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)


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.


Dependency Graph Usable Rule Argument Position Termination Proof Dependency Pair 
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.


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  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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