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Read e-book online A proof theory for general unification 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 technique orig­ inated by way of Herbrand and built in relation to typical first-order unifi­ cation by means of Martelli and Montanari. The formalism offers the unification computation as a suite of non-deterministic transformation ideas 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 homes of unification in a number of settings through supplying a fresh separation of the logical matters from the specification of procedural details, and quantities to a collection of 'inference ideas' for unification, for that reason the identify of this publication. We derive the set of modifications for normal E-unification and better­ order unification from an research of the feel during which phrases are 'the related' after program of a unifying substitution. In either instances, this ends up in an easy extension of the set of easy ameliorations given through Herbrand­ Martelli-Montanari for normal unification, and indicates in actual fact the fundamental relationships of the basic operations invaluable in every one case, and hence the underlying constitution of an important periods of time period unifi­ cation difficulties.

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O We now show that the existence of critical overlaps is a necessary condition for a system to be non-confluent. 19 (Knuth-Bendix) Let t, ft, and t2 be terms and let where 11 ~ rl and 12 ~ r2 are variants of rules from some set R. Then either there exists a confluence term w such that t2 ~R w ~R tl or there is a critical overlap of It ~ rl and 12 "":" r2 on t. 5 41 TERM REWRITING Proof. As regards the locations of the addresses 0' and 0:' in t, there are two cases. p,l w +--[a l h"':'r2,P21 tt, in other words, the rewrite steps commute, which may be illustrated: 1: (B) One of 0', 0" is a prefix of the other.

It is also possible that the procedure may run forever, even given certain fairness assumptions regarding the selection of equations and of critical pairs. KB ... 6 COMPLETION OF EQUATIONAL THEORIES 47 = is Eoo, Roo, where EOO Ui~O nj~i Ej is the set of persisting equations and ROO = Ui~O nj~i Rj is the set of persisting rewrite rules. A sequence is called successful if E oo = 0 and ROO is canonical. e. C8) is correct if EOO = 0 implies that ROO is canonical (whether the sequence is infinite or not), and it is complete if every sequence under the strategy is successful.

The first transformation simply removes trivial equations from E. EU{u== u},R ~ E,R. (I) We may reduce equations in E by rules in R, so that if u and if v - R W, E U {u == v}, R ~ E U {w == v}, R, (2a) E U {u == v}, R ~ E U {u == w}, R. (2b) - R W, Furthermore, we may inter-reduce rules in R. If v - R E,RU{u"":"'v} ~ E,RU{u==w}, w, then (3) and if I> is a well-founded ordering on rewrite rules and u - [a, ,. -r,p1 W, where u -=+ v I> 1-=+ r, then E,RU {u"":'" v} ~ Eu {w == v},R. ) If u >- v in the reduction ordering, then EU{u==v},R ~ E,RU{u-=+v}.

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