) 1188–1190.  
  4. J. Berstel, D. Perrin and C. Reutenauer, Codes and Automata, in Encycl. Math. Appl.129 (2009).  
  5. V.D. Blondel and V. Canterini, Undecidable problems for probabilistic automata of fixed dimension. Theor. Comput. Syst.36 (2003) 231–245.  
  6. V.D. Blondel and J.N. Tsitsiklis, When is a pair of matrices mortal?Inf. Proc. Lett.63 (1997) 283–286.  
  7. V.D. Blondel and J.N. Tsitsiklis, The boundedness of all products of a pair of matrices is undecidable. Syst. Control Lett.41 (2000) 135–140.  
  8. 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.  
  9. O. Bournez and M. Branicky, The mortality problem for matrices of low dimensions. Theor. Comput. Syst.35 (2002) 433–448.  
  10. J.L. Brenner and A. Charnow, Free semigroups of 2 × 2 matrices. Pacific J. Math.77 (1978) 57–69.  
  11. J. Cassaigne, T. Harju and J. Karhumäki, On the undecidability of freeness of matrix semigroups. Int. J. Algebra Comput.9 (1999) 295–305.  
  12. C. Choffrut and J. Karhumäki, Some decision problems on integer matrices. RAIRO – Theor. Inf. Appl.39 (2005) 125–131.  
  13. V. Claus, Some remarks on PCP(k) and related problems. Bull. Eur. Assoc. Theoret. Comput. Sci.12 (1980) 54–61.  
  14. A. Ehrenfeucht, J. Karhumäki and G. Rozenberg, The (generalized) Post correspondence problem with lists consisting of two words is decidable. Theoret. Comput. Sci.21 (1982) 119–144.  
  15. P. Gawrychowski, M. Gutan and A. Kisielewic, On the problem of freeness of multiplicative matrix semigroups. Theoret. Comput. Sci.411 (2010) 1115–1120.  
  16. 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.  
  17. Z. Grunschlag, Algorithms in Geometric Group Theory. Ph.D. thesis, Graduate division of the University of California at Berkeley (1999).  
  18. A. Grytczuk and M. Wójtowicz, Beardon’s diophantine equations and non-free Möbius groups. Bull. Lond. Math. Soc.32 (2000) 305–310.  
  19. L. Guyot, Private Communication (2008).  
  20. V. Halava and T. Harju, Mortality in matrix semigroups. Am. Math. Mon.108 (2001) 649–653.  
  21. V. Halava, T. Harju and M. Hirvensalo, Binary (generalized) Post correspondence problem. Theoret. Comput. Sci.276 (2002) 183–204.  
  22. V. Halava, T. Harju and M. Hirvensalo, Undecidability bounds for integer matrices using Claus instances. Int. J. Found. Comput. Sci.18 (2007) 931–948.  
  23. T. Harju and J. Karhumäki, Morphisms, in Handbook of formal languages, edited by G. Rozenberg and A. Salomaa 1 (1997) 439–510.  
  24. J.E. Hopcroft, R. Motwani and J.D. Ullman, Introduction to automata theory, languages, and computation, 2nd edition. Addison-Wesley (2001).  
  25. R.A. Horn and C.R. Johnson, Matrix analysis. Corrected reprint of the 1985 original. Cambridge University Press (1990).  
  26. G. Jacob, Un algorithme calculant le cardinal, fini ou infini, des demi-groupes de matrices. Theoret. Comput. Sci.5 (1977) 183–204.  
  27. G.J. Janusz, Algebraic number fields, in Pure Appl. Math.55 (1973).  
  28. M. Kambites, P.V. Silva and B. Steinberg, On the rational subset problem for groups. J. Algebra309 (2007) 622–639.  
  29. R. Kannan and R.J. Lipton, Polynomial-time algorithm for the orbit problem. J. Assoc. Comput. Mach.33 (1986) 808–821.  
  30. D.A. Klarner, J.-C. Birget and W. Satterfield, On the undecidability of the freeness of integer matrix semigroups. Int. J. Algebra Comput.1 (1991) 223–226.  
  31. M. Krom and M. Krom, More on mortality. Am. Math. Mon.97 (1990) 37–38.  
  32. M. Lothaire, Combinatorics on words, 2nd edition. Cambridge Mathematical Library, Cambridge University Press (1997).  
  33. M. Lothaire, Algebraic combinatorics on words, in Encycl. Math. Appl.90 (2002).  
  34. R.C. Lyndon and P.E. Schupp, Combinatorial group theory, in Ergebnisse der Mathematik und ihrer Grenzgebiete89 (1977).  
  35. A. Mandel and I. Simon, On finite semigroups of matrices. Theoret. Comput. Sci.5 (1977) 101–111.  
  36. Yu. Matiyasevich and G. Sénizergues, Decision problems for semi-Thue systems with a few rules. Theoret. Comput. Sci.330 (2005) 145–169.  
  37. F. Mazoit, Autour de quelques problèmes de décidabilité sur des semigroupes de matrices. Rapport de stage MIM 2. École normale supérieure de Lyon (1998).  
  38. M.A. Miller, Mortality for sets of 2 × 2 matrices. Math. Mag.67 (1994) 210–213.  
  39. F. Nicolas, (Generalized) Post correspondence problem and semi-Thue systems. Available at (2008).  URIhttp://arxiv.org/abs/0802.0726
  40. M.S. Paterson, Unsolvability in 3 × 3 matrices. Stud. Appl. Math.49 (1970) 105–107.  
  41. E.L. Post, A variant of a recursively unsolvable problem. Bull. Am. Math. Soc.52 (1946) 264–268.  
  42. E.L. Post, Recursive unsolvability of a problem of Thue. J. Symbolic Logic12 (1947) 1–11.  
  43. G. Richomme, Private Communication (2007).  
  44. J. Sakarovitch, Éléments de théorie des automates. Vuibert (2003).  
  45. P. Schultz, Mortality of 2 × 2 matrices. Am. Math. Mon.84 (1977) 463–464.  
  46. A. Tarski, A decision method for elementary algebra and geometry, 2nd edition. University of California Press (1951).  

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