A Breadth-First Strategy for Mating Search
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Mating search is a very general method for automating proof search; it specifies that one must find a complete mating, without specifying the way in which this is to be achieved. It is the foundation of TPS, an automated theorem-proving system for simply-typed lambda-calculus, and has proven very effective in discovering proofs of higher-order theorems. However, previous implementations of mating search have all relied on essentially the same mating search method: enumerating the paths through a matrix of literals. This is a depth-first strategy which is both computationally expensive and vulnerable to blind alleys in the search space; in addition, the incremental computation of unifiers which is required is, in the higher-order case, very inefficient. We describe a new breadth-first mating search method, called component search, in which matings are constructed by taking unions from a fixed list of smaller matings, whose unifiers are stored and manipulated as directed graphs. Component search is capable of handling much larger search spaces than were possible with path-enumeration search, and has produced fully automatic proofs of a number of interesting theorems which were previously intractable.
KeywordsOpen Path Large Search Space Minimal Component Enumeration Search Vertical Path
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- BA98.Matthew Bishop and Peter B. Andrews. Selectively instantiating definitions. In Claude Kirchner and Hélène Kirchner, editors, Proceedings of the 15th International Conference on Automated Deduction, volume 1421 of Lecture Notes in Artificial Intelligence, pages 365–380, Lindau, Germany, 1998. Springer-Verlag.Google Scholar
- Bar84.H. P. Barendregt. The λ-Calculus. Studies in logic and the foundations of mathematics, North-Holland, 1984.Google Scholar
- BBE+95.W. Bibel, S. Bruning, U. Egly, D. Korn, and T. Rath. Issues in theorem proving based on the connection method. In Peter Baumgartner, Reiner Hähnle, and Joachim Posegga, editors, Theorem Proving with Analytic Tableaux and Related Methods. 4th International Workshop. TABLEAUX’ 95, volume 918 of Lecture Notes in Artificial Intelligence, pages 1–16, Schloß Rheinfels, St. Goar, Germany, May 1995. Springer-Verlag.Google Scholar
- Bib82.Wolfgang Bibel. Automated Theorem Proving. Vieweg, Braunschweig, 1982.Google Scholar
- Bis99.Matthew Bishop. Mating Search Without Path Enumeration. PhD thesis, Department of Mathematical Sciences, Carnegie Mellon University, 1999.Google Scholar
- Brü94.S. Brüning. Techniques for Avoiding Redundancy in Theorem Proving Based on the Connection Method. PhD thesis, TH Darmstadt, 1994.Google Scholar
- Iss90.Sunil Issar. Path-focused duplication: A search procedure for general matings. In AAAI-90. Proceedings of the Eighth National Conference on Artificial Intelligence, volume 1, pages 221–226. AAAI Press/The MIT Press, 1990.Google Scholar
- Iss91.Sunil Issar. Operational Issues in Automated Theorem Proving Using Matings. PhD thesis, Carnegie Mellon University, 1991. 147 pp.Google Scholar
- Kna27.B. Knaster. Une théorème sur les fonctions d’ensembles. Annales Soc. Polonaise Math., 6:133–134, 1927.Google Scholar
- Pfe87.Frank Pfenning. Proof Transformations in Higher-Order Logic. PhD thesis, Carnegie Mellon University, 1987. 156 pp.Google Scholar
- Szá63.Gabor Szász. Introduction to Lattice Theory. Academic Press, New York and London, 1963.Google Scholar