New PDF release: A proof theory for general unification

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 technique orig­ inated by way of Herbrand and built in relation to normal first-order unifi­ cation via Martelli and Montanari. The formalism offers the unification computation as a collection of non-deterministic transformation principles for con­ verting a collection of equations to be unified into an specific illustration of a unifier (if such exists). this offers an summary and mathematically stylish technique of analysing the houses of unification in numerous settings by means of delivering a fresh separation of the logical concerns from the specification of procedural info, and quantities to a collection of 'inference ideas' for unification, accordingly the name of this e-book. We derive the set of ameliorations for normal E-unification and better­ order unification from an research of the feel during which phrases are 'the related' after software of a unifying substitution. In either instances, this ends up in an easy extension of the set of uncomplicated variations given via Herbrand­ Martelli-Montanari for traditional unification, and indicates in actual fact the fundamental relationships of the basic operations important in each one case, and therefore the underlying constitution of an important sessions of time period unifi­ cation difficulties.

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Extra info for A proof theory for general unification

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1 Let --+ be a binary relation on a set A, that is, --+ ~ A x A. The converse (or inverse) of the relation --+ is the relation denoted as _-lor +---, defined such that u +--- v iff v --+ u. The symmetric closure of - , denoted by - , is the relation --+ U +---. , ~, and ~. The n-fold composition of is denoted by ~. 2 Let --+ ~ T~(X) x T~(X) be a binary relation on terms. The relation is monotonic iff for any three terms s, t, and u, t, then u[o +- s] - u[a +- t] (somefor any 0 E Dom(u), if s times a monotonic relation is called a precongruence).

17 Suppose ta -[_""2, l"':"r p] t --+[Q 11 I 1 "':"r I, p] t1· 2 2 1 Then we say there is a critical overlap of the two rules on t if the projections of the non-variable symbols in each of the terms It and 12 onto t intersect, that is, if {a1,81,8 EN onVarDom(It}}n{a2"Y I"Y E NonVarDom(12)}:I 0. ~, Before we present the theorem, we need one lemma which generalizes a proposition from [73]. 18 Let R be a set of rewrite rules and (J and (J' be substitutions with D«(J) = D«(J') such that Vz E D«(J), (J(z) ~R (J'(z).

Xna we must have a Xl X2 Xn a, and so = = = = .. = UE(ax, xa) = {[a/x], faa/x], [aaa/x], ... }. Clearly this set is infinite, and none of the substitutions subsumes any other, since they are all ground, and so no most general E-unifier, nor even a finite set of "more general E-unifiers," can exist. We now discuss some notions needed to deal with this more complex situation. 3 Given a finite set E of equations and any set V of variables, we say that two substitutions u and 8 are equal modulo E over V, denoted by u =E 8[V], iff "Ix E V, u(x) ~E O(x).

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