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Read e-book online 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 procedure orig­ inated via Herbrand and built relating to common first-order unifi­ cation by way of 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 particular illustration of a unifier (if such exists). this offers an summary and mathematically stylish technique of analysing the homes of unification in a number of settings via supplying a fresh separation of the logical matters from the specification of procedural details, and quantities to a collection of 'inference principles' for unification, therefore the name of this e-book. We derive the set of changes for basic E-unification and better­ order unification from an research of the feel within which phrases are 'the comparable' after program of a unifying substitution. In either circumstances, this leads to an easy extension of the set of uncomplicated variations given by way of Herbrand­ Martelli-Montanari for normal unification, and exhibits truly the elemental relationships of the basic operations worthwhile in each one case, and hence the underlying constitution of an important sessions of time period unifi­ cation difficulties.

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R w ~ R t2 . We shall call such a w the confluence term for t 1 and t2, and shall write t1! t2 if there exists a confluence term for hand t2. The following definition and lemma provide an alternate characterization of confluence. 10 A system R is Church-Rosser if for any terms hand t2, t1 ~Rt2 implies that tt! t2. 11 A system R of rules is confluent iff it is Church-Rosser. Proof, The if part is trivial, since u ~ R t '::""R v implies that u ~R v, and hence that u! v. The only if part proceeds by induction on the 38 PRELIMINARIES number of rewrite steps in ~R.

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. Now consider the problem of E-unifying the two "strings" ax and xa. If Xl .. 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.

Tn»], where in the second step we applied the induction hypothesis. 9 For any set of equations E, T/ E FE. Proof Suppose I == r E E; we must prove that [/]~/E = [r]~/E for any assignment cp (which interpretes the universally quantified variables in I == r). But then there must exist some u as specified in the previous lemma (for example let u(x) be the least member of cp(x) wrt some total ordering on TE(X». 6. , [/]~/E=[r]~/E for any assignment cp. Thus let 32 PRELIMIN ARIES cp be the particular assignment which maps each z to [z], and let 0' be the identity on X.

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