Beyond Dependency Graphs

  • Martin Korp
  • Aart Middeldorp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5663)


The dependency pair framework is a powerful technique for proving termination of rewrite systems. One of the most frequently used methods within this framework is the dependency graph processor. In this paper we improve this processor by incorporating right-hand sides of forward closures. In combination with tree automata completion we obtain an efficient processor which can be used instead of the dependency graph approximations that are in common use in termination provers.


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  1. 1.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. TCS 236(1-2), 133–178 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  3. 3.
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (2002),
  4. 4.
    Dershowitz, N.: Termination of linear rewriting systems (preliminary version). In: Even, S., Kariv, O. (eds.) ICALP 1981. LNCS, vol. 115, pp. 448–458. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  5. 5.
    Durand, I., Middeldorp, A.: Decidable call-by-need computations in term rewriting. I&C 196(2), 95–126 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Genet, T.: Decidable approximations of sets of descendants and sets of normal forms. In: Nipkow, T. (ed.) RTA 1998. LNCS, vol. 1379, pp. 151–165. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  7. 7.
    Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: Finding finite automata that certify termination of string rewriting systems. IJFCS 16(3), 471–486 (2005)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: On tree automata that certify termination of left-linear term rewriting systems. I&C 205(4), 512–534 (2007)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Geupel, O.: Overlap closures and termination of term rewriting systems. Report MIP-8922, Universität Passau (1989)Google Scholar
  10. 10.
    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
  11. 11.
    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 (LNAI), vol. 3452, pp. 301–331. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  12. 12.
    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
  13. 13.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Mechanizing and improving dependency pairs. JAR 37(3), 155–203 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hirokawa, N., Middeldorp, A.: Automating the dependency pair method. I&C 199(1-2), 172–199 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hirokawa, N., Middeldorp, A.: Tyrolean termination tool: Techniques and features. I&C 205(4), 474–511 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Korp, M., Middeldorp, A.: Proving termination of rewrite systems using bounds. In: Baader, F. (ed.) RTA 2007. LNCS, vol. 4533, pp. 273–287. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Korp, M., Middeldorp, A.: Match-bounds with dependency pairs for proving termination of rewrite systems. In: Martín-Vide, C., Otto, F., Fernau, H. (eds.) LATA 2008. LNCS, vol. 5196, pp. 321–332. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  18. 18.
    Korp, M., Middeldorp, A.: Match-bounds revisited. I&C (to appear, 2009)Google Scholar
  19. 19.
    Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean termination tool 2. In: Proc. 20th RTA. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)Google Scholar
  20. 20.
    Kusakari, K.: Termination, AC-Termination and Dependency Pairs of Term Rewriting Systems. PhD thesis, JAIST (2000)Google Scholar
  21. 21.
    Kusakari, K., Toyama, Y.: On proving AC-termination by AC-dependency pairs. Research Report IS-RR-98-0026F, School of Information Science, JAIST (1998)Google Scholar
  22. 22.
    Middeldorp, A.: Approximating dependency graphs using tree automata techniques. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 593–610. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  23. 23.
    Middeldorp, A.: Approximations for strategies and termination. In: Proc. 2nd WRS. ENTCS, vol. 70, pp. 1–20 (2002)Google Scholar
  24. 24.
    Nagaya, T., Toyama, Y.: Decidability for left-linear growing term rewriting systems. I&C 178(2), 499–514 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Thiemann, R.: The DP Framework for Proving Termination of Term Rewriting. PhD thesis, RWTH Aachen (2007); available as technical report AIB-2007-17Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Martin Korp
    • 1
  • Aart Middeldorp
    • 1
  1. 1.Institute of Computer ScienceUniversity of InnsbruckAustria

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