Download A proof theory for general unification by W. Snyder PDF

By W. Snyder

ISBN-10: 0817635939

ISBN-13: 9780817635930

During this monograph we learn generalizations of ordinary unification, E-unification and higher-order unification, utilizing an summary process orig­ inated by way of Herbrand and built when it comes to ordinary first-order unifi­ cation by way of Martelli and Montanari. The formalism provides the unification computation as a suite of non-deterministic transformation principles for con­ verting a suite of equations to be unified into an specific illustration of a unifier (if such exists). this gives an summary and mathematically dependent technique of analysing the houses of unification in numerous settings through offering a fresh separation of the logical matters from the specification of procedural info, and quantities to a suite of 'inference principles' for unification, for that reason the name of this publication. We derive the set of alterations for normal E-unification and better­ order unification from an research of the feel within which phrases are 'the comparable' after software of a unifying substitution. In either situations, this ends up in an easy extension of the set of simple modifications given via Herbrand­ Martelli-Montanari for normal unification, and indicates in actual fact the fundamental relationships of the elemental operations invaluable in every one case, and hence the underlying constitution of an important periods of time period unifi­ cation difficulties.

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Extra resources for A proof theory for general unification

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Another difference from standard unification is that most general unifiers do not necessarily exist. In fact, it is 50 E-UNIFICATION possible that two terms have an infinite set of independent E-unifiers, as we now show. } U Eo, where Eo contains at least one constant symbol "a", and, using infix notation, let E = {(x' . y') . Zl == x' . (y' . Zl)}. This set axiomatizes non-empty strings over the set of constant symbols Eo, and so we represent terms as simply strings of constants and variables.

Yn}. 0 = = Since most uses of substitutions in this thesis are modulo renaming, this lemma will allow us to assume that substitutions are idempotent if necessary. We shall prove specific results related to the use of idempotent unifiers in later sections. Finally, we define the notion of a substitution instance and a matching substitution. 9 For any term t, a term (J(t) is called a substitution instance oft. Given two terms sand t, a substitution 0" is called a matching substitution of sand t iff s = O"(t).

3. 3 UNIFICATION BY TRANSFORMATIONS ON SYSTEMS 27 such as breadth-first search, could find the solved form. This form of completeness, which might be termed non-deterministic completeness, will be used in finding results on E-unification and higher-order unification, where the general problem is undecidable. In some contexts it may be useful to deal with idempotent unifiers which are renamed away from some set of 'protected' variables but which are most general over the set of variables in the original system.

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