NDLI: A Complete Mechanization of Second-Order Type Theory

Content Provider	ACM Digital Library
Author	Pietrzykowski, Tomasz
Copyright Year	1973
Abstract	A generalization of the resolution method for higher order logic is presented. The languages acceptable for the method are phrased in a theory of types of order $\textit{w}$ (all finite types)—including the λ-operator, propositional functors, and quantifiers. The resolution method is, of course, a machine-oriented theorem search procedure based on refutation. In order to make this method suitable for higher order logic, it was necessary to overcome two sorts of difficulties. The first is that the unifying substitution procedure—an essential feature of the classic first-order resolution—must be generalized (it is noted that for the higher order unification the proper notion of substitution will include λ-normalization). A general unification algorithm is produced and proved to be complete for second-order languages. The second difficulty arises because in higher order languages, semantic intent is essentially more “interwoven” in formulas than in first-order languages. Whereas quantifiers could be eliminated immediately in first-order resolution, their elimination must be deferred in the higher order case. The generalized resolution procedure which the author produces thus incorporates quantifier elimination along with the familiar features of unification and tautological reduction. It is established that the author's generalized resolution procedure is complete with respect to a natural notion of validity based on Henkin's general validity for type theory. Finally, there are presented examples of the application of the method to number theory and set theory.
Starting Page	333
Ending Page	364
Page Count	32
File Format	PDF
ISSN	00045411
e-ISSN	1557735X
DOI	10.1145/321752.321764
Journal	Journal of the ACM (JACM)
Volume Number	20
Issue Number	2
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	1973-04-01
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Content Type	Text
Resource Type	Article
Subject	Hardware and Architecture Information Systems Control and Systems Engineering Artificial Intelligence Software

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