Relational concurrent refinement part II: Internal operations and outputs

Abstract

Abstract. Two styles of description arise naturally in formal specification: state-based and behavioural. In state-based notations, a system is characterised by a collection of variables, and their values determine which actions may occur throughout a system history. Behavioural specifications describe the chronologies of actions -interactions between a system and its environment. The exact nature of such interactions is captured in a variety of semantic models with corresponding notions of refinement; refinement in state based systems is based on the semantics of sequential programs and is modelled relationally. Acknowledging that these viewpoints are complementary, substantial research has gone into combining the paradigms. The purpose of this paper is to do three things. First, we survey recent results linking the relational model of refinement to the process algebraic models. Specifically, we detail how variations in the relational framework lead to relational data refinement being in correspondence with traces-divergences, singleton failures and failures-divergences refinement in a process semantics. Second, we generalise these results by providing a general flexible scheme for incorporating the two main "erroneous" concurrent behaviours: deadlock and divergence, into relational refinement. This is shown to subsume previous characterisations. In doing this we derive relational refinement rules for specifications containing both internal operations and outputs that corresponds to failures-divergences refinement. Thirdly, the theory has been formally specified and verified using the interactive theorem prover KIV.

Introducing extra operations in refinement

“…the monograph [DB01] or the research papers [BDS09,BD10]. These include detailed descriptions in terms of binary relations, and derivations of formulations in Z for data refinement, using both upward and downward simulations, representing a wide range of commonly used refinement relations for state-based systems and process algebras.…”

“…A more general approach, also consistent with the process algebraic view, is to preserve or reduce any divergence that was already present in the abstract specification. This is worked out in detail in [BDS09], essentially for F although it could be done for NB as well, and the impact of differing notions of "livelock" or divergence is discussed in [BD09]. The semantic relation established in this case is roughly that for every concrete trace, an abstract trace exists that is consistent with it, with both traces' subsequences of external actions being identical 8 .…”

“…It also excludes reduction of granularity by introducing internal actions using the "perspicuous actions" conditions. However, the more general "weak refinement" rules can be used in combination with condition (3), as we have shown in [BDS09] for F, and in [DB99] for Z. This is explained by not being constrained to considering the concrete operation in isolation, but rather only considering it in the context of possible internal concrete behaviour.…”

Relational Concurrent Refinement: Timed Refinement

“…Our ongoing research on relational concurrent refinement [6,3,7,2,8] contributes to this agenda, by explicitly recording in a relational setting the observations characterising process algebraic models. This allows the verification of concurrent refinement through the standard relational method of simulations, and to interpret relational formalisms like Z in a concurrency model.…”

“…We derived simulation rules for process algebraic refinement (such as trace, failuresdivergences, readiness [6] etc), including also outputs and internal operations [3], for different models of divergence [2], as well as automaton-based refinements [8]. The current paper extends this by considering how time can be modelled in a relational context, and thus we derive simulation rules for some of the timed refinement preorders.…”