Complete Sets of Transformations for General E-Unification

In this thesis we study two generalizations of standard unification, E-unification and higher-order unification, using an abstract approach originated by Herbrand and developed in the case of standard first-order unification by Martelli and Montanari. The formalism presents the unification computation as a set of non-deterministic transformation rules for converting a term system consisting of pairs of terms to be unified into an explicit representation of a unifier (if such exists). This provides an abstract and mathematically elegant means of analysing the properties of unification in various settings by providing a clean separation of the logical issues from the specification of procedural information, and amounts to a set of 'inference rules' for unification. We derive the set of transformations for general E-unification and higher-order unification from an analysis of the sense in which terms are 'the same' after application of an unifying substitution. In both cases, this results in a simple extension of the set of basic transformations given by Herbrand-Martelli-Montanari for standard unification, and shows clearly the basic relationships of the fundamental operations necessary in each case, and thus the underlying structure of the most important classes of term unification problems. In addition to the presentation of a formalism which unifies and clarifies the diverse approaches currently being developed for E-unification and higher-order unification, we present the first rigorous analysis of a method for E-unification which is fully general in the sense that it is capable of enumerating a complete set of E-unifiers for arbitrary sets of equations E.

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