Transformation of Applicative Specifications into Imperative ...

Model Morphisms
7.4. AN INSTITUTION FOR RSLI
Let Σ = 〈A, OP, V 〉 and Σ ′ = 〈A ′ , OP ′ , V ′ 〉 be signatures, σ : Σ → Σ ′ a signature
morphism and m ′ = 〈m ′ A , m′ OP , s′ init 〉 a Σ′ -model. The model morphism
Mod(σ) : Mod(Σ ′ ) → Mod(Σ) of m ′ is the Σ-model m = 〈mA, mOP , sinit〉
where
• mA = m ′ A
• for mOP :
– dom mOP = dom OP = dom OP ′ = dom m ′ OP
– mOP (c) is defined in terms of m ′ OP (c) for c ∈ dom mOP . Some
examples of this is given below.
• sinit = s ′ init \(dom V ′ \dom V ). Note that sinit = [ ] in the applicative
case.
In the following examples of how mOP (f) is defined in terms of m ′ OP (f)
are given. In the examples it is assumed that the signature Σ is applicative
and the σ-transformation only has one type of interest tv with corresponding
variable v. Three cases are considered.
1. Observer
f : te1 × tv × te2 ∼ → tres
for all (ve1, x, ve2) ∈ Value A ∗ (te1×tv×te2):
mOP (f)(ve1, x, ve2)([ ]) ≡
case m ′ OP (f)(ve1, ve2)([ v ↦→ x ]) of
⊥ → ⊥,
(vres, [ v ↦→ x ′ ]) → (vres, [ ])
end
σ
f : te1 × te2 ∼ → read v tres
Note that [v ↦→ x] = [v ↦→ x ′ ] as m ′ OP (f) must write conform with the
empty set of write variables.
Explanation: mOP (f) returns chaos if m ′ OP (f) returns chaos. Otherwise
mOP (f) returns the same result as m ′ OP (f). When evaluating
m ′ OP
(f) a store s = [v ↦→ x] is kept reflecting the actual value x of
v. This value is not altered during the evaluation of f, since f is an
observer.
2. Generator
f : targ ∼ → te1 × tv × te2
σ
f : targ ∼ → write v te1 × te2
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