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On Tree Automata that Certify Termination of Left-Linear Term Rewriting Systems

  • Alfons Geser
  • Dieter Hofbauer
  • Johannes Waldmann
  • Hans Zantema
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3467)

Abstract

We present a new method for proving termination of term rewriting systems automatically. It is a generalization of the match bound method for string rewriting. To prove that a term rewriting system terminates on a given regular language of terms, we first construct an enriched system over a new signature that simulates the original derivations. The enriched system is an infinite system over an infinite signature, but it is locally terminating: every restriction of the enriched system to a finite signature is terminating. We then construct iteratively a finite tree automaton that accepts the enriched given regular language and is closed under rewriting modulo the enriched system. If this procedure stops, then the enriched system is compact: every enriched derivation involves only a finite signature. Therefore, the original system terminates. We present three methods to construct the enrichment: top heights, roof heights, and match heights. Top and roof heights work for left-linear systems, while match heights give a powerful method for linear systems. For linear systems, the method is strengthened further by a forward closure construction. Using these methods, we give examples for automated termination proofs that cannot be obtained by standard methods.

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References

  1. 1.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  2. 2.
    Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (1997–2001), Available at http://www.grappa.univ-lille3.fr/tata/
  3. 3.
    Contejean, E., Marché, C., Monate, B., Urbain, X.: Proving Termination of Rewriting with CiME. In: Rubio, A. (ed.) Proc. 6th Int. Workshop on Termination WST 2003, Universidad Politécnica de Valencia, Spain. Technical Report DSIC II/15/03, pp. 71–73 (2003)Google Scholar
  4. 4.
    Dershowitz, N.: Termination of linear rewriting systems. In: Even, S., Kariv, O. (eds.) ICALP 1981. LNCS, vol. 115, pp. 448–458. Springer, Heidelberg (1981)Google Scholar
  5. 5.
    Gécseg, F., Steinby, M.: Tree Languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 1–68. Springer, Heidelberg (1997)Google 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.: Match-bounded string rewriting systems. Appl. Algebra Engrg. Comm. Comput. 15(3-4), 149–171 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: Tree automata that certify termination of term rewriting systems. In: Codish, M., Middeldorp, A. (eds.) Proc. 7th Int. Workshop on Termination WST-04, Aachener Informatik Berichte AIB-2004-07, RWTH Aachen, Gemany, pp. 14–17 (2004)Google Scholar
  9. 9.
    Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: Tree automata that certify termination of left-linear term rewriting systems (2005), Full version available at http://www.imn.htwk-leipzig.de/~waldmann/pub/rta05/
  10. 10.
    Geser, A., Hofbauer, D., Waldmann, J., Zantema, H.: Finding finite automata that certify termination of string rewriting. In: Domaratzki, M., Okhotin, A., Salomaa, K., Yu, S. (eds.) CIAA 2004. LNCS, vol. 3317, pp. 134–145. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  11. 11.
    Geupel, O.: Overlap closures and termination of term rewriting systems. Technical Report MIP-8922, Universität Passau, Germany (1989)Google Scholar
  12. 12.
    Giesl, J., Thiemann, R., Schneider-Kamp, P., Falke, S.: Automated termination proofs with AProVE. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 210–220. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Hirokawa, N., Middeldorp, A.: Tsukuba Termination Tool. In: Nieuwenhuis, R. (ed.) RTA 2003. LNCS, vol. 2706, pp. 311–320. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  14. 14.
    Lankford, D.S., Musser, D.R.: A finite termination criterion. Unpublished draft, Information Sciences Institute, University of Southern California, Marina-del-Rey, CA (1978)Google Scholar
  15. 15.
    Marché, C., Rubio, A. (eds.): Termination Problems Data Base (2004), http://www.lri.fr/~marche/wst2004-competition/tpdb.html
  16. 16.
    Middeldorp, A.: Approximations for strategies and termination. In: Proc. 2nd Int. Workshop on Reduction Strategies in Rewriting and Programming. Electron. Notes Theor. Comput. Sci, vol. 70(6) (2002)Google Scholar
  17. 17.
    van Oostrom, V., de Vrijer, R.: Equivalences of Reductions. In: Terese, Term Rewriting Systems, pp. 301–474. Cambridge Univ. Press, Cambridge (2003)Google Scholar
  18. 18.
    Waldmann, J.: Matchbox: a tool for match-bounded string rewriting. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 85–94. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  19. 19.
    Zantema, H.: TORPA: Termination of rewriting proved automatically. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 95–104. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  20. 20.
    Zantema, H.: Termination. In: Terese, Term Rewriting Systems, pp. 181–259. Cambridge University Press, Cambridge (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Alfons Geser
    • 1
  • Dieter Hofbauer
    • 2
  • Johannes Waldmann
    • 3
  • Hans Zantema
    • 4
  1. 1.National Institute of AerospaceHamptonUSA
  2. 2. KasselGermany
  3. 3.Fb IMNHochschule für Technik, Wirtschaft und Kultur (FH) LeipzigLeipzigGermany
  4. 4.Faculteit Wiskunde en InformaticaTechnische Universiteit EindhovenEindhovenThe Netherlands

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