W. Snyder's A proof theory for general unification PDF

By W. Snyder

ISBN-10: 0817635939

ISBN-13: 9780817635930

During this monograph we learn generalizations of normal unification, E-unification and higher-order unification, utilizing an summary technique orig­ inated via Herbrand and constructed relating to regular first-order unifi­ cation via Martelli and Montanari. The formalism offers the unification computation as a suite 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 dependent technique of analysing the homes of unification in a number of settings by way of delivering a fresh separation of the logical concerns from the specification of procedural details, and quantities to a collection of 'inference ideas' for unification, as a result the name of this booklet. We derive the set of modifications for normal E-unification and better­ order unification from an research of the feel during which phrases are 'the comparable' after program of a unifying substitution. In either instances, this ends up in an easy extension of the set of easy modifications given via Herbrand­ Martelli-Montanari for normal unification, and exhibits in actual fact the fundamental relationships of the elemental operations worthy in every one case, and hence the underlying constitution of crucial periods of time period unifi­ cation difficulties.

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8 For any substitution 0" and set of variables W such that D(O") ~ W, there exists an idempotent substitution 0"' such that D(O") = D( 0"'), 0" ~ 0"', and u' ~ O"[W]. Proof. Let D(u) n 1(0") = {x}, ... ,x n }, let {Yl, ... ,Yn} be a set of new variables disjoint from Wand 1(0"), let Pl = [yt/Xl, ... , Yn/xn], and let P2 [Xt/Yl, ... ,xn/Yn]. Now let 0"' U 0 Pl, so that clearly 0" ~ 0"' and D(O") = D(u') as required. Since Pl 0 P2 = Id[W U 1(0")], then 0" = 0" 0 Pl 0 P2 = 0"' 0 P2[W], and thus u' ~ u[W].

4 Equational Logic In this section we review the basic notions of the model and proof theory of equational logic. First we recall some simple concepts regarding relations on terms. 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.

18 = pi(ld· But then by monotonicity, we must have This result shows us that the test for local confluence may be restricted to critical overlaps. But the variant assumption for the use of rules allows us to sharpen this test so that we may confine our search for critical overlaps to an examination of the rules themselves. :... :... r2 are two variants of (not necessarily distinct) rewrite rules. Then there exists a critical overlap of the two rules on t iff there exists a substitution u = mgu(ldf3, 12), where ~ E NonVarDom(h).

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A proof theory for general unification by W. Snyder

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