Read e-book online A proof theory for general unification PDF

By W. Snyder

ISBN-10: 0817635939

ISBN-13: 9780817635930

During this monograph we research generalizations of ordinary unification, E-unification and higher-order unification, utilizing an summary procedure orig­ inated through Herbrand and built relating to normal first-order unifi­ cation via Martelli and Montanari. The formalism provides the unification computation as a suite of non-deterministic transformation principles for con­ verting a collection of equations to be unified into an particular illustration of a unifier (if such exists). this offers an summary and mathematically based technique of analysing the homes of unification in quite a few settings through delivering a fresh separation of the logical matters from the specification of procedural details, and quantities to a suite of 'inference principles' for unification, as a result the name of this booklet. 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 related' after software of a unifying substitution. In either circumstances, this ends up in an easy extension of the set of simple variations given through Herbrand­ Martelli-Montanari for normal unification, and indicates basically the elemental relationships of the elemental operations beneficial in every one case, and therefore the underlying constitution of an important sessions of time period unifi­ cation difficulties.

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Extra resources for A proof theory for general unification

Example text

E E, and u and v be two terms. We have the following transformations. Trivial: {u~u}uS ~ (1) S Term Decomposition: For any ! E En for some n > 0, Variable Elimination: {z~v}US ~ {z~v}Uq(S), where z ~ v is not in solved form, x f/. Var(v), (3) and u = [viz]. 3 25 UNIFICATION BY TRANSFORMATIONS ON SYSTEMS equation to be transformed. We shall say that Unify(S) = exists some sequence of transformations S ~ (J iff there ... :::::} S', where S' is in solved form and (J = US' . ) Clearly, by choosing S = {u ~ v} , we can attempt to find a unifier for two terms u, and v, as the following example shows.

We say that u is more general modulo Ethan 0 over V, denoted by u ~E O(V], iff there exists some substitution 'fJ such that 0 =E u 0 'fJ(V]. When V is the set of all variables, we drop the notation (V], and similarly we drop the subscript E when E = 0. 4 If 0 =E u then for any system S, 0 E UE(S) iff u E UE(S). Proof For any equation u ~ v in S, a simple induction on the structure of u and v suffices to show that 8('11) ~E O(v) iff u(u) ~E u(v). 5 If u E UE(S) and u ~E O(Var(S)] then 8 E UE(S).

In other words, two terms match if the first is a substitution instance of the second. , whether such a 0" exists) is called the matching problem for sand t. 3 Unification by Transformations on Systems We now define unification of terms and present an abstract view of the unification process as a set of non-deterministic rules for transforming a unification problem into an explicit representation of its solution, if such exists; in Chapters §5 and §6 this will be extended to E-unification, and in Chapter §7 to higher-order unification.

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A proof theory for general unification by W. Snyder

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