A Completion Method to Decide Reachability in Rewrite Systems

  • Guillaume BurelEmail author
  • Gilles Dowek
  • Ying Jiang
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9322)


The Knuth-Bendix method takes in argument a finite set of equations and rewrite rules and, when it succeeds, returns an algorithm to decide if a term is equivalent to another modulo these equations and rules. In this paper, we design a similar method that takes in argument a finite set of rewrite rules and, when it succeeds, returns an algorithm to decide not equivalence but reachability modulo these rules, that is if a term reduces to another. As an application, we give new proofs of the decidability of reachability in finite ground rewrite systems and in pushdown systems.


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  1. 1.
    Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: Application to model-checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  2. 2.
    Burel, G., Dowek, G., Jiang, Y.: Automata, resolution and cut-elimination (manuscript) (2015)Google Scholar
  3. 3.
    Dauchet, M., Tison, S.: Decidability of confluence for ground term rewriting systems. In: Budach, L. (ed.) FCT 1985. LNCS, vol. 199, pp. 80–89. Springer, Heidelberg (1985)CrossRefGoogle Scholar
  4. 4.
    Dershowitz, N.: Completion and its applications. In: Aït-Kaci, H., Nivat, M. (eds.) Resolution of Equations in Algebraic Structures, vol. 2, chapter 2, pp. 31–86. Academic Press (1989)Google Scholar
  5. 5.
    Dershowitz, N., Manna, Z.: Proving termination with multiset orderings. Communications of the ACM 22(8), 465–476 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dowek, G.: What is a theory? In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 50–64. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  7. 7.
    Dowek, G.: Confluence as a cut elimination property. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 2–13. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Dowek, G.: Polarized resolution modulo. In: Calude, C.S., Sassone, V. (eds.) TCS 2010. IFIP AICT, vol. 323, pp. 182–196. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Hirokawa, N.: Commutation and signature extension. In: Tiwari, A., Aoto, T. (eds.) International Workshop on Confluence (2015)Google Scholar
  10. 10.
    Huet, G.: Confluent reductions: abstract properties and applications to term rewriting systems. Journal of the Association of Computing Machinery 27(4), 797–821 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jouannaud, J.-P., Kirchner, H.: Completion of a set of rules modulo a set of equations. SIAM Journal of Computing 15(4), 1155–1194 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kamin, S., Lévy, J.-J.: Attempts for generalizing the recursive path ordering (unpublished manuscript)Google Scholar
  13. 13.
    Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: Computational Problems in Abstract Algebra, pp. 263–297, Pergamon (1970)Google Scholar
  14. 14.
    Lankford, D.S.: Canonical inference. Technical report, Louisiana Tech. University (1975)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Ensiie, 1 square de la RésistanceÉvryFrance
  2. 2.InriaParis Cedex 13France
  3. 3.State Key Laboratory of Computer Science, Institute of SoftwareChinese Academy of SciencesBeijingChina

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