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Quantifiers and the System KE: Some Surprising Results

  • Uwe Egly
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1584)

Abstract

In this paper, we consider the free-variable variant of the calculus KE and investigate the effect of different preprocessing activities to the proof length of variants of KE. In this context, skolemization is identified to be harmful as compared to the δ-rule. This does not only have consequences for proof length in KE, but also for the “efficiency” of some structural translations. Additionally, we investigate the effect of quantifier-shifting and quantifier-distributing rules, respectively. In all these comparisons, we identify classes of formulae for which a non-elementary difference in proof length occur.

Keywords

Automate Deduction Input Formula Proof Complexity Negation Normal Form Signed Formula 
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.
    Baaz, M., Egly, U., Leitsch, A.: Extension Methods in Automated Deduction. In: Bibel, W., Schmitt, P. (eds.) Automated Deduction– A Basis for Applications, part 4, ch. 12, vol. II, pp. 331–360. Kluwer Academic Press, Dordrecht (1998)Google Scholar
  2. 2.
    Baaz, M., Fermüller, C., Leitsch, A.: A Non-Elementary Speed Up in Proof Length by Structural Clause Form Transformation. In: LICS 1994, pp. 213–219. IEEE Computer Society Press, Los Alamitos (1994)Google Scholar
  3. 3.
    Baaz, M., Leitsch, A.: On Skolemization and Proof Complexity. Fundamenta Informaticae 20, 353–379 (1994)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Beckert, B., Posegga, J.: leanTAP: Lean Tableau-based Deduction. J. Automated Reasoning 15(3), 339–358 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    D’Agostino, M.: Are Tableaux an Improvement on Truth-Tables? Cut-Free Proofs and Bivalence. J. Logic, Language and Information 1, 235–252 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D’Agostino, M., Mondadori, M.: The Taming of the Cut. Classical Refutations with Analytic Cut. J. Logic and Computation 4, 285–319 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Eder, E.: Relative Complexities of First Order Calculi. Vieweg (1992)Google Scholar
  8. 8.
    Egly, U.: On the Value of Antiprenexing. In: Pfenning, F. (ed.) Proceedings of the International Conference on Logic Programming and Automated Reasoning, pp. 69–83. Springer, Heidelberg (1994)Google Scholar
  9. 9.
    Egly, U.: On Different Structure-preserving Translations to Normal Form. J. Symbolic Computation 22, 121–142 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Egly, U.: Cuts in Tableaux. In: Bibel, W., Schmitt, P. (eds.) Automated Deduction– A Basis for Applications, part 1, ch. 4, vol. I, pp. 103–132. Kluwer Academic Press, Dordrecht (1998)Google Scholar
  11. 11.
    Fitting, M.: First-Order Logic and Automated Theorem Proving, 2nd edn. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  12. 12.
    Leitsch, A.: The Resolution Calculus. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  13. 13.
    Lifschitz, V.: Computing Circumscription. In: Proceedings of IJCAI-1985, Los Altos, CA., pp. 121–127. Morgan Kaufmann, San Francisco (1985)Google Scholar
  14. 14.
    McCarthy, J.: Circumscription – A Form of Non-Monotonic Reasoning. Artificial Intelligence 13, 27–39 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mondadori, M.: Classical Analytical Deduction. Technical Report Annali dell’Università di Ferrara, Sezione III, Discussion Paper 1, Università di Ferrara (1988)Google Scholar
  16. 16.
    Orevkov, V.P.: Lower Bounds for Increasing Complexity of Derivations after Cut Elimination. Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im V. A. Steklova AN SSSR 88, 137–161 (1979); English translation in J. Soviet Mathematics, pp. 2337–2350 (1982)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Sieg, W., Kauffmann, B.: Unification for Quantified Formulae. PHIL 44, Carnegie Mellon University (1993)Google Scholar
  18. 18.
    Tseitin, G.S.: On the Complexity of Derivation in Propositional Calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic, Part II, Leningrad. Seminars in Mathematics, vol. 8, pp. 234–259 (1968); English translation: Consultants Bureau, New York, pp. 115–125 (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Uwe Egly
    • 1
  1. 1.Institut für Informationssysteme E184.3Technische Universität WienWienAustria

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