Variable elimination and chaining in a resolution-based prover for inequalities

  • W. W. Bledsoe
  • Larry M. Hines
Wednesday Morning
Part of the Lecture Notes in Computer Science book series (LNCS, volume 87)

Abstract

A modified resolution procedure is described which has been designed to prove theorems about general linear inequalities. This prover uses "variable elimination", and a modified form of inequality chaining (in which chaining is allowed only on so called "shielding terms"), and a decision procedure for proving ground inequality theorems. These techniques and others help to avoid the explicit use of certain axioms, such as the transitivity and interpolation axioms for inequalities, in order to reduce the size of the search space and to speed up proofs. Several examples are given along with results from a computer implementation. In particular this program has proved some difficult theorems such as: The sum of two continuous functions is continuous.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    W.W. Bledsoe. A Resolution-based Prover for General Inequalities. University of Texas, Math Department Memo ATP-52, July 1979.Google Scholar
  2. 2.
    J.R. Slagle and L. Norton. Experiments with an Automatic Theorem Prover Having Partial Ordering Rules. CACM 16 (1973), 683–688.Google Scholar
  3. 3.
    J.C. King. A Program Verifier. Ph.D. Thesis, Carnegie-Mellon University, 1969.Google Scholar
  4. 4.
    Greg Nelson and Derek Oppen. A Simplifier Based on Efficient Decision Algorithms. Proc. 5th ACM Symp. on Principles of Programming Languages, 1978.Google Scholar
  5. 5.
    Robert Shostak. A Practical Decision Procedure for Arithmetic with Function Symbols. JACM, April 1979.Google Scholar
  6. 6.
    W.W. Bledsoe, Peter Bruell and Robert Shostak. A Prover for General Inequalities. University of Texas, Math Department Memo ATP-40A, February 1979. Also IJCAI-79, Tokyo, Japan, August 1979.Google Scholar
  7. 7.
    Louis Hodes. Solving Programs by Formula Manipulation in Logic and Linear Inequality. Proc. IJCAI-71, London, 1971, pages 553–559.Google Scholar
  8. 8.
    M.A. Stickel. A Complete Unification for Associative-Commutative Functions, IJCAI-75, Tbilisi, USSR, 1975, pages 71–76.Google Scholar
  9. 9.
    J.A. Robinson. A Machine-oriented Logic Based on the Resolution Principle. JACM 12 (1965), 23–41.CrossRefGoogle Scholar
  10. 10.
    W.W. Bledsoe. Splitting and Reduction Heuristics in Automatic Theorem Proving. AEJ 2 (1971), 55–77.Google Scholar
  11. 11.
    L. Wos and G.A. Robinson. Paramodulation and Set of Support. Proc. Symp. Automatic Demonstration, Versailles, France. Springer-Verlag, New York, 1968, 276–310.Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • W. W. Bledsoe
    • 1
  • Larry M. Hines
    • 1
  1. 1.The University of TexasAustin

Personalised recommendations