First-order applicative term rewrite systems provide a natural framework for modeling higher-order aspects. In this paper we present a transformation from untyped applicative term rewrite systems to functional term rewrite systems that preserves and reflects termination. Our transformation is less restrictive than other approaches. In particular, head variables in right-hand sides of rewrite rules can be handled. To further increase the applicability of our transformation, we present a version for dependency pairs.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aoto, T., Yamada, T.: Termination of simply typed term rewriting 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: HOR 2004. Technical Report AIB-2004-03, RWTH Aachen. 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.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236(1-2), 133–178 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  6. 6.
    Blanqui, F., Jouannaud, J.-P., Rubio, A.: HORPO with computability closure: A reconstruction. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 138–150. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Dershowitz, N.: 33 Examples of termination. In: French Spring School of Theoretical Computer Science. LNCS, vol. 909, pp. 16–26. Springer, Heidelberg (1995)Google Scholar
  8. 8.
    Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of rewrite systems. Journal of Automated Reasoning 40(2-3), 195–220 (2008)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 2004. LNCS, vol. 3452, pp. 301–331. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. Journal of Automated Reasoning 37(3), 155–203 (2006)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. Information and Computation 199(1-2), 172–199 (2005)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: Techniques and features. Information and Computation 205(4), 474–511 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Jouannaud, J.P., Rubio, A.: Polymorphic higher-order recursive path orderings. Journal of the ACM 54(1) (2007)Google Scholar
  15. 15.
    Kennaway, R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Comparing curried and uncurried rewriting. Journal of Symbolic Computation 21(1), 15–39 (1996)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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
  17. 17.
    Thiemann, R.: The DP Framework for Proving Termination of Term Rewriting. PhD thesis, RWTH Aachen, Available as technical report AIB-2007-17 (2007)Google Scholar
  18. 18.
    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
  19. 19.
    Xi, H.: Towards automated termination proofs through freezing. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 271–285. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  20. 20.
    Zantema, H.: Termination. In: Terese (ed.) Term Rewriting Systems 2003. Cambridge Tracts in Theoretical Computer Science, vol. 55, pp. 181–259. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nao Hirokawa
    • 1
  • Aart Middeldorp
    • 2
  • Harald Zankl
    • 2
  1. 1.School of Information ScienceJapan Advanced Institute of Science and TechnologyJapan
  2. 2.Institute of Computer ScienceUniversity of InnsbruckAustria

Personalised recommendations