Investigating Z

Abstract

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.

A Logic for Schema-Based Program Development

“…Similarly, in schemas, the observations have values in an implicit state (binding); this point was discussed in section 2.1 (and discussed at length in our other papers e.g. [6], [8]) where we remarked that the states become explicit in derivations which ultimately take place in the underlying logical system Z C . It would be very much more pleasant if we could avoid mention of the state entirely, even in derivations which eventually require calculation in the core logic.…”

“…This logic is a conservative extension of the basis logic Z C . Full details may be found in [8] though, for convenience, we have included a summary in appendix A. The appendix also contains useful information regarding the notation we have employed in the paper.…”

“…The methods we adopt here for the state schema calculus are those we have previously introduced and, since the details have been thoroughly investigated before, we now present an overview. The reader is invited to consult [8] for a comprehensive treatment. The explanation of unfamiliar notation, and further details of the underlying logic Z C , are give in appendix A 2.1.…”

“…In this definition Pre U z holds if z is a binding in the precondition of the schema U (see [8]). The same relaxed frame axiom applies in this interpretation.…”

“…We will show how it can be constructed as a conservative extension of our existing logic for Z (see [6], [7], [8]) and illustrate the use of the theory in practice with a number of examples. The basis of our approach is to model a specification as a set of legitimate implementations.…”