By W. Snyder
During this monograph we learn generalizations of normal unification, E-unification and higher-order unification, utilizing an summary strategy orig inated via Herbrand and constructed relating to usual 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 suite of equations to be unified into an particular illustration of a unifier (if such exists). this gives an summary and mathematically based technique of analysing the houses of unification in a number of settings through delivering a fresh separation of the logical matters from the specification of procedural info, and quantities to a suite of 'inference ideas' for unification, as a result the name of this publication. We derive the set of differences for basic 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 ends up in an easy extension of the set of uncomplicated adjustments given via Herbrand Martelli-Montanari for traditional unification, and indicates sincerely the elemental relationships of the basic operations useful in each one case, and hence the underlying constitution of an important periods of time period unifi cation difficulties.
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Additional resources for A proof theory for general unification
We are interested of course in using orderings on terms to prove termination of rewrite systems, and, more generally, to do inductive proofs. The most general of these orderings are based on the notion of syntactic simplification. 6 A strict ordering -< has the subterm property if s -< 1( . ) for every term 1( . ,s, ... ). A simplification ordering -< is a strict ordering that is monotonic and has the subterm property (since we are considering symbols having a fixed rank, the deletion property is superfluous, as noted in Dershowitz ).
Now, since 0' is a mgu, it must be the case that 0' ~ PI U P2, and so there must be some 1] such that PI U P2 = 0' 0 1]. Also, by the variant assumption, D(pt} n Var(r2) 0 and D(P2) n Var(h) 0. 8 +- r2]) = 1]( 0'( IdP +- r2])) = 1](p). Finally, by the variant assumption, D(P2) n Var(rt) = tda = = = = 0, and so PI(rI) PI U p2(rI) 1](O'(rd) 1](q). o PRELIMINARIES 44 We may now bring all these results together in presenting the major theorem of this section, which shows that the test for local confluence may be restricted to the critical pairs of a system of rewrite rules.
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.