Abstract
This is an expository introduction to an approach to theorem proving in higher-order logic based on establishing appropriate connections between subformulas of an expanded form of the theorem to be proved. Expansion trees and expansion proofs play key roles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andrews, Peter B., ‘Resolution in Type Theory’, Journal of Symbolic Logic 36, 414–432 (1971).
Andrews, Peter B., ‘Resolution and the Consistency of Analysis’, Notre Dame Journal of Formal Logic 15, 73–84 (1974).
Andrews, Peter B., ‘Refutations by Matings’, IEEE Transactions on Computers C-25, 801–807 (1976).
Andrews, Peter B., ‘Transforming Matings into Natural Deduction Proofs,’ in 5th Conference on Automated Deduction. edited by W. Bibel and R. Kowalski, Les Arcs, France, Lecture Notes in Computer Science 87, Springer-Verlag, 1980, pp. 281–292.
Andrews, Peter B., ‘Theorem Proving via General Matings’, Journal of the ACM 28, 193–214 (1981).
Andrews, Peter B., Miller, Dale A., Cohen, Eve Longini, and Pfenning, Frank, ‘Automating Higher-Order Logic’, in Automated Theorem Proving: After 25 Years, edited by W. W. Bledsoe and D. W. Loveland, Contemporary Mathematics series, vol. 29, American Mathematical Society, 1984, pp. 169–192.
Andrews, Peter B., An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, Academic Press, 1986.
Andrews, Peter B., ‘Connections and Higher-Order Logic’, in 8th International Conference on Automated Deduction, edited by Jorg H. Siekmann, Oxford, England, Lecture Notes in Computer Science 230, Springer-Verlag, 1986, pp. 1–4.
Bibel Wolfgang, and Schreiber J., §lsProof search in a Gentzen-like system of first-order logic’, in International Computing Symposium 1975, edited by E. Gelenbe and D. Potier Eds, North-Holland, Amsterdam, 1975, pp. 205–212.
Bibel, Wolfgang, ‘Tautology Testing with a Generalized Matrix Reduction Method’, Theoretical Computer Science 8, 31–44 (1979).
Bibel, Wolfgang, ‘On Matrices with Connections’, Journal of the ACM, 28, 633–645 (1981).
Bibel, Wolfgang, Automated Theorem Proving, Vieweg, Braunschweig, 1982.
Bibel, Wolfgang, ‘A Comparative Study of Several Proof Procedures’, Artificial Intelligence, 18, 269–293 (1982).
Bibel, Wolfgang, ‘Matings in Matrices’, Communications of the ACM, 26, 844–852 (1983).
Bibel, Wolfgang, and Buchberger, Bruno, ‘Towards a Connection Machine for Logical Inference’, Future Generations Computer Systems 1, 177–185 (1984–1985).
Bledsoe, W. W., ‘A Maximal Method for Set Variables in Automatic Theorem Proving’, in Machine Intelligence 9, Ellis Harwood Ltd., Chichester, and John Wiley & Sons, 1979, pp. 53–100.
Bledsoe, W. W., ‘Using Examples to Generate Instantiations of Set Variables’, in Proceedings of IJCAI-83, Karlsruhe, Germany, Aug 8–12, 1983, pp. 892–901.
Church, Alonzo, ‘A formulation of the Simple Theory of Types’, Journal of Symbolic Logic 5, 56–68 (1940).
Davis, Martin, ‘Eliminating the Irrelevant from Mechanical Proofs’, in Experimental Arithmetic, High Speed Computing and Mathematics, Proceedings of Symposia in Applied Mathematics XV, American Mathematical Society, 1963, pp. 15–30.
Dreben, Burton, Andrews, Peter, and Aanderaa, Stal, ‘False Lemmas in Herbrand’, Bulletin of the American Mathematical Society, 69, 699–706 (1963).
Dreben, Burton and Denton, John, ‘A Supplement to Herbrand’, Journal of Symbolic Logic 31, 393–398 (1966).
Huet, Gérard P., ‘A Mechanization of Type Theory’, in Proceedings of the Third International Joint Conference on Artificial Intelligence, IJCAI, 1973, pp. 139–146.
Huet, Gérard P., ‘A Unification Algorithm for Typed λ-Calculus’, Theoretical Computer Science 1, 27–57 (1975).
Jensen, D. C., and Pietrzykowski, T., ‘Mechanizing ω-Order Type Theory Through Unification’, Theoretical Computer Science 3, 123–171 (1976).
Miller, Dale A., ‘Proofs in Higher-Order Logic’, Ph.D. Thesis, Carnegie Mellon University, 1983, 81 pp.
Miller, Dale A., ‘Expansion Tree Proofs and Their Conversion to Natural Deduction Proofs’, in 7th International Conference on Automated Deduction, edited by R. E. Shostak, Napa, California, USA, Lecture Notes in Computer Science 170, Springer-Verlag, 1984, pp. 375–393.
Miller, Dale A., ‘A Compact Representation of Proofs’, Studia Logica 46, 347–370 (1987).
Pfenning, Frank, ‘Analytic and Non-analytic Proofs’, in 7th International Conference on Automated Deduction, edited by R. E. Shostak, Napa, California, USA. Lecture Notes in Computer Science 170, Springer-Verlag, 1984, pp. 394–413.
Pfenning, Frank, ‘Proof Transformations in Higher-Order Logic’, Ph.D. Thesis, Carnegie Mellon University, 1987, 156 pp.
Prawitz, Dag, ‘Advances and Problems in Mechanical Proof Procedures’, in Machine Intelligence 4, Edinburgh University Press, 1969, pp. 59–71.
Prawitz, Dag, ‘A proof procedure with matrix reduction’, in Symposium on Automatic Demonstration, Versailles, France, edited by M. Laudet, D. Lacombe, L. Nolin, and M. Schutzenberger, Lecture Notes in Mathematics 125, Springer-Verlag, 1970, pp. 207–214.
Shostak, Robert E., ‘Refutation Graphs’, Artificial Intelligence 7, 51–64 (1976).
Author information
Authors and Affiliations
Additional information
This is an extended version of a lecture presented to the 8th International Conference on Automated Deduction in Oxford, England on 27 July 1986. This material is based upon work supported by the National Science Foundation under grants DCR-8402532 and CCR-8702699.
Rights and permissions
About this article
Cite this article
Andrews, P.B. On connections and higher-order logic. J Autom Reasoning 5, 257–291 (1989). https://doi.org/10.1007/BF00248320
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00248320