Get A proof theory for general unification PDF

By W. Snyder

ISBN-10: 0817635939

ISBN-13: 9780817635930

During this monograph we examine generalizations of normal unification, E-unification and higher-order unification, utilizing an summary process orig­ inated by means of Herbrand and built relating to general first-order unifi­ cation via Martelli and Montanari. The formalism provides the unification computation as a suite of non-deterministic transformation ideas for con­ verting a suite of equations to be unified into an particular illustration of a unifier (if such exists). this offers an summary and mathematically based technique of analysing the houses of unification in a variety of settings through delivering a fresh separation of the logical concerns from the specification of procedural details, and quantities to a suite of 'inference ideas' for unification, accordingly the name of this e-book. We derive the set of variations 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 situations, this leads to an easy extension of the set of easy variations given by means of Herbrand­ Martelli-Montanari for traditional unification, and indicates truly the elemental relationships of the elemental operations important in each one case, and therefore the underlying constitution of crucial periods of time period unifi­ cation difficulties.

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Extra info for A proof theory for general unification

Example text

It is easy to show that for any poset (S, >-) we have associated posets (M, ~) (where M is the set of all finite multisets of members of S) and (BB, >-'ez) for n > O. Furthermore >- is total (respectively, well-founded) iff >-'ez (for any n) is total (respectively, well-founded) iff ~ is total (respectively, well-founded). 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.

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], ... ,un) for some i, 1 ~ i ~ n, and by hypothesis E F Ui[S] == Ui[t] , and then (since the rest of the context is identical) clearly = = = E F f(U1' = = = ...

Ut, ... (Vt, ... ,vn) for some! E E, and O(Ui) = O(Vi) for 1 $ i $ n; or (iii) u is a variable not in V ar( v) or vice versa. If u is a variable and u f/. V ar( v), then [vlu] E U(u, v) and [vlu] $ O. By extending this analysis to account for systems of equations, we have a set of transformations for finding mgu's. 5 (The set of transformation rules ST) Let S denote any system (possibly empty), ! E E, and u and v be two terms. We have the following transformations. Trivial: {u~u}uS ~ (1) S Term Decomposition: For any !

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