Specification, horizontal composition and parameterization of algebraic implementations

Loose specifications of abstract data types (ADTs) have many non-isomorphic algebras as models. An implementation between two loose Specifications should therefore consider many abstraction functions together with their source and target algebras. Just like specifications are stepwise refined to restrict their class of models, implementations should be stepwise refinable to restrict the class of abstraction functions. In this scenario specifications and implementations can be developed interwovenly. We suggest to have implementation specifications analogously to loose ADT specifications: Implementations have signatures, models, axioms and sentences thus constituting an institution. Implementation specifications are the theories of this institution and refinements between implementation specifications are its theory morphisms. In this framework, implementations between parameterized specifications and horizontal composition of implementations turn out to be special cases of the more powerful concept of parameterized implementations, which allow to instantiate an implementation by substituting a subimplementation by another implementation