Transformation of Applicative Specifications into Imperative ...

Sentences
7.4. AN INSTITUTION FOR RSLI
An RSLI Σ-sentence is a boolean value expression, which is well-formed
according to the static semantics of RSL and is within the subset of RSLI.
Sentence Morphisms
As a specification represents a signature and a collection of sentences, the
transformation rules do not only define a transformation between the signatures
of the specifications but also a transformation between sentences. An
RSLI sentence morphism Sen(σ) : Sen(Σ) → Sen(Σ ′ ) should be defined
such that it respects the defined transformation rules for sentences.
Sentence Functor
The functor Sen : Sign → Set is a functor that maps signatures Σ in Sign to
the set Sen(Σ) of Σ-sentences and maps each signature morphism σ : Σ → Σ ′
in Sign to the sentence morphism Sen(σ) : Sen(Σ) → Sen(Σ ′ ) as described
above.
Models
Let Σ = 〈A, OP, V 〉 be a signature. An RSLI Σ-model is a triple m =
〈mA, mOP , sinit〉 where
where
• mA : Id →m Types, such that dom mA = dom A and mA(a) =
M(A ∗ (a)). That is mA maps type identifiers into the types their abbreviations
denote.
• mOP : Id →m Values, such that dom mOP = dom OP and mOP (op) ∈
M(OP (op)). That is mOP maps value identifiers into the values they
denote. These values must be in the types denoted by their type expressions.
• sinit ∈ Store. That is sinit denotes the initial store that maps variable
identifiers to their initial values.
• A ∗ is a function that takes a type expression to its canonical form,
i.e. a type expression not referring to any type identifiers, by recursively
expanding type identifiers according to their abbreviations.
• M : T (A, V ) → Types is a meaning function mapping type expressions
into the types they denote: M(t) = Value A ∗ (t).
A type is a set of values, also known as a value domain. Valuet denotes
the value domain for a type expression t ∈ T (∅, V ), e.g.:
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