A Machine-Oriented Logic Based on the Resolution Principle
Journal of the ACM (JACM)
Journal of Symbolic Computation
Unification: a multidisciplinary survey
ACM Computing Surveys (CSUR)
Higher-order unification revisited: Complete sets of transformations
Journal of Symbolic Computation
Semi-Automatic Program Construction from Specifications Using Library Modules
IEEE Transactions on Software Engineering
Higher Order Unification 30 Years Later
TPHOLs '02 Proceedings of the 15th International Conference on Theorem Proving in Higher Order Logics
Context Unification and Traversal Equations
RTA '01 Proceedings of the 12th International Conference on Rewriting Techniques and Applications
Stratified Context Unification Is in PSPACE
CSL '01 Proceedings of the 15th International Workshop on Computer Science Logic
Handbook of automated reasoning
IJCAI'77 Proceedings of the 5th international joint conference on Artificial intelligence - Volume 1
A mechanization of type theory
IJCAI'73 Proceedings of the 3rd international joint conference on Artificial intelligence
Interpretation and inference with maximal referential terms
Journal of Computer and System Sciences
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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 w (all finite types)—including the &lgr;-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 &lgr;-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.