Incompleteness of relational simulations in the blocking paradigm
Science of Computer Programming
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This paper presents a new strand of investigation which complements our previous investigation of refinement for specifications whose semantics is given by partial relations (using Z as a linguistic vehicle for this semantics). It revolves around extending our mathematical apparatus so as to continue our quest for examining mathematically the essence of the lifted-totalisation semantics (which underlies the de facto standard notion of refinement in Z) and the role of the semantic elements $$\perp$$ in model-theoretic refinement, but this time in the abortive paradigm. The analysis is given in two salient parts. In the first part, we consider the simpler framework of operation-refinement: we examine the (de facto) standard account of operation-refinement in this regime by introducing a simpler, normative theory which captures the notion of firing-conditions refinement directly in the language and in terms of the natural properties of preconditions and postconditions. In the second part, we generalise our analysis to a more intricate investigation of simulation-based data-refinement. The proof-theoretic approach we undertake in the formal analysis provides us with a mathematical apparatus which enables us to examine precisely the relationships amongst the various theories of refinement. This enables us to examine the general mathematical role that the $$\perp$$ values play in model-theoretic refinement in the abortive paradigm, as well as the significance of the unique interaction of these values with the notions of lifting (of data simulations) and lifted-totalisation (of operations) in this regime. Furthermore, we generalise this mathematical analysis to a more conceptual one which also involves extreme specifications.