By W. Snyder
During this monograph we learn generalizations of ordinary unification, E-unification and higher-order unification, utilizing an summary strategy orig inated via Herbrand and built relating to average 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 dependent technique of analysing the houses of unification in quite a few settings by way of delivering a fresh separation of the logical concerns from the specification of procedural info, and quantities to a collection of 'inference ideas' for unification, accordingly the name of this ebook. We derive the set of ameliorations for basic 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 leads to an easy extension of the set of simple differences given via Herbrand Martelli-Montanari for traditional unification, and exhibits basically the elemental relationships of the elemental operations worthy in each one case, and hence the underlying constitution of an important sessions of time period unifi cation difficulties.
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Extra info for A proof theory for general unification
Now because A F I == r, we have [l]~ = [7']~' and so [p(l)]~ = [I]~ = [r]~ = [p(r)]~, and since cp was arbitrary, we conclude that E F p(l) == p(r). Next we show that if E 1= S == t , then E F u[s] == u[t] for any context u, by induction on lui. For the base case, if lui 1, then u[s] sand u[t] t and the result is trivial. Now suppose lui> 1 . Again, if u[s] s and u[t] t the result is trivial. Otherwise, if u f( Ul, ... , un), then u[s] = f(ut, ... ,Ui[S], ... ,un) and u[t] = f(U1, ... ,u;[t], ...
The common practice is to require I to be a proper sub term of u or a proper substitution instance of u, but it can also be advantagous to consider the age of the rules. In no case can a rule be used to reduce its own left-hand side. ) The soundness of the transformations is given by this next theorem, whose proof (although tedious) is not hard and is omitted. 2 (Soundness of the set ICB) If E, R .. -EuR = .. => K B E', R' then -E'UR' . A completion procedure is a strategy for applying these transformations; specifically, it takes as input a finite set of equations E and a reduction ordering >- and generates a sequence of pairs It should be obvious that the systems R 1 , R2, ...
For any such A and cp, let () : X -+ A be the assignment such that for every z, ()(z) [p(x)]~ (or, equivalently, () = po rp, where rp : T~(X) -+ A is the homomorphic extension of cp). We claim that [p(t)]~ [t]~ for any t E TE(X) , and proceed by [c] [c]~ ; and if t = induction on Itl. If t c E Eo , then [c]~ = C,A x EX, then [x]~ = cp(x) [z]~. •• , tn) for n > O. Then = = = = = = = [p(f(t1, ... ,tnm~ = [f(P(t1), ... ,p(tn))]~ = f,A ([P(td]~, ... , [p(tn)]~) = f,A([t1]~' ... ,[tn]~) = [/(t1, ...
A proof theory for general unification by W. Snyder