Argument Filterings and Usable Rules for Simply Typed Dependency Pairs

  • Takahito Aoto
  • Toshiyuki Yamada
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5749)

Abstract

Simply typed term rewriting (Yamada, 2001) is a framework of higher-order term rewriting without bound variables based on Lisp-like syntax. The dependency pair method for the framework has been obtained by extending the first-order dependency pair method and subterm criterion in (Aoto & Yamada, 2005). In this paper, we incorporate termination criteria using reduction pairs and related refinements into the simply typed dependency pair framework using recursive path orderings for S-expression rewriting systems (Toyama, 2008). In particular, we incorporate the usable rules criterion with respect to argument filterings, which is a key ingredient to prove the termination in a modular way. The proposed technique has been implemented in a termination prover and an experimental result is reported.

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References

  1. 1.
    Aoto, T., Yamada, T.: Termination of simply typed term rewriting systems by translation and labelling. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 380–394. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Aoto, T., Yamada, T.: Termination of simply-typed applicative term rewriting systems. In: Proc. of HOR 2004, pp. 61–65 (2004)Google Scholar
  3. 3.
    Aoto, T., Yamada, T.: Dependency pairs for simply typed term rewriting. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 120–134. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Aoto, T., Yamada, T.: Argument filterings and usable rules for simply typed dependency pairs (extended abstract). In: Proc. of HOR 2007, pp. 21–27 (2007)Google Scholar
  5. 5.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236(1-2), 133–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Codish, M., Lagoon, V., Stuckey, P.J.: Solving partial order constraints for LPO termination. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 4–18. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Giesl, J., Schneider-Kamp, P., Thiemann, R.: AProVE 1.2: Automatic termination proofs in the dependency pair framework. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 281–286. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Giesl, J., Thiemann, R., Schneider-Kamp, P.: Proving and disproving termination of higher-order functions. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 216–231. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Giesl, J., Thiemann, R., Schneider-Kamp, P.: Mechanizing and improving dependency pairs. Journal of Automated Reasoning 37(3), 155–203 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: Techniques and features. Information and Computation 205(4), 474–511 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hirokawa, N., Middeldorp, A., Zankl, H.: Uncurrying for termination. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 667–681. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Kennaway, R., Klop, J.W., Sleep, R., de Vries, F.-J.: Comparing curried and uncurried rewriting. Journal of Symbolic Computation 21, 57–78 (1996)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kusakari, K.: On proving termination of term rewriting systems with higher-order variables. IPSJ Transactions on Programming 42(SIG. 7 PRO. 11), 35–45 (2001)Google Scholar
  14. 14.
    Kusakari, K.: Higher-order path orders based on computability. IEICE Trans. on Inf. & Sys., E87–D(2), 352–359 (2004)Google Scholar
  15. 15.
    Kusakari, K., Sakai, M.: Enhancing dependency pair method using strong computability in simply-typed term rewriting. Applicable Algebra in Engineering, Communication and Computing 18(5), 407–431 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kusakari, K., Sakai, M.: Static dependency pair method for simply-typed term rewriting and related techniques. IEICE Trans. on Inf. & Sys., E92–D(2), 235–247 (2009)Google Scholar
  17. 17.
    Sakurai, T., Kusakari, K., Sakai, M., Sakabe, T., Nishida, N.: Usable rules and labeling product-typed terms for dependency pair method in simply-typed term rewriting systems (in Japanese). IEICE Trans. on Inf. & Sys., J90–D(4), 978–989 (2007)Google Scholar
  18. 18.
    Schneider-Kamp, P., Thiemann, R., Annov, E., Codish, M., Giesl, J.: Proving termination using recursive path orders and SAT solving. In: Konev, B., Wolter, F. (eds.) FroCos 2007. LNCS (LNAI), vol. 4720, pp. 267–282. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Toyama, Y.: Termination of S-expression rewriting systems: Lexicographic path ordering for higher-order terms. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 40–54. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Toyama, Y.: Termination proof of S-expression rewriting systems with recursive path relations. In: Voronkov, A. (ed.) RTA 2008. LNCS, vol. 5117, pp. 381–391. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  21. 21.
    Urbain, X.: Modular & incremental automated termination proofs. Journal of Automated Reasoning 32, 315–355 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yamada, T.: Confluence and termination of simply typed term rewriting systems. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 338–352. Springer, Heidelberg (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Takahito Aoto
    • 1
  • Toshiyuki Yamada
    • 2
  1. 1.Research Institute of Electrical CommunicationTohoku UniversityJapan
  2. 2.Graduate School of EngineeringMie UniversityJapan

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