By Ariya Isihara.
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Additional resources for Algorithmic term rewriting systems
In the present case, since we allow infinitely long reductions, strong →-compatibility is too strong to characterize the WN property. So, we think of replacing root reductions → by , and the other lower-level reductions by . Thus, an appropriate quasiorder should satisfy the following conditions: 1. If t → s, then t → -compatible’. s. This property is temporarily called ‘strongly 2. If t → s, then t →-compatible’. s. 4. Inductive height 43 3. The quasiorder is well-founded. In fact, we will see that an algorithmic term rewriting system is WN if and only if there exists a strongly → -compatible and weakly →-compatible well-founded quasiorder on initially proper ground terms (Theorems 96 and 97).
Proposition 60 A quasiorder is R1 . . Rn | R 1 . . R m -compatible if and only if the quasiorder is Ri | ∅-compatible for every i = 1, . . , n and ∅ | R j -compatible for every j = 1, . . , m. 4 Inductive height In this section we define the notion of inductive height. This notion arises when we characterise properness of a term t by a well-founded quasiorder on dom(t). First, we define the immediate subterm relation as follows. Definition 61 Let t be a term. Then, for every i = 1, . . , ar(t( )), we write t t/i.
Let t Θ = t Θ. Ω As mentioned above, we do not have to specify a function Θ: Lemma 77 Let A, (−)A be a semantics. Then, the interpretations − − Θ agree on each other for all applicable Θ and Θ . Θ and Proof: Let the semantics be an M, L semantics. Fix an applicable pair of Θ, Θ : SC → A, and an arbitrary positive real number ε. We show that dAsrt(t) ( t Θ , t Θ )<ε which will suffice. Since t Θ t Θ k k∈N and k k∈N , as in Definition 76 are both Cauchy sequences, respectively leading to t Θ and t Θ , we can find k0 ∈ N such that k k0 implies dAsrt(t) ( t Θ , t Θ k ), dAsrt(t) ( t Θ , t Θ k ) < ε/3 On the other hand, we can find k1 ∈ N such that M Lk1 < ε/3.
Algorithmic term rewriting systems by Ariya Isihara.