By W. Snyder

During this monograph we learn generalizations of normal unification, E-unification and higher-order unification, utilizing an summary technique orig inated by means of Herbrand and constructed on the subject of common first-order unifi cation through Martelli and Montanari. The formalism provides the unification computation as a collection of non-deterministic transformation principles for con verting a suite of equations to be unified into an particular illustration of a unifier (if such exists). this gives an summary and mathematically stylish technique of analysing the homes of unification in numerous settings by means of supplying a fresh separation of the logical matters from the specification of procedural info, and quantities to a collection of 'inference principles' for unification, therefore the identify of this e-book. We derive the set of adjustments for normal E-unification and better order unification from an research of the experience during which phrases are 'the related' 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 essentially the fundamental relationships of the basic operations beneficial in each one case, and therefore the underlying constitution of an important periods of time period unifi cation difficulties.

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**Example text**

This was the approach taken in the pioneering work of [131], and extended to a large number of specific theories since then. , the E-unification problem may be decidable, or it may be possible always to find finite CSUs. A good survey of results of this sort may be found in [145] or [83]. In this book we are interested in more general forms of E-unification, and so we shall not discuss this ad-hoc approach further. The second approach developed for E-unification is called narrowing and depends on the special characteristics of canonical sets of rewrite rules.

If u = OIVar(S) satisfies condition (i), then we have our result trivially. Otherwise, if 1(0) = {Xl, ... ,x n } then let {Ylt ... ,Yn} be a set of new variables disjoint from the variables in W, D(O), 1(0), and Var(S). Now define the renaming substitutions Pl = [yt/Xlt ... , Yn/xn] and P2 [xt/Ylt ... , xn/Yn], and then let u 0 PIIVar(S). Clearly u satisfies (i), and since u = 00 pdVar(S)], we have the second part of (iii). Now since PIOP2 Id[Var(S)Ul(O)], we must have 0 OOPI 0P2[V areS)]. But then by the fact that u OopdVar(S)] we have 0 uop2[Var(S)], proving the first part of (iii).

R S! = t1 ~R t. 5 TERM REWRITING 39 exist some term w such that S~R w~Rt. But then by the previous lemma, w! is the unique normal form of w, s, and t. 0 This theorem gives us a simple decision procedure for the word problem in canonical theories: we simply reduce both terms to their unique normal forms and compare. Confluence is undecidable in general [77], but, in fact, if we can prove that the system has the termination property, it is possible to localize the test for confluence to single applications of rewrite rules.