This is the first of a series of papers devoted to the thorough investigation of (total correctness) refinement based on an underlying partial relational model. In this paper we restrict attention to operation refinement. We explore four theories of refinement based on an underlying partial relation model for specifications, and we show that they are all equivalent. This, in particular, sheds some light on the relational completion operator (lifted-totalisation) due to Woodcock which underlies data refinement in, for example, the specification language Z. It further leads to two simple alternative models which are also equivalent to the others.

In this paper we introduce and investigate an improved kernel logic Z C for the specification language Z. Unlike standard accounts, this logic is consistent and is easily shown to be sound. We show how a complete schema calculus can be derived within this logic and in doing so we reveal a high degree of logical organization within the language. Finally, our approach eschews all non-standard concepts introduced in the standard approach, notably object level notions of substitution and entities which share properties both of constants and variables. We show, in addition, that these unusual notions are derivable in Z C and are, therefore, unnecessary innovations.

Stepwise design involves the process of deriving a concrete model of a software system from a given abstract one. This process is sometimes known as refinement.There

This is the first of two related papers. We introduce a simple specification logic Z C comprising a logic and a semantics (in ZF set theory) within which the logic is sound. We then provide an interpretation for (a rational reconstruction of) the specification language Z within Z C. As a result we obtain a sound logic for Z, including a basic schema calculus.

Abstract. This is the second of two related papers. In “Revising Z: Part I - logic and semantics” (this journal) we introduced a simple specification logic Z C comprising a logic and a semantics (in ZF set theory). We then provided an interpretation for (a rational reconstruction of) the specification language Z within Z C . As a result we obtained a sound logic for Z, including the basic schema calculus. In this paper we extend the basic framework with more sophisticated features (including schema operations) and we mount a critique of a number of concepts used in Z. We further demonstrate that the complications and confusions which these concepts introduce can be avoided without compromising expressibility.