A proof theory for general unification by W. Snyder PDF

By W. Snyder

ISBN-10: 0817635939

ISBN-13: 9780817635930

During this monograph we learn generalizations of ordinary unification, E-unification and higher-order unification, utilizing an summary method orig­ inated by means of Herbrand and constructed in terms of average first-order unifi­ cation by way of Martelli and Montanari. The formalism provides the unification computation as a collection of non-deterministic transformation principles for con­ verting a collection of equations to be unified into an specific illustration of a unifier (if such exists). this gives an summary and mathematically based technique of analysing the houses of unification in quite a few settings by means of delivering a fresh separation of the logical matters from the specification of procedural details, and quantities to a suite of 'inference principles' for unification, therefore the identify 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 similar' after software of a unifying substitution. In either circumstances, this leads to an easy extension of the set of simple changes given via Herbrand­ Martelli-Montanari for traditional unification, and indicates sincerely the elemental relationships of the basic operations priceless in every one case, and therefore the underlying constitution of an important periods of time period unifi­ cation difficulties.

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Extra info for A proof theory for general unification

Example text

In fact, it is 50 E-UNIFICATION possible that two terms have an infinite set of independent E-unifiers, as we now show. } U Eo, where Eo contains at least one constant symbol "a", and, using infix notation, let E = {(x' . y') . Zl == x' . (y' . Zl)}. This set axiomatizes non-empty strings over the set of constant symbols Eo, and so we represent terms as simply strings of constants and variables. Now consider the problem of E-unifying the two "strings" ax and xa. If Xl .. Xna we must have a Xl X2 Xn a, and so = = = = ..

For any system S, let us define a complexity measure peS) = < n, m > , where n is the number of unsolved variables and m is the sum of the sizes of all the terms in the system. Then the lexicographic ordering on < n, m > is wellfounded,4 and each transformation produces a new system with a measure strictly smaller under this ordering: Trivial and Term Decomposition must decrease m and can not increase n, and Variable Elimination must decrease n. Therefore the relation ==> is well-founded, and every transformation sequence must end in some system to which no transformation applies.

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 [36]). A reduction ordering -< is a strict ordering that is monotonic, stable, and such that >- is well-founded. With a slight abuse of language, we will also say that the converse >- of a strict ordering -< is a simplification ordering (or a reduction ordering). The importance of these orderings is shown by this next fundamental result, from [35].

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A proof theory for general unification by W. Snyder

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