An Analysis of Total Correctness Refinement Models for Partial Relation Semantics I

Abstract

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.

An alternative approach to handling nondeterminism in model-oriented data-refinement

“…We then define the notions of implementation by means of this operator and establish the induced theories of data-refinement (sections 4.3 and 4.4). In section 5, we show that each of the implementation-based theories is equivalent to the corresponding standard characterisation of data-refinement, demonstrating that reallocation of nondeterminism does not modify the intentions manifested by our normative characterisation of data-refinement (shown to be equivalent to standard account in [8,9,7]). Our paper concludes with a review of what has been established and an agenda for further investigation (section 6).…”

“…This paper provides a generalisation, to data refinement, of an approach we investigated for operation-refinement in [11,12]. There, in the spirit of constructive theories of program development, e.g.…”

“…in [11,12]) as well as data-refinement (e.g. in [8,9,7]), answering various questions raised by the technical machinery; in particular, precisely explaining the extent to which this approach is canonical and the extent to which it is arbitrary. The modus operandi we have employed in that work is to investigate the relationship amongst a variety of refinement theories, each of which is based on a distinct notion of relational completion.…”

Results on formal stepwise design in Z

“…Additional details can be found in appendix A; however, the reader may need to consult [9] and [4] in order to fully understand the notational and meta-notational conventions.…”

“…We show in appendix A how terms are incorporated in the existing types of [9], in order to obtain a conservative extension framework, [4]. Furthermore, we present the notion of the natural carrier set for each type, which excludes terms or, in case of a schema type, bindings that contain at least one observation bound to .…”