Advertisement

System Description: The Proof Transformation System CERES

  • Tsvetan Dunchev
  • Alexander Leitsch
  • Tomer Libal
  • Daniel Weller
  • Bruno Woltzenlogel Paleo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6173)

Abstract

Cut-elimination is the most prominent form of proof transformation in logic. The elimination of cuts in formal proofs corresponds to the removal of intermediate statements (lemmas) in mathematical proofs. The cut-elimination method CERES (cut-elimination by resolution) works by extracting a set of clauses from a proof with cuts. Any resolution refutation of this set then serves as a skeleton of an ACNF, an LK-proof with only atomic cuts.

The system CERES, an implementation of the CERES-method has been used successfully in analyzing nontrivial mathematical proofs (see 4).In this paper we describe the main features of the CERES system with special emphasis on the extraction of Herbrand sequents and simplification methods on these sequents. We demonstrate the Herbrand sequent extraction and simplification by a mathematical example.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Andrews, P.B.: Resolution in Type Theory. J. of Symbolic Logic 36, 414–432 (1971)zbMATHCrossRefGoogle Scholar
  2. 2.
    Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: Proof Transformation by CERES. In: Borwein, J.M., Farmer, W.M. (eds.) MKM 2006. LNCS (LNAI), vol. 4108, pp. 82–93. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: System Description: The Cut-Elimination System CERES. In: Proc. ESCoR 2006, pp. 159–167 (2006)Google Scholar
  4. 4.
    Baaz, M., Hetzl, S., Leitsch, A., Richter, C., Spohr, H.: CERES: An analysis of Fürstenberg’s proof of the infinity of primes. Th. Co. Sci. 403, 160–175 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baaz, M., Leitsch, A.: Cut-Elimination and Redundancy-Elimination by Resolution. Journal of Symbolic Computation 29, 149–176 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Baaz, M., Leitsch, A.: Towards a Clausal Analysis of Cut-Elimination. Journal of Symbolic Computation 41, 381–410 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Dunchev, T.: Simplification of Herbrand Sequents. Master Thesis (2009)Google Scholar
  8. 8.
    Gentzen, G.: Untersuchungen über das logische Schliessen. Mathematische Zeitschrift 39, 176–210, 405–431 (1934-1935) Google Scholar
  9. 9.
    Hetzl, S., Leitsch, A., Weller, D., Woltzenlogel Paleo, B.: Herbrand sequent extraction. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC 2008, Calculemus 2008, and MKM 2008. LNCS (LNAI), vol. 5144, pp. 462–477. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Hetzl, S., Leitsch, A., Weller, D., Woltzenlogel Paleo, B.: A Clausal Approach to Proof Analysis in Second-Order Logic. In: Logical Foundations of Computer Sci. (2009)Google Scholar
  11. 11.
    Woltzenlogel Paleo, B.: Herbrand Sequent Extraction. VDM-Verlag (2008)Google Scholar
  12. 12.
    Woltzenlogel Paleo, B.: A General Analysis of Cut-Elimination by CERes. PhD Thesis (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Tsvetan Dunchev
    • 1
  • Alexander Leitsch
    • 1
  • Tomer Libal
    • 1
  • Daniel Weller
    • 1
  • Bruno Woltzenlogel Paleo
    • 1
  1. 1.Institute of Computer Languages (E185)Vienna University of TechnologyViennaAustria

Personalised recommendations