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Show (i) V~£1; D(~) .Q V & I(~) n W = 0 {protection of W) (ii) ~ .Q UE(A,8} {correctness) (iii} V~E UE(A,8} :=leE~ 8 ~E ~ (V] (completeness). In addition, ~ is said to be minimal if and only if it also satisfies the condition: v~ ,eE~ ~-¥- e => ~ t:;E e [Vl (minimality). An E-unification algorithm is complete if it generates a complete set of E-unifiers for all E-unifiable input terms. 0 Remarks: 1. The introduction of W is technical; this is because it is necessary to separate the variables introduced by unification from the variables that are still in use. 2. In fact there is no reason to introduce W -¥- V as long as we study E-unification by itself. But in many algorithms, unification needs to be performed on subterms and it is necessary to separate the variables introduced by unification from those of the context. This is the case for resolution in equational theories [92], and for the generalization of the Knuth-Bendix completion procedure in congruence classes of terms [89]. 3. Complete sets of E-unifiers for any theory always exist. This can be shown by taking all the E-unifiers satisfying {i). Complete sets of minimal E-unifiers, however, may not always exist for a theory, and when they exist, they may not always be finite [28, 47, 109, 11 O]. 4. In general, it is desirable to obtain complete sets of minimal Eunifiers. The minimality property, however, is much harder to obtain than completeness. Definition 2-6: A term rewriting system R is a set of pairs of terms ak - > flk, such that V(flk) .Q V{ak}. We say that a term A reduces at address u to a term 8 using the rule ak - > flk and write A ->R[u,k] 8, if there exists a matching substitution .,, from ak to A/u and B = A[u <- vJ{flk)]. We may omit R and, sometimes omit k or [u,k], and write A - > u B, or A - > 8, respectively. -> is called the reduction relation on T{F,V). We say that A/u is a reducible subterm of A (or simply, u is a reducible address) and denote by U{A) the set of all addresses v such that A/v is a reducible subterm of A. 18 |
