On the Theory of One-Step Rewriting in Trace Monoids

  • Dietrich Kuske
  • Markus Lohrey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2380)

Abstract

We prove that the first-order theory of the one-step rewriting relation associated with a trace rewriting system is decidable and give a nonelementary lower bound for the complexity. The decidability extends known results on semi-Thue systems but our proofs use new methods; these new methods yield the decidability of local properties expressed in first-order logic augmented by modulo-counting quantifiers. Using the main decidability result, we describe a class of trace rewriting systems for which the confluence problem is decidable. The complete proofs can be found in the Technical Report [14].

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References

  1. 1.
    P. Cartier and D. Foata. Problèmes combinatoires de commutation et réarrangements. Lecture Notes in Mathematics vol. 85. Springer, Berlin-Heidelberg-New York, 1969.MATHGoogle Scholar
  2. 2.
    D. Caucal. On the regular structure of prefix rewriting. Theoretical Computer Science, 106:61–86, 1992.CrossRefMathSciNetGoogle Scholar
  3. 3.
    K. Compton and C. Henson. A uniform method forproving lower bounds on the computational complexity of logical theories. Annals of Pure and Applied Logic, 48:1–79, 1990.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    B. Courcelle. The monadic second-order logic of graphs, II: Infinite graphs of bounded width. Mathematical Systems Theory, 21:187–221, 1989.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Proceedings of the 5th Annual IEEE Symposium on Logic in Computer Science (LICS’ 90), pages 242–256. IEEE Computer Society Press, 1990.Google Scholar
  6. 6.
    V. Diekert. On the Knuth-Bendix completion for concurrent processes. In Th. Ottmann, editor, Proceedings of the14th International Colloquium on Automata, Languages and Programming (ICALP 87), Karlsruhe (Germany), number 267 in Lecture Notes in Computer Science, pages 42–53. Springer, 1987.Google Scholar
  7. 7.
    V. Diekert. Combinatorics on Traces. Number 454 in Lecture Notes in Computer Science. Springer, 1990.MATHGoogle Scholar
  8. 8.
    V. Diekert, Y. Matiyasevich, and A. Muscholl. Solving word equations modulo partial commutations. Theoretical Computer Science, 224(1–2):215–235, 1999.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    V. Diekert and G. Rozenberg, editors. The Book of Traces. World Scientific, Singapore, 1995.Google Scholar
  10. 10.
    C. Duboc. On some equations in free partially commutative monoids. Theoretical Computer Science, 46:159–174, 1986.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    C. Frougny and J. Sakarovitch. Synchronized rational relations of finite and infinite words. Theoretical Computer Science, 108(1):45–82, 1993.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium’ 81, pages 105–135, 1982, North Holland.Google Scholar
  13. 13.
    F. Jacquemard. Automates d’arbres et Réécriture de termes. PhD thesis, Université de Paris-Sud, 1996.Google Scholar
  14. 14.
    D. Kuske and M. Lohrey. On the theory of one-step rewriting in trace monoids. Technical Report 2002-01, Department of Mathematics and Computer Science, University of Leicester. Available at http://www.mcs.le.ac.uk/~dkuske/pub-rest.html#UNP9.
  15. 15.
    L. Libkin. Logics capturing local properties. ACM Transactions on Computational Logic. To appear.Google Scholar
  16. 16.
    M. Lohrey. On the confluence of trace rewriting systems. In V. Arvind and R. Ramanujam, editors, Proceedings of the 18th Conference on Foundations of Software Technology and Theoretical Computer Science, (FSTTCS’98), Chennai (India), number 1530 in Lecture Notes in Computer Science, pages 319–330. Springer, 1998.Google Scholar
  17. 17.
    M. Lohrey. Confluence problems for trace rewriting systems. Information and Computation. 170:1–25, 2001.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    A. Markov. On the impossibility of certain algorithms in the theory of associative systems. Doklady Akademii Nauk SSSR, 55, 58:587–590, 353–356, 1947.Google Scholar
  19. 19.
    Y. Matiyasevich. Some decision problems for traces. In S. Adian and A. Nerode, editors, Proceedings of the 4th International Symposium on Logical Foundations of Computer Science (LFCS’97), Yaroslavl (Russia), number 1234 in Lecture Notes in Computer Science, pages 248–257. Springer, 1997.Google Scholar
  20. 20.
    A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical report, DAIMI Report PB-78, Aarhus University, 1977.Google Scholar
  21. 21.
    A. Meyer. Weak monadic second order theory of one successor is not elementary recursive. In Proc. Logic Colloquium, Lecture Notes in Mathematics vol. 453, pages 132–154. Springer, 1975.Google Scholar
  22. 22.
    P. Narendran and F. Otto. Preperfectness is undecidable for Thue systems containing only length-reducing rules and a single commutation rule. Information Processing Letters, 29:125–130, 1988.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    J. Nurmonen. Counting modulo quantifiers on finite structures. Information and Computation, 160:62–87, 2000. LICS 1996, Part I (New Brunswick, NJ).CrossRefMathSciNetGoogle Scholar
  24. 24.
    E. Post. Recursive unsolvability of aproblemofThue. Journal of Symbolic Logic, 12(1):1–11, 1947.CrossRefMathSciNetGoogle Scholar
  25. 25.
    H. Straubing, D. Thérien, and W. Thomas. Regular languages defined with generalized quantifiers. Information and Computation, 118:289–301, 1995.MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Terese. Term Rewriting Systems. To appear with Cambridge University Press, 2001.Google Scholar
  27. 27.
    W. Thomas. On logical definability of trace languages. In V. Diekert, editor, Proceedings of a workshop of the ESPRIT Basic Research Action No 3166: Algebraic and Syntactic Methods in Computer Science (ASMICS), Kochel am See (Germany), Report TUM-I9002, Technical University of Munich, pages 172–182, 1990.Google Scholar
  28. 28.
    A. Thue. Probleme über die Veränderungen von Zeichenreihen nach gegebenen Regeln. Skr. Vid. Kristiania, I Math. Natuv. Klasse, No. 10, 34 S., 1914.Google Scholar
  29. 29.
    R. Treinen. The first-order theory of linear one-step rewriting is undecidable. Theoretical Computer Science, 208(1–2): 149–177, 1998.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Dietrich Kuske
    • 1
  • Markus Lohrey
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of LeicesterLeicesterUK
  2. 2.Universität Stuttgart, Institut für InformatikStuttgartGermany

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