Transformation of Applicative Specifications into Imperative ...

CHAPTER 7. CORRECTNESS OF TRANSFORMATION RULES
7.4 An Institution for RSLI
In [Lin04] and [LH04] an institution for mRSL is defined. As mentioned
above mRSL is an applicative subset of RSL. It is in the following outlined
how an institution for the imperative subset RSLI can be defined and in that
way an outline of a verification of the transformation rules is given. To simplify
matters sort definitions, variant definitions and short record definitions
are not considered, nor are hidden and implicit observers or hidden and implicit
generators. In a proper definition they must be included. Furthermore,
only maximal type expressions are considered.
Signatures
An RSLI signature is a triple Σ = 〈A, OP, V 〉 where
• A = Id →m T (A, V ) is a map, which represents a set of non-cyclic abbreviation
definitions. It maps type identifiers to the type expressions
they abbreviate.
• OP = Id →m T (A, V ) is a map, which represents a set of value declarations.
It maps value identifiers to their declared type expressions.
• V = Id →m T (A, V ) is a map, which represents a set of variable definitions.
It maps variable identifiers to the corresponding type expressions.
such that dom A, dom OP and dom V are disjoint.
T (A, V ) is a set of RSL type expressions referring to type identifiers in
dom A and variable identifers in dom V .
A signature is said to be applicative if V = [ ], since there are no variable
declarations in the applicative case.
Signature Morphisms
An RSLI signature morphism σ : Σ → Σ ′ is defined as the transformation
from the transformable specification SP to the specification SP ′ such that
Sign(SP ) = Σ and Sign(SP ′ ) = Σ ′ . Composition is defined in the obvious
way and the identities are defined such that a signature is mapped to itself.
Recalling the definition of the transformation rules, note that for a signature
morphism σ : Σ → Σ ′ , where Σ = 〈A, OP, V 〉 and Σ ′ = 〈A ′ , OP ′ , V ′ 〉:
• A = A ′ , since the abbreviation definitions are the same
• dom OP = dom OP ′ , since the value declarations are transformed
into value declarations of the same names
• dom V ⊆ dom V ′ , since variables cannot be removed, only added
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