A survey of modal logics characterising behavioural equivalences for non-deterministic and stochastic systems

Abstract

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.

Logical characterization of fluid equivalences

“…In [18,15], on sequential and concurrent Markovian process calculi SMPC (MPC) and CMPC, the logical characterizations of Markovian trace and bisimulation equivalences have been accomplished with the modal logics HM L M T r and HM L M B , based on Hennessy-Milner Logic (HML) [52]. In [19], on (sequential) Markovian process calculus MPC, the characterizations of Markovian trace and bisimulation equivalences have been constructed with the HML-based modal logics HM L N P M T r and HM L M B .…”

“…The formal definition of fluid trace equivalence resembles that of ordinary Markovian trace equivalence, proposed on transition-labeled CTMCs in [83], on sequential and concurrent Markovian process calculi SMPC and CMPC in [14,18,15,16,19] and on Uniform Labeled Transition Systems (ULTraS) in [21,22,17]. While defining fluid trace equivalence, we additionally have to take into account the fluid flow rates in the corresponding discrete markings of two compared LFSPNs.…”

“…Note also that our notion of fluid trace equivalence is based rather on that of Markovian trace equivalence from [83], since there the average sojourn times in the states "surrounding" the actions of the corresponding traces of the equivalent processes should coincide while in the definition of the mentioned equivalence from [14,18,15,16,19], the shorter average sojourn time may simulate the longer one. If we would adopt such a simulation then the smaller average potential fluid change volume (the product of the average sojourn time and potential fluid flow rate) would model the bigger one, since the potential fluid flow rate remains constant while residing in a discrete marking.…”

“…Therefore, our definition of the trace equivalence on the discrete markings of LFSPNs is similar to that of ordinary (that with the absolute time counter or with the countdown timer) Markovian trace equivalence [83] on transition-labeled CTMCs. Ordinary Markovian trace equivalence and its variants from [83] have been later investigated and enhanced on interactive Markov chains (IMCs) in [84], on sequential and concurrent Markovian process calculi SMPC and CMPC in [14,18,15,16,19], on Uniform Labeled Transition Systems (ULTraS) in [21,22,17], on continuous time Markov decision processes (CTMDPs) in [66] and on Markov automata (MAs) in [67]. As for the continuous markings of the two LFSPNs, we further select the paths with the same extracted action sequence and the same sequence of the extracted average sojourn times (exit rates) by counting the execution probabilities only of those paths additionally having the same sequence of extracted potential fluid flow rates of the respective continuous places (we assume that each compared LFSPN has only one continuous place) in the corresponding discrete markings.…”

“…Thus, our definition of the bisimulation equivalence on the discrete markings of LFSPNs is similar to that of the performance bisimulation equivalences [28,29] on labeled CTSPNs and labeled generalized SPNs (GSPNs) [60,61,12,13], as well as the strong equivalence from [53] on stochastic process algebra PEPA. All these relations belong to the family of Markovian bisimulation equivalences, investigated on sequential and concurrent Markovian process calculi SMPC and CMPC in [14,18,15,16,19], as well as on Uniform Labeled Transition Systems (ULTraS) in [21,22,17]. As for the continuous markings, we require that, for every pair of the Markovian bisimilar discrete markings, the fluid flow rate of the continuous place in the first LFSPN should coincide with that of the continuous place in the second LFSPN (again, we compare only LFSPNs with a single continuous place each).…”