Specification of Parametrized Programs - Persistency Revisited

The important distinction between parameterized specifications and specifications of parameterized programs has been originally pointed out in [12]. An important difference concerns the objects which the two kinds of parameterization allow one to reuse. A parameterized specification, “PSP”, e.g., of stacks parameterized by elements, Stack(El), allows one, under some conditions, to instantiate the formal parameter specification El with an actual parameter specification, sayNat, to obtain a new specification Stack(Nat). What is reused is the text of the parameterized specification. Except for that, the model classes of Stack(El) and Stack(Nat) are just model classes of two different specifications, related merely by the existence of a reduct functor from the latter to the former, e.g. [2]. Specification of a parameterized data type, “PDT” [12, 13], on the other hand, is a specification which requires a reusable implementation. E.g., specification of a data type stack parameterized by elements, Stack[El], requires an implementation of a data type with a parameter, i.e., one capable of taking any implementation of El and resulting in an implementation of Stack[El]. The model class of such a specification should be thus seen as consisting of some – perhaps all – functors sending models of the formal parameter specification X to models of the parameterized specification P[X] : FMod(P[X]) ⊆ {F : Mod(X) → Mod(P[X])}. (1)