Selectively instantiating definitions

  • Matthew Bishop
  • Peter B. Andrews
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1421)


When searching for proofs of theorems which contain definitions, it is a significant problem to decide which instances of the definitions to instantiate. We describe a method called dual instantiation, which is a partial solution to the problem in the context of the connection method; the same solution may also be adaptable to other search procedures. Dual instantiation has been implemented in TPS, a theorem prover for classical type theory, and we provide some examples of theorems that have been proven using this method. Dual instantiation has the desirable properties that the search for a proof cannot possibly fail due to insufficient instantiation of definitions, and that the natural deduction proof which results from a successful search will contain no unnecessary instantiations of definitions. Furthermore, the time taken by a proof search using dual instantiation is in general comparable to the time taken by a search in which exactly the required instances of each definition have been instantiated. We also describe how this technique can be applied to the problem of instantiating set variables.


Theorem Prove Automate Reasoning Predicate Symbol Automate Deduction Plan Line 
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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  1. 1.
    Peter B. Andrews. Theorem Proving via General Matings. Journal of the ACM, 28:193–214, 1981.zbMATHCrossRefGoogle Scholar
  2. 2.
    Peter B. Andrews. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Academic Press, 1986.Google Scholar
  3. 3.
    Peter B. Andrews. On Connections and Higher-Order Logic. Journal of Automated Reasoning, 5:257–291, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Peter B. Andrews, Matthew Bishop, Sunil Issar, Dan Nesmith, Frank Pfenning, and Hongwei Xi. TPS: A Theorem Proving System for Classical Type Theory. Journal of Automated Reasoning, 16:321–353, 1996.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Sidney C. Bailin and Dave Barker-Plummer. Z-match: An Inference Rule for Incrementally Elaborating Set Instantiations. Journal of Automated Reasoning, 11:391–428, 1993. Errata: JAR 12:411–412, 1994.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Dave Barker-Plummer. Gazing: An Approach to the Problem of Definition and Lemma Use. Journal of Automated Reasoning, 8:311–344, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    W. W. Bledsoe. Using Examples to Generate Instantiations of Set Variables. In Proceedings of IJCAI-83, Karlsruhe, Germany, pages 892–901, Aug 8–12, 1983.Google Scholar
  8. 8.
    W. W. Bledsoe and Peter Bruell. A Man-Machine Theorem-Proving System. Artificial Intelligence, 5(1):51–72, 1974.CrossRefMathSciNetGoogle Scholar
  9. 9.
    W. W. Bledsoe and Gohui Feng. Set-Var. Journal of Automated Reasoning, 11:293–314, 1993.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Alonzo Church. A Formulation of the Simple Theory of Types. Journal of Symbolicogic, 5:56–68, 1940.zbMATHMathSciNetGoogle Scholar
  11. 11.
    Fausto Giunchiglia and Adolfo Villafiorita. ABSFOL: a Proof Checker with Abstraction. In M.A. McRobbie and J.K. Slaney, editors, CADE-13: Proceedings of the 13th International Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1104, pages 136–140. Springer-Verlag, 1996.Google Scholar
  12. 12.
    Fausto Giunchiglia and Toby Walsh. Theorem Proving with Definitions. In Proceedings of AISB 89, Society for the Study of Artificial Intelligence and Simulation of Behaviour, 1989.Google Scholar
  13. 13.
    Fausto Giunchiglia and Toby Walsh. A Theory of Abstraction. Artificial Intelligence, 57(2–3):323–389, 1992.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Gerard P. Huet. A Unification Algorithm for Typed λ-Calculus. Theoretical Computer Science, 1:27–57, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ignace I. Kolodner. Fixed Points. American Mathematical Monthly, 71:906, 1964.CrossRefMathSciNetGoogle Scholar
  16. 16.
    Dale A. Miller. A compact representation of proofs. Studia Logica, 46(4):347–370, 1987.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D. Pastre. Automatic Theorem Proving in Set Theory. Artificial Intelligence, 10:1–27, 1978zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Frank Pfenning. Proof Transformations in Higher-Order Logic. PhD thesis, Carnegie Mellon University, 1987. 156 pp.Google Scholar
  19. 19.
    D.A. Plaisted. Abstraction Mappings in Mechanical Theorem Proving. In 5th Conference on Automated Deduction, Lecture Notes in Computer Science 87, pages 264–280. Springer-Verlag, 1980.Google Scholar
  20. 20.
    D.A. Plaisted. Theorem Proving with Abstraction. Artificial Intelligence, 16:47–108, 1981.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dave Plummer. Gazing: Controlling the Use of Rewrite Rules. PhD thesis, Dept. of Artificial Intelligence, University of Edinburgh, 1987.Google Scholar
  22. 22.
    K. Warren. Implementation of a Definition Expansion Mechanism in a Connection Method Theorem Prover. Master's thesis, Dept. of Artificial Intelligence, Univ. of Edinburgh, 1987.Google Scholar
  23. 23.
    Larry Wos. The Problem of Definition Expansion and Contraction. Journal of Automated Reasoning, 3:433–435, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Matthew Bishop
    • 1
  • Peter B. Andrews
    • 1
  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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