AC Completion with Termination Tools

  • Sarah Winkler
  • Aart Middeldorp
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6803)


We present masco tt, a tool for Knuth-Bendix completion modulo the theory of associative and commutative operators. In contrast to classical completion tools, masco tt does not rely on a fixed AC-compatible reduction order. Instead, a suitable order is implicitly constructed during a deduction by collecting all oriented rules in a similar fashion as done in the tool Slothrop. This allows for convergent systems which cannot be completed using standard orders. We outline the underlying inference system and comment on implementation details such as the use of multi-completion, term indexing techniques, and critical pair criteria.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with mu-term. In: Johnson, M., Pavlovic, D. (eds.) AMAST 2010. LNCS, vol. 6486, pp. 201–208. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Alarcón, B., Lucas, S., Meseguer, J.: A dependency pair framework for AC-termination. In: Ölveczky, P.C. (ed.) WRLA 2010. LNCS, vol. 6381, pp. 35–51. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Bachmair, L.: Canonical Equational Proofs. Progress in Theoretical Computer Science. Birkhäuser (1991)Google Scholar
  4. 4.
    Bachmair, L., Chen, T., Ramakrishnan, I.V.: Associative-commutative discrimination nets. In: Gaudel, M.-C., Jouannaud, J.-P. (eds.) TAPSOFT 1993. LNCS, vol. 668, pp. 61–74. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  5. 5.
    Ben Cherifa, A., Lescanne, P.: Termination of rewriting systems by polynomial interpretations and its implementation. SCP 9(2), 137–159 (1987)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Contejean, E., Marché, C.: CiME: Completion modulo E. In: Ganzinger, H. (ed.) RTA 1996. LNCS, vol. 1103, pp. 416–419. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  7. 7.
    Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. JAR 40(2-3), 195–220 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Fortenbacher, A.: An algebraic approach to unification under associativity and commutativity. JSC 3(3), 217–229 (1987)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Gehrke, W.: Detailed catalogue of canonical term rewrite systems generated automatically. Technical report, RISC Linz (1992)Google Scholar
  10. 10.
    Giesl, J., Kapur, D.: Dependency pairs for equational rewriting. In: Middeldorp, A. (ed.) RTA 2001. LNCS, vol. 2051, pp. 93–108. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  11. 11.
    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
  12. 12.
    Kapur, D., Musser, D.R., Narendran, P.: Only prime superpositions need be considered in the Knuth-Bendix completion procedure. JSC 6(1), 19–36 (1988)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Korp, M., Sternagel, C., Zankl, H., Middeldorp, A.: Tyrolean Termination Tool 2. In: Treinen, R. (ed.) RTA 2009. LNCS, vol. 5595, pp. 295–304. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Kurihara, M., Kondo, H.: Completion for multiple reduction orderings. JAR 23(1), 25–42 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kusakari, K.: AC-Termination and Dependency Pairs of Term Rewriting Systems. PhD thesis, JAIST (2000)Google Scholar
  16. 16.
    Lincoln, P., Christian, J.: Adventures in associative-commutative unification. JSC 8, 393–416 (1989)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Marché, C., Urbain, X.: Modular and incremental proofs of AC-termination. JSC 38(1), 873–897 (2004)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Peterson, G.E., Stickel, M.E.: Complete sets of reductions for some equational theories. JACM 28(2), 233–264 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Sato, H., Winkler, S., Kurihara, M., Middeldorp, A.: Multi-completion with termination tools (system description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 306–312. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Wehrman, I., Stump, A., Westbrook, E. M.: slothrop: Knuth-Bendix completion with a modern termination checker. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 287–296. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  21. 21.
    Winkler, S., Sato, H., Middeldorp, A., Kurihara, M.: Optimizing mkbTT (system description). In: Proc. 21st RTA. LIPIcs, vol. 6, pp. 373–384 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sarah Winkler
    • 1
  • Aart Middeldorp
    • 1
  1. 1.Institute of Computer ScienceUniversity of InnsbruckAustria

Personalised recommendations