Toward Formal Development of Programs from Algebraic Speciications: Implementations Revisited 1

The program development process is viewed as a sequence of implementation steps leading from a spe-ciication to a program. Based on an elementary notion of reenement, two notions of implementation are studied: constructor implementations which involve a construction \on top of" the implementing speciication, and abstractor implementations which additionally provide for abstraction from some details of the implemented speciication. These subsume most formal notions of implementation in the literature. Both kinds of implementations satisfy a vertical composition and a (modiied) horizontal composition property. All the deenitions and results are shown to generalise to the framework of an arbitrary institution, and a way of changing institutions during the implementation process is introduced. All this is illustrated by means of simple concrete examples.