By Ariya Isihara.

ISBN-10: 9086594425

ISBN-13: 9789086594429

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**Example text**

A consequence of the above is that the algebraic interpretations of this section assume some order on the domain. More specifically, we will use so-called coinductive domains (see Definition 67), of which the prime example in our context is the partial order of countable ordinals. 5. Algebraic interpretation 47 semantical interpretation employs complete metric spaces. The method of algebraic interpretation plays an important role to ensure termination of a finitary rewriting system [46, Chapter 7].

Then, a continuous Σ-algebra A, [−] consists of a function [f] : An → A for each n-ary function symbol f, satisfying the following condition: For every n-tuple of sets B1 , . . , Bn ⊆ A, if f ∈ C and Bi is singleton for every i such that in(f, i) ∈ SI , then lub{[f](b1 , . . , bn ) | bi ∈ Bi } = [f](lub B1 , . . , lub Bn ) Given a continuous Σ-algebra A, [−] , for each ordinal α Ω, define the interpretation [−]α : {t ∈ G | t < α} → A by transfinite induction on α as follows: 1. For α = 0, the domain {t ∈ G | t < 0} is empty.

3. The semantics is adequate. Proof: (1⇒2) Follows from Corollary 79, using Compression Lemma (Lemma 38). (2⇒3) Since t → → → cnf(t), we have t = cnf(t) . f. Figure 7). Definition 82 Let a semantical interpretation − : V → A (that can be regarded as a semantics under assumption D = ∅) be given. Then: 1. The semantics is full if for every a ∈ A there exists some t ∈ V such that t = a. 2. The semantics is strongly sound if, for every t, s ∈ V , t = s implies t ≡ s. 3. The semantics is weakly sound if, for every n-ary defined symbol f and every t1 , .

### Algorithmic term rewriting systems by Ariya Isihara.

by Anthony

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