By W. Snyder

During this monograph we examine generalizations of normal unification, E-unification and higher-order unification, utilizing an summary method orig inated through Herbrand and built on the subject of commonplace 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 particular illustration of a unifier (if such exists). this offers an summary and mathematically stylish technique of analysing the homes of unification in quite a few settings by way of offering a fresh separation of the logical concerns from the specification of procedural details, and quantities to a collection of 'inference ideas' for unification, consequently the name of this ebook. We derive the set of variations for normal E-unification and better order unification from an research of the experience 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 simple modifications given via Herbrand Martelli-Montanari for traditional unification, and exhibits essentially the fundamental relationships of the basic operations worthwhile 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**

If this test fails, we may attempt to complete the system of rules by adding the critical pairs to the system as rules, if they can be oriented with respect to the termination ordering. Since each pair is a logical consequence of the original system, this is a conservative extension of the theory, and each of the original critical pairs is now trivially confluent. Unfortunately, new critical pairs may be created. 14) to solve the word problem in E. Of course, since the word problem for arbitrary E is not decidable in general, this method will not work in every case.

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). The relation --+ is stable (under substitution) if s --+ t implies that u(s) --+ u(t) for every su bstitution u. 3 Let E ~ T~(X) x T~(X) be a set of equations. We define the relation - E over T~(X) as the smallest symmetric, stable, and monotonic relation that contains E.

Proof. We simply observe that (J(t) = (J(t)[a +- (J(t/a)] = O(t[O' +- 00 O(t/O')]) = (J(t[O' +- O(t/O')]). o We now formally justify that we may often restrict our attention to idempotent substitutions without loss of generality, by showing that any substitution is equivalent (over an arbitrary superset of its support) up to renaming with an idempotent substitution. 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].