Termination of Narrowing Using Dependency Pairs

  • María Alpuente
  • Santiago Escobar
  • José Iborra
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5366)

Abstract

In this work, we extend the dependency pair approach for automated proofs of termination in order to prove the termination of narrowing. Our extension of the dependency pair approach generalizes the standard notion of dependency pairs by taking specifically into account the dependencies between the left-hand side of a rewrite rule and its own argument subterms. We demonstrate that the new narrowing dependency pairs exactly capture the narrowing termination behavior and provide an effective termination criterion which we prove to be sound and complete. Finally, we discuss how the problem of analyzing narrowing chains can be recast as a standard analysis problem for traditional (rewriting) chains, so that the proposed technique can be effectively mechanized by reusing the standard DP infrastructure.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alarcón, B., Gutiérrez, R., Iborra, J., Lucas, S.: Proving termination of context-sensitive rewriting with Mu–Term. ENTCS 188, 105–115 (2007)MATHGoogle Scholar
  2. 2.
    Alpuente, M., Escobar, S., Iborra, J.: Modular Termination of Basic Narrowing. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 1–16. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  3. 3.
    Alpuente, M., Escobar, S., Iborra, J.: Dependency Pairs for the Termination of Narrowing. Technical Report DSIC-II/08/08, DSIC-UPV (2008)Google Scholar
  4. 4.
    Alpuente, M., Escobar, S., Iborra, J.: Termination of Narrowing revisited. Theor. Comput. Sci. (to appear, 2008)Google Scholar
  5. 5.
    Alpuente, M., Falaschi, M., Vidal, G.: Compositional Analysis for Equational Horn Programs. In: Rodríguez-Artalejo, M., Levi, G. (eds.) ALP 1994. LNCS, vol. 850, pp. 77–94. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  6. 6.
    Arts, T., Giesl, J.: Termination of Term Rewriting using Dependency Pairs. Theor. Comput. Sci. 236(1-2), 133–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Christian, J.: Some termination criteria for narrowing and e-narrowing. In: Kapur, D. (ed.) CADE 1992. LNCS, vol. 607, pp. 582–588. Springer, Heidelberg (1992)Google Scholar
  8. 8.
    Escobar, S., Meadows, C., Meseguer, J.: A Rewriting-Based Inference System for the NRL Protocol Analyzer and its Meta-Logical Properties. Theor. Comput. Sci. 367(1-2), 162–202 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Giesl, J., Thiemann, R., Schneider-Kamp, P.: The dependency pair framework: Combining techniques for automated termination proofs. In: Baader, F., Voronkov, A. (eds.) LPAR 2005. LNCS, vol. 3452, pp. 301–331. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Automated termination proofs with AProVe. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 210–220. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and Improving Dependency Pairs. J. Autom. Reasoning 37(3), 155–203 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hanus, M.: The Integration of Functions into Logic Programming: From Theory to Practice. J. Log. Program. 19-20, 583–628 (1994)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hirokawa, N., Middeldorp, A.: Dependency pairs revisited. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 249–268. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  14. 14.
    Hullot, J.-M.: Canonical Forms and Unification. In: Bibel, W. (ed.) CADE 1980. LNCS, vol. 87, pp. 318–334. Springer, Heidelberg (1980)Google Scholar
  15. 15.
    Kirchner, C., Kirchner, H., Santana de Oliveira, A.: Analysis of Rewrite-Based Access Control Policies. In: 3rd Int’l Workshop on Security and Rewriting Techniques, SecreT 2008. ENTCS (to appear, 2008)Google Scholar
  16. 16.
    Meseguer, J., Thati, P.: Symbolic reachability analysis using narrowing and its application to verification of cryptographic protocols. Higher-Order and Symbolic Computation 20(1-2), 123–160 (2007)CrossRefMATHGoogle Scholar
  17. 17.
    Nguyen, M.T., Schneider-Kamp, P., de Schreye, D., Giesl, J.: Termination Analysis of Logic Programs based on Dependency Graphs. In: King, A. (ed.) LOPSTR 2007. LNCS, vol. 4915, pp. 8–22. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Nishida, N., Miura, K.: Dependency graph method for proving termination of narrowing. In: 8th Int’l Workshop on Termination, WST 2006 (2006)Google Scholar
  19. 19.
    Nishida, N., Sakai, M., Sakabe, T.: Narrowing-based simulation of term rewriting systems with extra variables. ENTCS 86(3) (2003)Google Scholar
  20. 20.
    Nishida, N., Vidal, G.: Termination of Narrowing via Termination of Rewriting (2008), http://www.dsic.upv.es/~gvidal
  21. 21.
    TeReSe (ed.): Term Rewriting Systems. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • María Alpuente
    • 1
  • Santiago Escobar
    • 1
  • José Iborra
    • 1
  1. 1.Technical University of Valencia (UPV)Spain

Personalised recommendations