# Connections and higher-order logic

## Abstract

Theorem proving is difficult and deals with complex phenomena. The difficulties seem to be compounded when one works with higher-order logic, but the rich expressive power of Church's formulation [10] [3] of this language makes research on theorem proving in this realm very worthwhile. In order to make significant progress on this problem, we need to try many approaches and ideas, and explore many questions. A highly relevant question is "What makes a logical formula valid?".

One approach to this question is semantic. Theorems are true because they express essential truths and thus are true in all models of the language in which they are expressed. Truth can be perceived from many perspectives, so there may be many essentially different proofs of theorems. This point of view is very appealing, but it does not shed much light on the basic question of what makes certain sentences true in all models, while others are not.

Of course, theorems are formulas which have proofs, and every proof in any logical system may provide some insight. This suggests seeing what one can learn by studying the forms proofs can take. While this may be helpful, many of the most prominent features of proofs seem to be influenced as much by the logical system in which the proof is given as by the theorem that is being proved.

We focus on trying to understand what there is about the syntactic structures of theorems that makes them valid. In the case of formulas of propositional calculus, one can test a formula for being a tautology in an explicit syntactic way. However, simply checking each line of a truth table is not really very enlightening, and we may still find ourselves asking "What is there about the structure of this formula which makes it a tautology?". Clearly the pattern of occurrences of positive and negative literals in such a formula is very important, and this leads to the theory of *connections* or *matings* [4] [1] [6]. A connection is a pair of literal-occurrences, and a mating is a set of connections, i.e., a relation between occurrences of literals. A mating is *acceptable* if its structure guarantees that the formula is a tautology. Perhaps much more can be said about criteria for matings to be acceptable.

## Keywords

Theorem Prove Automate Deduction Negative Literal Essential Truth Primitive Substitution## Preview

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## References

- 1.Peter B. Andrews,
*Theorem Proving via General Matings*, Journal of the ACM**28**(1981), 193–214.CrossRefGoogle Scholar - 2.Peter B. Andrews, Dale A. Miller, Eve Longini Cohen, Frank Pfenning, "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, 169–192.Google Scholar - 3.Peter B. Andrews,
*An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof*, Academic Press, 1986.Google Scholar - 4.Wolfgang Bibel,
*On Matrices with Connections*, Journal of the ACM**28**(1981), 633–645.CrossRefGoogle Scholar - 5.Wolfgang Bibel,
*Automated Theorem Proving*, Vieweg, Braunschweig, 1982.Google Scholar - 6.Wolfgang Bibel,
*Matings in Matrices*, Communications of the ACM**26**(1983), 844–852.CrossRefGoogle Scholar - 7.Wolfgang Bibel and Bruno Buchberger,
*Towards a Connection Machine for Logical Inference*, Future Generations Computer Systems**1**(1984–1985).Google Scholar - 8.W. W. Bledsoe, "A Maximal Method for Set Variables in Automatic Theorem Proving," in
*Machine Intelligence 9*, Ellis Harwood Ltd., Chichester, 1979, pp. 53–100.Google Scholar - 9.W. W. Bledsoe. Using Examples to Generate Instantiations for Set Variables, ATP-67, University of Texas at Austin, July 1982, 44 ppGoogle Scholar
- 10.Alonzo Church,
*A Formulation of the Simple Theory of Types*, Journal of Symbolic Logic**5**(1940), 56–68.Google Scholar - 11.Gérard P. Huet, "A Mechanization of Type Theory," in
*Proceedings of the Third International Joint Conference on Artificial Intelligence*, IJCAI, 1973, 139–146.Google Scholar - 12.Gérard P. Huet,
*A Unification Algorithm for Typed λ-Calculus*, Theoretical Computer Science**1**(1975), 27–57.CrossRefGoogle Scholar - 13.D. C. Jensen and T. Pietrzykowski,
*Mechanizing ω-Order Type Theory Through Unification*, Theoretical Computer Science**3**(1976), 123–171.CrossRefGoogle Scholar - 14.Dale A. Miller.
*Proofs in Higher-Order Logic*, Ph.D. Thesis, Carnegie-Mellon University, October, 1983. 81 pp.Google Scholar - 15.Dale A. Miller, "Expansion Tree Proofs and Their Conversion to Natural Deduction Proofs," in
*7th International Conference on Automated Deduction, Napa, California, USA*, edited by R. E. Shostak, Lecture Notes in Computer Science 170, Springer-Verlag, May 14–16, 1984, 375–393.Google Scholar - 16.Frank Pfenning, "Analytic and Non-analytic Proofs," in
*7th International Conference on Automated Deduction, Napa, California, USA*, edited by R. E. Shostak, Lecture Notes in Computer Science 170, Springer-Verlag, May 14–16, 1984, 394–413.Google Scholar - 17.Frank Pfenning.
*Proof Transformations in Higher-Order Logic*, Ph.D. Thesis, Carnegie-Mellon University, 1986.Google Scholar