Satisfying KBO Constraints

  • Harald Zankl
  • Aart Middeldorp
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4533)

Abstract

This paper presents two new approaches to prove termination of rewrite systems with the Knuth-Bendix order efficiently. The constraints for the weight function and for the precedence are encoded in (pseudo-)propositional logic and the resulting formula is tested for satisfiability. Any satisfying assignment represents a weight function and a precedence such that the induced Knuth-Bendix order orients the rules of the encoded rewrite system from left to right.

Keywords

Weight Function Function Symbol Conjunctive Normal Form Propositional Formula Satisfying Assignment 
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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References

  1. 1.
    Arts, T., Giesl, J.: Termination of term rewriting using dependency pairs. Theoretical Computer Science 236, 133–178 (2000)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  3. 3.
    Codish, M., Lagoon, V., Stuckey, P.: Solving partial order constraints for LPO termination. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 4–18. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Codish, M., Schneider-Kamp, P., Lagoon, V., Thiemann, R., Giesl, J.: SAT solving for argument filterings. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 30–44. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Dick, J., Kalmus, J., Martin, U.: Automating the Knuth-Bendix ordering. Acta. Infomatica 28, 95–119 (1990)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Eén, N.: Personal conversation, Google Group on Minisat (2007)Google Scholar
  7. 7.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Eén, N., Sörensson, N.: Translating pseudo-boolean constraints into SAT. Journal on Satisfiability, Boolean Modeling and Computation 2, 1–26 (2006)MATHGoogle Scholar
  9. 9.
    Endrullis, J., Waldmann, J., Zantema, H.: Matrix interpretations for proving termination of term rewriting. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 574–588. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R., Zankl, H.: SAT solving for termination analysis with polynomial interpretations. In: Proc. 10th International Conference on Theory and Applications of Satisfiability Testing. LNCS, vol. 4501, pp. 340–354. Springer, Heidelberg (2007)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.
    Hirokawa, N., Middeldorp, A.: Tyrolean termination tool. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 175–184. Springer, Heidelberg (2005)Google Scholar
  13. 13.
    Hofbauer, D., Waldmann, J.: Termination of string rewriting with matrix interpretations. In: Pfenning, F. (ed.) RTA 2006. LNCS, vol. 4098, pp. 328–342. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Knuth, D.E., Bendix, P.: Simple word problems in universal algebras. In: Leech, J. (ed.) Computational Problems in Abstract Algebra, pp. 263–297. Pergamon Press, New York (1970)Google Scholar
  15. 15.
    Korovin, K., Voronkov, A.: Orienting rewrite rules with the Knuth-Bendix order. Information and Computation 183, 165–186 (2003)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Kurihara, M., Kondo, H.: Efficient BDD encodings for partial order constraints with application to expert systems in software verification. In: Orchard, B., Yang, C., Ali, M. (eds.) IEA/AIE 2004. LNCS (LNAI), vol. 3029, pp. 827–837. Springer, Heidelberg (2004)Google Scholar
  17. 17.
    Manquinho, V., Roussel, O.: Pseudo-boolean evaluation (2007), http://www.cril.univartois.fr/PB07/
  18. 18.
    Marché, C.: Termination problem data base (TPDB), version 3.2 (June 2006), www.lri.fr/~marche/tpdb
  19. 19.
    Steinbach, J.: Extensions and comparison of simplification orders. In: Dershowitz, N. (ed.) Rewriting Techniques and Applications. LNCS, vol. 355, pp. 434–448. Springer, Heidelberg (1989)Google Scholar
  20. 20.
    Tseitin, G.: On the complexity of derivation in propositional calculus. In: Studies in Constructive Mathematics and Mathematical Logic, Part 2, pp. 115–125 (1968)Google Scholar
  21. 21.
    Zankl, H.: SAT techniques for lexicographic path orders. Seminar report (2006), Available at http://arxiv.org/abs/cs.SC/0605021
  22. 22.
    Zankl, H., Hirokawa, N., Middeldorp, A.: Constraints for argument filterings. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds.) SOFSEM 2007. LNCS, vol. 4362, pp. 579–590. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  23. 23.
    Zankl, H., Middeldorp, A.: KBO as a satisfaction problem. In: Proc. 8th International Workshop on Termination, pp. 55–59 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Harald Zankl
    • 1
  • Aart Middeldorp
    • 1
  1. 1.Institute of Computer Science, University of InnsbruckAustria

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