Link to this article: http://journals.cambridge.org/abstract_S0960129507006408How to cite this article: MARCO BERNARDO and STEFANIA BOTTA (2008). A survey of modal logics characterising behavioural equivalences for non-deterministic and stochastic systems.Behavioural equivalences are a means of establishing whether computing systems possess the same properties. The specific set of properties that are preserved by a specific behavioural equivalence clearly depends on how the system behaviour is observed and can usually be characterised by means of a modal logic. In this paper we consider three different approaches to the definition of behavioural equivalences -bisimulation, testing and traceapplied to three different classes of systems -non-deterministic, probabilistic and Markovian -and we survey the nine resulting modal logic characterisations, each of which stems from the Hennessy-Milner logic. We then compare the nine characterisations with respect to the logical operators, in order to emphasise the differences between the three approaches in the definition of behavioural equivalences and the regularities within each of them. In the probabilistic and Markovian cases we also address the issue of whether the probabilistic and temporal aspects should be treated in a local or global way and consequently whether the modal logic interpretation should be qualitative or quantitative.
Markovian testing and trace equivalences have been recently proposed as reasonable alternatives to Markovian bisimilarity, as both of them induce at the Markov chain level an aggregation strictly coarser than ordinary lumping that is still exact. In this paper we provide a modal logic characterization for each of the two non-bisimulation-based Markovian behavioral equivalences, which relies on a quantitative interpretation of a variant of the Hennessy-Milner logic. We also show that, unlike the Markovian bisimilarity case, such modal logic characterizations exist only when probabilistic and temporal aspects are treated in a global way
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