Refinement and the Z Schema Calculus

Abstract

It is well known that the principal operators in the Z schema calculus are not monotonic with respect to refinement, which limits their usefulness in software development. The usual reaction to this observation is to remove all schema operators before performing any kind of refinement, or to move to a different formalism such as B or the refinement calculus. This paper examines the interaction between refinement and the schema calculus more closely, showing exactly how non-monotonicity arises, and identifying various conditions under which components of schema expressions can be safely replaced by their refinements. This analysis uses a decomposition of the standard refinement relation into two simpler relations that allow us to study refinements that modify a specification in different ways.