Nominal Completion for Rewrite Systems with Binders

  • Maribel Fernández
  • Albert Rubio
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7392)


We design a completion procedure for nominal rewriting systems, based on a generalisation of the recursive path ordering to take into account alpha equivalence. Nominal rewriting generalises first-order rewriting by providing support for the specification of binding operators. Completion of rewriting systems with binders is a notably difficult problem; the completion procedure presented in this paper is the first to deal with binders in rewrite rules.


nominal syntax rewriting orderings completion 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arts, T., Giesl, J.: Termination of Term Rewriting Using Dependency Pairs. Theoretical Computer Science 236, 133–178 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press, Great Britain (1998)Google Scholar
  3. 3.
    Bachmair, L., Dershowitz, N., Hsiang, J.: Orderings for equational proofs. In: Proc. Symp. Logic in Computer Science, Boston, pp. 346–357 (1986)Google Scholar
  4. 4.
    Blanqui, F.: Termination and Confluence of Higher-Order Rewrite Systems. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833, pp. 47–61. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Bachmair, L., Dershowitz, N., Plaisted, D.A.: Completion Without Failure. In: Ait-Kaci, H., Nivat, M. (eds.) Resolution of Equations in Algebraic Structures. Rewriting Techniques, vol. 2, ch. 1. Academic Press, New York (1989)Google Scholar
  6. 6.
    Blanqui, F., Jouannaud, J.-P., Rubio, A.: The Computability Path Ordering: The End of a Quest. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 1–14. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Borralleras, C., Ferreira, M., Rubio, A.: Complete Monotonic Semantic Path Orderings. In: McAllester, D. (ed.) CADE 2000. LNCS (LNAI), vol. 1831, pp. 346–364. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  8. 8.
    Breazu-Tannen, V., Gallier, J.: Polymorphic rewriting conserves algebraic strong normalization. Theoretical Computer Science 83(1) (1991)Google Scholar
  9. 9.
    Calvès, C., Fernández, M.: Matching and Alpha-Equivalence Check for Nominal Terms. Journal of Computer and System Sciences, Special issue: Selected papers from WOLLIC 2008 (2009)Google Scholar
  10. 10.
    Cheney, J.: A Dependent Nominal Type Theory. Logical Methods in Computer Science 8(1) (2012)Google Scholar
  11. 11.
    Clouston, R.A., Pitts, A.M.: Nominal Equational Logic. In: Cardelli, L., Fiore, M., Winskel, G. (eds.) Computation, Meaning and Logic. Articles dedicated to Gordon Plotkin. Electronic Notes in Theoretical Computer Science, vol. 1496. Elsevier (2007)Google Scholar
  12. 12.
    Dershowitz, N.: Orderings for Term-Rewriting Systems. Theoretical Computer Science 17(3), 279–301 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Fernández, M., Gabbay, M.J.: Nominal rewriting. Information and Computation 205(6) (2007)Google Scholar
  14. 14.
    Fernández, M., Gabbay, M.J.: Closed nominal rewriting and efficiently computable nominal algebra equality. In: Proc. 5th Int. Workshop on Logical Frameworks and Metalanguages: Theory and Practice (LFMTP 2010). EPTCS, vol. 34 (2010)Google Scholar
  15. 15.
    Fernández, M., Gabbay, M.J., Mackie, I.: Nominal rewriting systems. In: Proceedings of the 6th ACM-SIGPLAN Symposium on Principles and Practice of Declarative Programming (PPDP 2004), Verona, Italy. ACM Press (2004)Google Scholar
  16. 16.
    Gabbay, M.J., Mathijssen, A.: Nominal universal algebra: equational logic with names and binding. Journal of Logic and Computation 19(6), 1455–1508 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Gabbay, M.J., Pitts, A.M.: A new approach to abstract syntax with variable binding. Formal Aspects of Computing 13, 341–363 (2001)CrossRefGoogle Scholar
  18. 18.
    Hamana, M.: Semantic Labelling for Proving Termination of Combinatory Reduction Systems. In: Escobar, S. (ed.) WFLP 2009. LNCS, vol. 5979, pp. 62–78. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  19. 19.
    Jouannaud, J.-P., Okada, M.: Executable higher-order algebraic specification languages. In: Proceedings, Sixth Annual IEEE Symposium on Logic in Computer Science, pp. 350–361. IEEE Computer Society Press (1991)Google Scholar
  20. 20.
    Jouannaud, J.-P., Rubio, A.: Polymorphic Higher-Order Recursive Path Orderings. Journal of ACM 54(1) (2007)Google Scholar
  21. 21.
    Khasidashvili, Z.: Expression reduction systems. In: Proceedings of I. Vekua Institute of Applied Mathematics, Tbilisi, vol. 36, pp. 200–220 (1990)Google Scholar
  22. 22.
    Klop, J.-W., van Oostrom, V., van Raamsdonk, F.: Combinatory reduction systems, introduction and survey. Theoretical Computer Science 121, 279–308 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Knuth, D., Bendix, P.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, Oxford (1970)Google Scholar
  24. 24.
    Kusakari, K., Chiba, Y.: A higher-order Knuth-Bendix procedure and its applications. IEICE Transactions on Information and Systems E90-D(4), 707–715 (2007)CrossRefGoogle Scholar
  25. 25.
    Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. Theoretical Computer Science 192, 3–29 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Newman, M.H.A.: On theories with a combinatorial definition of equivalence. Annals of Mathematics 43(2), 223–243 (1942)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Nipkow, T., Prehofer, C.: Higher-Order Rewriting and Equational Reasoning. In: Bibel, W., Schmitt, P. (eds.) Automated Deduction — A Basis for Applications. Volume I: Foundations. Applied Logic Series, vol. 8, pp. 399–430. Kluwer (1998)Google Scholar
  28. 28.
    Pitts, A.M.: Nominal logic, a first order theory of names and binding. Information and Computation 186, 165–193 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Pitts, A.M.: Structural Recursion with Locally Scoped Names. Journal of Functional Programming 21(3), 235–286 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    van de Pol, J.C.: Termination of higher-order rewrite systems. PhD thesis, Utrecht University, Utrecht, The Netherlands (December 1996)Google Scholar
  31. 31.
    Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press (2003)Google Scholar
  32. 32.
    Urban, C., Pitts, A.M., Gabbay, M.J.: Nominal unification. Theoretical Computer Science 323, 473–497 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Maribel Fernández
    • 1
  • Albert Rubio
    • 2
  1. 1.Dept. of InformaticsKing’s College LondonUK
  2. 2.LSITechnical University of CataloniaBarcelonaSpain

Personalised recommendations