Selectively instantiating definitions
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.
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- 2.Peter B. Andrews. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Academic Press, 1986.Google Scholar
- 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
- 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.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
- 18.Frank Pfenning. Proof Transformations in Higher-Order Logic. PhD thesis, Carnegie Mellon University, 1987. 156 pp.Google Scholar
- 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
- 21.Dave Plummer. Gazing: Controlling the Use of Rewrite Rules. PhD thesis, Dept. of Artificial Intelligence, University of Edinburgh, 1987.Google Scholar
- 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.Larry Wos. The Problem of Definition Expansion and Contraction. Journal of Automated Reasoning, 3:433–435, 1987.Google Scholar