Semigroups with a Context-Free Word Problem

  • Michael Hoffmann
  • Derek F. Holt
  • Matthew D. Owens
  • Richard M. Thomas
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)


The word problem is of fundamental interest in group theory and has been widely studied. One important connection between group theory and theoretical computer science has been the consideration of the word problem as a formal language; a pivotal result here is the classification by Muller and Schupp of groups with a context-free word problem. Duncan and Gilman have proposed a natural extension of the notion of the word problem as a formal language from groups to semigroups and the question as to which semigroups have a context-free word problem then arises. Whilst the depth of the Muller-Schupp result and its reliance on the geometrical structure of Cayley graphs of groups suggests that a generalization to semigroups could be very hard to obtain we have been able to prove some results about this intriguing class of semigroups.


Word Problem Formal Language Cayley Graph Regular Language Simple Semigroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anīsīmov, A.V.: Certain algorithmic questions for groups and context-free languages. Kibernetika (Kiev) (2), 4–11 (1972)Google Scholar
  2. 2.
    Autebert, J.-M., Boasson, L., Sénizergues, G.: Groups and NTS languages. J. Comput. System Sci. 35(2), 243–267 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ayik, H., Rus̆kuc, N.: Generators and relations of Rees matrix semigroups. Proc. Edinburgh Math. Soc. (2) 42(3), 481–495 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Berstel, J.: Transductions and context-free languages. Leitfäden der Angewandten Mathematik und Mechanik, vol. 38. B. G. Teubner, Stuttgart (1979)zbMATHGoogle Scholar
  5. 5.
    Clifford, A.H., Preston, G.B.: The algebraic theory of semigroups.Vol. I. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1961)Google Scholar
  6. 6.
    Clifford, A.H., Preston, G.B.: The algebraic theory of semigroups. Vol. II. Mathematical Surveys, vol. 7. American Mathematical Society, Providence (1967)Google Scholar
  7. 7.
    Duncan, A., Gilman, R.H.: Word hyperbolic semigroups. Math. Proc. Cambridge Philos. Soc. 136(3), 513–524 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math. 81(3), 449–457 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ginsburg, S.: The mathematical theory of context-free languages. McGraw-Hill Book Co., New York (1966)zbMATHGoogle Scholar
  10. 10.
    Harrison, M.A.: Introduction to formal language theory. Addison-Wesley Publishing Co., Reading (1978)Google Scholar
  11. 11.
    Herbst, T.: On a subclass of context-free groups. RAIRO Inform. Théor. Appl. 25(3), 255–272 (1991)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Herbst, T., Thomas, R.M.: Group presentations, formal languages and characterizations of one-counter groups. Theoret. Comput. Sci. 112(2), 187–213 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hoffmann, M., Kuske, D., Otto, F., Thomas, R.M.: Some relatives of automatic and hyperbolic groups. In: Semigroups, Algorithms, Automata and Languages (Coimbra, 2001), pp. 379–406. World Sci. Publ., River Edge (2002)CrossRefGoogle Scholar
  14. 14.
    Holt, D.F., Owens, M.D., Thomas, R.M.: Groups and semigroups with a one-counter word problem. J. Aust. Math. Soc. 85(2), 197–209 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Holt, D.F., Rees, S., Röver, C.E., Thomas, R.M.: Groups with context-free co-word problem. J. London Math. Soc. (2) 71(3), 643–657 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Co., Reading (1979)zbMATHGoogle Scholar
  17. 17.
    Howie, J.M.: Fundamentals of semigroup theory. London Mathematical Society Monographs. New Series, vol. 12. The Clarendon Press, Oxford University Press, New York (1995)zbMATHGoogle Scholar
  18. 18.
    Jura, A.: Determining ideals of a given finite index in a finitely presented semigroup. Demonstratio Math. 11(3), 813–827 (1978)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Lallement, G.: Semigroups and combinatorial applications. John Wiley & Sons, New York (1979)zbMATHGoogle Scholar
  20. 20.
    Muller, D.E., Schupp, P.E.: Groups, the theory of ends, and context-free languages. J. Comput. System Sci. 26(3), 295–310 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Pelletier, M., Sakarovitch, J.: Easy multiplications. II. Extensions of rational semigroups. Inform. and Comput. 88(1), 18–59 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Rosenberg, A.L.: A machine realization of the linear context-free languages. Information and Control 10(2), 175–188 (1967)zbMATHCrossRefGoogle Scholar
  23. 23.
    Sakarovitch, J.: Easy multiplications. I. The realm of Kleene’s theorem. Inform. and Comput. 74(3), 173–197 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Stallings, J.: Group theory and three-dimensional manifolds. Yale University Press, New Haven (1971)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Michael Hoffmann
    • 1
  • Derek F. Holt
    • 2
  • Matthew D. Owens
    • 2
  • Richard M. Thomas
    • 1
  1. 1.Department of Computer ScienceUniversity of LeicesterLeicesterEngland
  2. 2.Department of MathematicsUniversity of WarwickCoventryEngland

Personalised recommendations