Advertisement

Modular Church-Rosser Modulo

  • Jean-Pierre Jouannaud
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4098)

Abstract

In [12], Toyama proved that the union of two confluent term-rewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [5,11].

In this paper we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies in two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions; a modularity property of ordered completion, that allows to pairwise match the caps and alien substitutions of two equivalent terms.

We then show that Toyama’s modularity result scales up to rewriting modulo equations in all considered cases.

Keywords

Normal Form Induction Hypothesis Function Symbol Critical Pair Ground Term 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Dershowitz, N., Jouannaud, J.-P.: Rewrite systems. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 243–309. North-Holland, Amsterdam (1990)Google Scholar
  2. 2.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. Journal of the ACM 27(4), 797–821 (1980)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Jouannaud, J.-P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM Journal on Computing 15(4), 1155–1194 (1986)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Jouannaud, J.-P., van Raasdon, F., Rubio, A.: Rewriting with types and arities (2005), available from the webGoogle Scholar
  5. 5.
    Klop, J.W., Middeldorp, A., Toyama, Y., de Vrijer, R.: Modularity of confluence: A simplified proof. Information Processing Letters 49(2), 101–109 (1994)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Lankford, D.S., Ballantyne, A.M.: Decision procedures for simple equational theories with permutative axioms: Complete sets of permutative reductions. Research Report Memo ATP-37, Department of Mathematics and Computer Science, University of Texas, Austin, Texas, USA (August 1977)Google Scholar
  7. 7.
    Marché, C.: Normalised rewriting and normalised completion. In: Proc. 9th IEEE Symp. Logic in Computer Science, pp. 394–403 (1994)Google Scholar
  8. 8.
    Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. Theoretical Computer Science 192(1), 3–29 (1998)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Middeldorp, A.: Modular aspects of properties of term rewriting systems related to normal forms. In: Dershowitz, N. (ed.) RTA 1989. LNCS, vol. 355, pp. 263–277. Springer, Heidelberg (1989)Google Scholar
  10. 10.
    Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. Journal of the ACM 28(2), 233–264 (1981)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bezem, M., Kop, J.W., de Vrijer, R. (eds.): Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science 55. Cambridge University Press, Cambridge (2003)Google Scholar
  12. 12.
    Toyama, Y.: On the Church-Rosser property for the direct sum of term rewriting systems. Journal of the ACM 34(1), 128–143 (1987)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jean-Pierre Jouannaud
    • 1
  1. 1.École Polytechnique, LIX, UMR CNRS 7161PalaiseauFrance

Personalised recommendations