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The program development process is viewed as a sequence of implementation steps leading from a specification to a program. Based on an elementary notion of refinement, two notions of implementation are studied: constructor implementations which involve a construction "on top of" the implementing specification, and abstractor implementations which additionally provide for abstraction from some details of the implemented specification. These subsume most formal notions of implementation in the literature. Both kinds of implementations satisfy a vertical composition and a (modified) horizontal composition property. All the definitions and results generalise to the framework of an arbitrary institution.

The program development process is viewed as a sequence of implementation steps leading from a specification to a program. Based on an elementary notion of refinement, two notions of implementation are studied: constructor implementations which involve a construction "on top of" the implementing specification, and abstractor implementations which additionally provide for abstraction from some details of the implemented specification. These subsume most formal notions of implementation in the literature. Both kinds of implementations satisfy a vertical composition and a (modified) horizontal composition property. All the definitions and results generalise to the framework of an arbitrary institution. Unfortunately, for lack of space we are not able to cover this topic here; the interested reader should consult [ST 87] for a detailed treatment of this subject as well as for all the proofs, examples and full discussion which we are forced to omit here. 2 Algebraic preliminaries Due to lack of space we omit the definitions of the following standard algebraic notions: signature (E), signature morphism (a), the category Sign of signatures with initial (empty) signature ~.