Decidability and Complexity in Automatic Monoids

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

Abstract

We prove several complexity and decidability results for automatic monoids: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, we show that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [4].

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References

  1. 1.
    Blumensath, A., Grädel, E.: Automatic structures. In: Proc. LICS 2000, pp. 51–62. IEEE Computer Society Press, Los Alamitos (2000)Google Scholar
  2. 2.
    Blumensath, A., Grädel, E.: Finite presentations of infinite structures: Automata and interpretations. In: Proc. CiAD 2002 (2002)Google Scholar
  3. 3.
    Book, R.V., Otto, F.: String–Rewriting Systems. Springer, Heidelberg (1993)MATHGoogle Scholar
  4. 4.
    Cai, J.-Y.: Parallel computation over hyperbolic groups. In: Proc. STOC 1992, pp. 106–115. ACM Press, New York (1992)Google Scholar
  5. 5.
    Calbrix, H., Knapik, T.: A string-rewriting characterization of Muller and Schupp’s context-free graphs. In: Arvind, V., Sarukkai, S. (eds.) FST TCS 1998. LNCS, vol. 1530, pp. 331–342. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Campbell, C.M., Robertson, E.F., Ruškuc, N., Thomas, R.M.: Automatic semigroups. Theor. Comput. Sci. 250(1-2), 365–391 (2001)MATHCrossRefGoogle Scholar
  7. 7.
    Compton, K.J., Henson, C.W.: A uniform method for proving lower bounds on the computational complexity of logical theories. Ann. Pure Appl. Logic 48, 1–79 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Coornaert, M., Delzant, T., Papadopoulos, A.: Géométrie et théorie des groupes. In: Lecture Notes in Mathematics. Springer, Heidelberg (1990)Google Scholar
  9. 9.
    Dahlhaus, E., Warmuth, M.K.: Membership for growing context-sensitive grammars is polynomial. J. Comput. Syst. Sci. 33, 456–472 (1986)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Duncan, A., Gilman, R.H.: Word hyperbolic semigroups. Math. Proc. Cambridge Philos. Soc. 136, 513–524 (2004)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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, Boston (1992)MATHGoogle Scholar
  12. 12.
    Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)MATHCrossRefGoogle Scholar
  13. 13.
    Hoffmann, M.: Automatic semigroups. PhD thesis, University of Leicester, Department of Mathematics and Computer Science (2000)Google Scholar
  14. 14.
    Hoffmann, M., Thomas, R.M.: Notions of automaticity in semigroups. Semigroup Forum 66(3), 337–367 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hoffmann, M., Kuske, D., Otto, F., Thomas, R.M.: Some relatives of automatic and hyperbolic groups. In: Workshop on Semigroups, algorithms, automata and languages 2001, pp. 379–406. World Scientific, Singapore (2002)CrossRefGoogle Scholar
  16. 16.
    Hudson, J.F.P.: Regular rewrite systems and automatic structures. In: Semigroups, Automata and Languages, pp. 145–152. World Scientific, Singapore (1998)Google Scholar
  17. 17.
    Kelarev, A.V., Quinn, S.J.: A combinatorial property and Cayley graphs of semigroups. Semigroup Forum 66(1), 89–96 (2003)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1994)Google Scholar
  19. 19.
    Knapik, T., Calbrix, H.: Thue specifications and their monadic second-order properties. Fundam. Inform. 39, 305–325 (1999)MATHMathSciNetGoogle Scholar
  20. 20.
    Kuske, D., Lohrey, M.: Logical aspects of Cayley-graphs: the group case. In: Ann. Pure Appl. Logic (to appear)Google Scholar
  21. 21.
    Kuske, D., Lohrey, M.: Decidable theories of Cayley-graphs. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 463–474. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  22. 22.
    Lohrey, M.: Automatic structures of bounded degree. In: Y. Vardi, M., Voronkov, A. (eds.) LPAR 2003. LNCS (LNAI), vol. 2850, pp. 344–358. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  23. 23.
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Heidelberg (1977)MATHGoogle Scholar
  24. 24.
    Muller, D.E., Schupp, P.E.: Groups, the theory of ends, and context-free languages. J. Comput. Syst. Sci. 26, 295–310 (1983)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Muller, D.E., Schupp, P.E.: The theory of ends, pushdown automata, and second-order logic. Theor. Comput. Sci. 37(1), 51–75 (1985)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Otto, F., Ruškuc, N.: Confluent monadic string-rewriting systems and automatic structures. J. Autom. Lang. Comb. 6(3), 375–388 (2001)MATHMathSciNetGoogle Scholar
  27. 27.
    Ruzzo, W.L.: Tree–size bounded alternation. J. Comput. Syst. Sci. 21, 218–235 (1980)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Schupp, P.E.: Groups and graphs: Groups acting on trees, ends, and cancellation diagrams. Math. Intell. 1, 205–222 (1979)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Sénizergues, G.: Formal languages and word-rewriting. In: Comon, H., Jouannaud, J.-P. (eds.) TCS School 1993. LNCS, vol. 909, pp. 75–94. Springer, Heidelberg (1993)Google Scholar
  30. 30.
    Silva, P.V., Steinberg, B.: A geometric characterization of automatic monoids. Technical Report CMUP 2000-03, University of Porto (2001)Google Scholar
  31. 31.
    Silva, P.V., Steinberg, B.: Extensions and submonoids of automatic monoids. Theor. Comput. Sci. 289, 727–754 (2002)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Sudborough, I.H.: On the tape complexity of deterministic context–free languages. J. Assoc. Comput. Mach. 25(3), 405–414 (1978)MATHMathSciNetGoogle Scholar
  33. 33.
    Venkateswaran, H.: Properties that characterize LOGCFL. J. Comput. Syst. Sci. 43, 380–404 (1991)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Markus Lohrey
    • 1
  1. 1.Lehrstuhl für Informatik IRWTH AachenGermany

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