On Transfinite Knuth-Bendix Orders

  • Laura Kovács
  • Georg Moser
  • Andrei Voronkov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6803)

Abstract

In this paper we discuss the recently introduced transfinite Knuth-Bendix orders. We prove that any such order with finite subterm coefficients and for a finite signature is equivalent to an order using ordinals below ωω, that is, finite sequences of natural numbers of a fixed length. We show that this result does not hold when subterm coefficients are infinite. However, we prove that in this general case ordinals below \(\omega^{\omega^{\omega}}\) suffice. We also prove that both upper bounds are tight. We briefly discuss the significance of our results for the implementation of first-order theorem provers and describe relationships between the transfinite Knuth-Bendix orders and existing implementations of extensions of the Knuth-Bendix orders.

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References

  1. 1.
    Akbarpour, B., Paulson, L.C.: MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions. J. of Automated Reasoning 44(3), 175–205 (2010)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Dershowitz, N., Jouannaud, J.P.: Rewrite Systems, pp. 245–319 (1990)Google Scholar
  4. 4.
    Gaillourdet, J.-M., Hillenbrand, T., Löchner, B., Spies, H.: The New Waldmeister Loop at Work. In: Proc. of CADE, pp. 317–321 (2003)Google Scholar
  5. 5.
    Jech, T.: Set Theory. Springer, Heidelberg (2002)MATHGoogle Scholar
  6. 6.
    Kirby, L., Paris, J.: Accessible Independence Results for Peano Arithmetic. Bulletin London Mathematical Society 4, 285–293 (1982)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Korovin, K., Voronkov, A.: Orienting Equalities with the Knuth-Bendix Order. In: Proc. of LICS, pp. 75–84 (2003)Google Scholar
  8. 8.
    Kovacs, L., Voronkov, A.: Finding Loop Invariants for Programs over Arrays Using a Theorem Prover. In: Chechik, M., Wirsing, M. (eds.) FASE 2009. LNCS, vol. 5503, Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Kovács, L., Voronkov, A.: Interpolation and symbol elimination. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 199–213. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Löchner, B.: Things to Know when Implementing KBO. J. of Automated Reasoning 36(4), 289–310 (2006)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Ludwig, M., Waldmann, U.: An extension of the knuth-bendix ordering with LPO-like properties. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 348–362. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  12. 12.
    McCune, B.: Private Communication (September 2004)Google Scholar
  13. 13.
    McCune, W.W.: OTTER 3.0 Reference Manual and Guide. Technical Report ANL-94/6, Argonne National Laboratory (January 1994)Google Scholar
  14. 14.
    Moser, G.: The Hydra Battle and Cichon’s Principle. AAECC 20(2), 133–158 (2009)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Riazanov, A., Voronkov, A.: The Design and Implementation of Vampire. AI Communications 15(2-3), 91–110 (2002)MATHGoogle Scholar
  16. 16.
    Schulz, S.: System Description: E 0.81. In: Proc. of IJCAR, pp. 223–228 (2004)Google Scholar
  17. 17.
    Sutcliffe, G.: The CADE-16 ATP System Competition. J. of Automated Reasoning 24(3), 371–396 (2000)CrossRefMATHGoogle Scholar
  18. 18.
    Sutcliffe, G.: The TPTP Problem Library and Associated Infrastructure. The FOF and CNF Parts, v3.5.0. J. of Automated Reasoning 43(4), 337–362 (2009)CrossRefMATHGoogle Scholar
  19. 19.
    Vampire’s homepage, http://www.vprover.org/
  20. 20.
    Weidenbach, C., Schmidt, R.A., Hillenbrand, T., Rusev, R., Topic, D.: System description: spass version 3.0. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 514–520. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  21. 21.
    Wos, L.: Milestones for Automated Reasoning with Otter. Int. J. on Artificial Intelligence Tools 15(1), 3–20 (2006)CrossRefGoogle Scholar
  22. 22.
    Zankl, H., Hirokawa, N., Middeldorp, A.: KBO orientability. JAR 43(2), 173–201 (2009)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Laura Kovács
    • 1
  • Georg Moser
    • 2
  • Andrei Voronkov
    • 3
  1. 1.TU ViennaAustria
  2. 2.Institute of Computer ScienceUniversity of InnsbruckAustria
  3. 3.University of ManchesterUK

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