On the Theory of One-Step Rewriting in Trace Monoids
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 .
Unable to display preview. Download preview PDF.
- 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.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
- 9.V. Diekert and G. Rozenberg, editors. The Book of Traces. World Scientific, Singapore, 1995.Google Scholar
- 12.H. Gaifman. On local and nonlocal properties. In J. Stern, editor, Logic Colloquium’ 81, pages 105–135, 1982, North Holland.Google Scholar
- 13.F. Jacquemard. Automates d’arbres et Réécriture de termes. PhD thesis, Université de Paris-Sud, 1996.Google Scholar
- 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.L. Libkin. Logics capturing local properties. ACM Transactions on Computational Logic. To appear.Google Scholar
- 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
- 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.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.A. Mazurkiewicz. Concurrent program schemes and their interpretation. Technical report, DAIMI Report PB-78, Aarhus University, 1977.Google Scholar
- 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
- 26.Terese. Term Rewriting Systems. To appear with Cambridge University Press, 2001.Google Scholar
- 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.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