Abstract
Reactive systems \`a la Leifer and Milner, an abstract categorical framework for rewriting, provide a suitable framework for deriving bisimulation congruences. This is done by synthesizing interactions with the environment in order to obtain a compositional semantics. We enrich the notion of reactive systems by conditions on two levels: first, as in earlier work, we consider rules enriched with application conditions and second, we investigate the notion of conditional bisimilarity. Conditional bisimilarity allows us to say that two system states are bisimilar provided that the environment satisfies a given condition. We present several equivalent definitions of conditional bisimilarity, including one that is useful for concrete proofs and that employs an up-to-context technique, and we compare with related behavioural equivalences. We consider examples based on DPO graph rewriting, an instantiation of reactive systems.
Highlights
Behavioural equivalences, such as bisimilarity, relate system states with the same behaviour
We enrich the notion of reactive systems by conditions on two levels: first, as in earlier work, we consider rules enriched with application conditions and second, we investigate the notion of conditional bisimilarity
We present several equivalent definitions of conditional bisimilarity, including one that is useful for concrete proofs and that employs an up-to-context technique, and we compare with related behavioural equivalences
Summary
Behavioural equivalences, such as bisimilarity, relate system states with the same behaviour. We are in particular interested in conditional bisimilarity, which allows us to say that two states a, b are bisimilar provided that the environment satisfies a condition C Work on such conditional bisimulations appears somewhat scattered in the literature (see for instance [Lar[86], HL95, Fit[02], BKKS17]). We state our results in a very general setting: reactive systemsa la Leifer and Milner [LM00], a categorical abstract framework for rewriting, which provides a suitable framework for deriving bisimulation congruences (Transitions labelled with such a minimal context will be called representative steps in the sequel They are related to the idem pushout steps of [LM00].) Here, we rely on the notion of saturated bisimilarity introduced in [BKM06] and we consider reactive system rules with application conditions, generalizing [HK12].
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