Refinement of Parameterized Algebraic Specifications

A refinement relation for parameterized algebraic specifications is introduced in which the body specification is refined covariantly (i.e., specialized) while the parameter specification is refined contravariantly (i.e., generalized). This refinement is similar to the subtyping relation between function types. Assuming a few relatively weak axioms about parameterized specifications, refinement is shown to satisfy the expected properties for software development in-the-large: independent refinement of body and parameter, and closure under sequential composition, instantiation and a suitable form of `horizontal' composition. Keywords Parameterized specifications, mixed-variance refinement, compositionality 1 INTRODUCTION The primary subject of this paper is a general notion of refinement for parameterized specifications. Many notions of refinement have been previously proposed in the literature (Ehrig & Kreowski 1982, Ganzinger 1983, Goguen & Meseguer 1982, Sannella & Tarlecki 1988, S..