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International Conference on Developments in Language Theory

DLT 2004: Developments in Language Theory pp 296–307Cite as

Context-Sensitive Decision Problems in Groups

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3340))

Abstract

There already exist classifications of those groups which have regular, context-free or recursively enumerable word problem. The only remaining step in the Chomsky hierarchy is to consider those groups with a context-sensitive word problem. In this paper we consider this problem and prove some results about these groups. We also establish some results about other context-sensitive decision problems in groups.

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References

  1. Anisimov, A.V.: Group languages. Kibernetika 4, 18–24 (1971)

    MathSciNet  Google Scholar 

  2. Anisimov, A.V.: Some algorithmic problems for groups and context-free languages. Kibernetika 8, 4–11 (1972)

    MathSciNet  Google Scholar 

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

    Article  MATH  Google Scholar 

  4. Avenhaus, J., Madlener, K.: The Nielsen reduction and P-complete problems in free groups. Theoret. Comp. Sci. 32, 61–76 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. Boone, W.W., Higman, G.: An algebraic characterization of groups with a solvable word problem. J. Australian Math. Soc. 18, 41–53 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  6. Collins, D.J., Miller III, C.F.: The conjugacy problem and subgroups of finite index. Proc. London Math. Soc. 34, 535–556 (1977)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Epstein, D.B.A., Cannon, J.W., Holt, D.F., Levy, S.V.F., Paterson, M.S., Thurston, W.P.: Word processing in groups. Jones and Bartlett (1991)

    Google Scholar 

  9. Haring-Smith, R.H.: Groups and simple languages. Trans. Amer. Math. Soc. 279, 337–356 (1983)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Higman, G.: Subgroups of finitely presented groups. Proc. Roy. Soc. Ser. A 262, 455–475 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  12. Immerman, N.: Nondeterministic space is closed under complementation. SIAM J. Comput. 17, 935–938 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Ito, M., Kari, L., Thierren, G.: Insertion and deletion closure of languages. Theoret. Comp. Sci. 183, 3–19 (1997)

    Article  MATH  Google Scholar 

  14. Lakin, S.R.: Context-sensitive decision problems in groups PhD thesis, University of Leicester (2002)

    Google Scholar 

  15. Lipton, R.J., Zalcstein, Y.: Word problems solvable in logspace. J. Assoc. Comput. Mach. 24, 522–526 (1977)

    MATH  MathSciNet  Google Scholar 

  16. Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. Springer, Heidelberg (1977)

    MATH  Google Scholar 

  17. Magnus, W., Karrass, A., Solitar, D.: Combinatorial group theory. Dover (1976)

    Google Scholar 

  18. Mihailova, K.A.: The occurrence problem for direct products of groups. Dokl. Akad. Nauk SSSR 119, 1103–1105 (1958)

    MathSciNet  Google Scholar 

  19. Miller III, C.F.: On group-theoretic decision problems and their classification. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  20. Miller III, C.F.: Decision problems in groups - surveys and reflections. In: Baumslag, G., Miller III, C.F. (eds.) Algorithms and classification in combinatorial group theory, vol. 23, pp. 1–59. MSRI Publications, Springer (1982)

    Google Scholar 

  21. Muller, D.E., Schupp, P.E.: Groups, the theory of ends, and context-free languages. J. Comp. System Sci. 26, 295–310 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  22. Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata and second-order logic. Theoret. Comp. Sci. 37, 51–75 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  23. Röver, C.E., Holt, D.F., Rees, S.E., Thomas, R.M.: Groups with a context-free co-word problem, J. London Math. Soc. (to appear)

    Google Scholar 

  24. Parkes, D.W., Thomas, R.M.: Groups with context-free reduced word problem. Comm. Algebra 30, 3143–3156 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Parkes, D.W., Thomas, R.M.: Reduced and irreducible word problems of groups MCS Technical Report 1999/4, University of Leicester (1999)

    Google Scholar 

  26. Szelepcsényi, R.: The method of forcing for nondeterministic automata. Bull. EATCS 33, 96–100 (1987)

    MATH  Google Scholar 

  27. Shapiro, M.: A note on context-sensitive languages and word problems. Internat. J. Algebra Comput. 4, 493–497 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  28. Stillwell, J.C.: Classical topology and combinatorial group theory. Springer, Heidelberg (1993)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Cogent Computing Group, School of Mathematical and Information Sciences, Coventry University, Coventry, UK

    Stephen R. Lakin

  2. Department of Computer Science, University of Leicester, Leicester, UK

    Richard M. Thomas

Authors
  1. Stephen R. Lakin
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  2. Richard M. Thomas
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Cristian S. Calude

  2. Computer Science, Institute of Information and Mathematical Sciences, Massey University Albany, Auckland, New Zealand

    Elena Calude

  3. Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Michael J. Dinneen

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© 2004 Springer-Verlag Berlin Heidelberg

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Lakin, S.R., Thomas, R.M. (2004). Context-Sensitive Decision Problems in Groups. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds) Developments in Language Theory. DLT 2004. Lecture Notes in Computer Science, vol 3340. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30550-7_25

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  • DOI: https://doi.org/10.1007/978-3-540-30550-7_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24014-3

  • Online ISBN: 978-3-540-30550-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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