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International Conference on Developments in Language Theory

DLT 2012: Developments in Language Theory pp 97–108Cite as

Semigroups with a Context-Free Word Problem

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7410))

Abstract

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.

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References

  1. Anīsīmov, A.V.: Certain algorithmic questions for groups and context-free languages. Kibernetika (Kiev) (2), 4–11 (1972)

    Google Scholar 

  2. Autebert, J.-M., Boasson, L., Sénizergues, G.: Groups and NTS languages. J. Comput. System Sci. 35(2), 243–267 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ayik, H., Rus̆kuc, N.: Generators and relations of Rees matrix semigroups. Proc. Edinburgh Math. Soc. (2) 42(3), 481–495 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berstel, J.: Transductions and context-free languages. Leitfäden der Angewandten Mathematik und Mechanik, vol. 38. B. G. Teubner, Stuttgart (1979)

    MATH  Google Scholar 

  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. 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. Duncan, A., Gilman, R.H.: Word hyperbolic semigroups. Math. Proc. Cambridge Philos. Soc. 136(3), 513–524 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math. 81(3), 449–457 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ginsburg, S.: The mathematical theory of context-free languages. McGraw-Hill Book Co., New York (1966)

    MATH  Google Scholar 

  10. Harrison, M.A.: Introduction to formal language theory. Addison-Wesley Publishing Co., Reading (1978)

    Google Scholar 

  11. Herbst, T.: On a subclass of context-free groups. RAIRO Inform. Théor. Appl. 25(3), 255–272 (1991)

    MathSciNet  MATH  Google Scholar 

  12. Herbst, T., Thomas, R.M.: Group presentations, formal languages and characterizations of one-counter groups. Theoret. Comput. Sci. 112(2), 187–213 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hopcroft, J.E., Ullman, J.D.: Introduction to automata theory, languages, and computation. Addison-Wesley Publishing Co., Reading (1979)

    MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  18. Jura, A.: Determining ideals of a given finite index in a finitely presented semigroup. Demonstratio Math. 11(3), 813–827 (1978)

    MathSciNet  MATH  Google Scholar 

  19. Lallement, G.: Semigroups and combinatorial applications. John Wiley & Sons, New York (1979)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pelletier, M., Sakarovitch, J.: Easy multiplications. II. Extensions of rational semigroups. Inform. and Comput. 88(1), 18–59 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rosenberg, A.L.: A machine realization of the linear context-free languages. Information and Control 10(2), 175–188 (1967)

    Article  MATH  Google Scholar 

  23. Sakarovitch, J.: Easy multiplications. I. The realm of Kleene’s theorem. Inform. and Comput. 74(3), 173–197 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Stallings, J.: Group theory and three-dimensional manifolds. Yale University Press, New Haven (1971)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Leicester, Leicester, England

    Michael Hoffmann & Richard M. Thomas

  2. Department of Mathematics, University of Warwick, Coventry, England

    Derek F. Holt & Matthew D. Owens

Authors
  1. Michael Hoffmann
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  2. Derek F. Holt
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  3. Matthew D. Owens
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  4. Richard M. Thomas
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Editor information

Editors and Affiliations

  1. Deptartment of Electrical Engineering, National Taiwan University, 106, Taipei, Taiwan

    Hsu-Chun Yen

  2. Department of Computer Science, University of California, 93106, Santa Barbara, CA, USA

    Oscar H. Ibarra

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© 2012 Springer-Verlag Berlin Heidelberg

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Hoffmann, M., Holt, D.F., Owens, M.D., Thomas, R.M. (2012). Semigroups with a Context-Free Word Problem. In: Yen, HC., Ibarra, O.H. (eds) Developments in Language Theory. DLT 2012. Lecture Notes in Computer Science, vol 7410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31653-1_10

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  • DOI: https://doi.org/10.1007/978-3-642-31653-1_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31652-4

  • Online ISBN: 978-3-642-31653-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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