Automated reasoning contributes to mathematics and logic

  • L. Wos
  • S. Winker
  • W. McCune
  • R. Overbeek
  • E. Lusk
  • R. Stevens
  • R. Butler
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 449)

Abstract

In this article, we present some results of our research focusing on the use of our newest automated reasoning program OTTER to prove theorems from Robbins algebra, equivalential calculus, implicational calculus, combinatory logic, and finite semigroups. Included among the results are answers to open questions and new shorter and less complex proofs to known theorems. To obtain these results, we relied upon our usual paradigm, which heavily emphasizes the role of demodulation, subsumption, set of support, weighting, paramodulation, hyperresolution, and UR-resolution. Our position is that all of these components are essential, even though we can shed little light on the relative importance of each, the coupling of the various components, and the metarules for making the most effective choices. Indeed, without these components, a program will too often offer inadequate control over the redundancy and irrelevancy of deduced information. We include experimental evidence to support our position, examples producing success when the paradigm is employed, and examples producing failure when it is not. In addition to providing evidence that automated reasoning has made contributions to both mathematics and logic, the theorems we discuss also serve nicely as challenge problems for testing the merits of a new idea or a new program and provide interesting examples for comparing different paradigms.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Barendregt, H.P. The Lambda Calculus: Its Syntax and Semantics, North-Holland, Amsterdam, 1981.Google Scholar
  2. [2]
    Henkin, L., J. Monk, and A. Tarski. Cylindric Algebras, Part I, North-Holland, Amsterdam, 1971.Google Scholar
  3. [3]
    Huntington, E. New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell's Principia Mathematica, Trans. of AMS 35 (1933), 274–304.Google Scholar
  4. [4]
    Lusk E. and R. McFadden. Using automated reasoning tools: a study of the semigroup {F2B2}, Semigroup Forum 36, 1 (1987), 75–88.Google Scholar
  5. [5]
    McCune, W. OTTER 2.0 Users Guide, Argonne National Laboratory Report ANL-90/9, 1990.Google Scholar
  6. [6]
    Ohlbach, H.-J. and G. Wrightson. Solving a problem in relevance logic with an automated theorem prover, Proc. of CADE-7, Springer-Verlag Lecture Notes in Computer Science, Vol. 170, ed. R. Shostak, New York, 1984, 496–508.Google Scholar
  7. [7]
    Peterson J. The Possible Shortest Single Axioms for EC-Tautologies, Aukland University Department of Mathematics Report Series No. 105, 1977.Google Scholar
  8. [8]
    Pfenning, F. Single axioms in the implicational propositional calculus, Proc. of CADE-9, Springer-Verlag Lecture Notes in Computer Science, Vol. 310, ed. E. Lusk and R. Overbeek, New York, 1988, 710–713.Google Scholar
  9. [9]
    Slaney, J. K. and E. Lusk. Parallelizing the closure computation in automated deduction, to appear in the proceedings of CADE-10, 1990.Google Scholar
  10. [10]
    Smullyan, R. To Mock a Mockingbird, A. Knopf, New York, 1985.Google Scholar
  11. [11]
    Wos, L., R. Overbeek, E. Lusk, and J. Boyle. Automated Reasoning: Introduction and Applications, Prentice-Hall, New York, 1984.Google Scholar
  12. [12]
    Wos, L., S. Winker, B. Smith, R. Veroff, and L. Henschen. A new use of an automated reasoning assistant: open questions in equivalential calculus and the study of infinite domains, Artificial Intelligence 22 (1984), 303–356.Google Scholar
  13. [13]
    Wos, L., R. Overbeek, and E. Lusk. Subsumption, a sometimes undervalued procedure, preprint MCS-P93-0789, Argonne National Laboratory, Argonne, Ill., July 1989.Google Scholar
  14. [14]
    Wos, L., S. Winker, W. McCune, R. Overbeek, E. Lusk, R. Stevens, and R. Butler. OTTER Experiments Pertinent to CADE-10, Argonne National Laboratory Report ANL-89/39, to appear.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • L. Wos
    • 1
  • S. Winker
    • 1
  • W. McCune
    • 1
  • R. Overbeek
    • 1
  • E. Lusk
    • 1
  • R. Stevens
    • 1
  • R. Butler
    • 2
  1. 1.Mathematics and Computer Science DivisionArgonne National LaboratoryArgonne
  2. 2.Division of Computer and Information ScienceUniversity of North FloridaJacksonville

Personalised recommendations