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Register a new userLogical, Metric, and Algorithmic Characterisations of Probabilistic Bisimulation
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Added by
Yuxin Deng
Publication date
2011
fields
Informatics Engineering
and research's language is
English
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Many behavioural equivalences or preorders for probabilistic processes involve a lifting operation that turns a relation on states into a relation on distributions of states. We show that several existing proposals for lifting relations can be reconciled to be different presentations of essentially the same lifting operation. More interestingly, this lifting operation nicely corresponds to the Kantorovich metric, a fundamental concept used in mathematics to lift a metric on states to a metric on distributions of states, besides the fact the lifting operation is related to the maximum flow problem in optimisation theory. The lifting operation yields a neat notion of probabilistic bisimulation, for which we provide logical, metric, and algorithmic characterisations. Specifically, we extend the Hennessy-Milner logic and the modal mu-calculus with a new modality, resulting in an adequate and an expressive logic for probabilistic bisimilarity, respectively. The correspondence of the lifting operation and the Kantorovich metric leads to a natural characterisation of bisimulations as pseudometrics which are post-fixed points of a monotone function. We also present an on the fly algorithm to check if two states in a finitary system are related by probabilistic bisimilarity, exploiting the close relationship between the lifting operation and the maximum flow problem.
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Bisimulation metrics provide a robust and accurate approach to study the behavior of nondeterministic probabilistic processes. In this paper, we propose a logical characterization of bisimulation metrics based on a simple probabilistic variant of the Hennessy-Milner logic. Our approach is based on the novel notions of mimicking formulae and distance between formulae. The former are a weak version of the well known characteristic formulae and allow us to characterize also (ready) probabilistic simulation and probabilistic bisimilarity. The latter is a 1-bounded pseudometric on formulae that mirrors the Hausdorff and Kantorovich lifting the defining bisimilarity pseudometric. We show that the distance between two processes equals the distance between their own mimicking formulae.
The topological interpretation of modal logics provides descriptive languages and proof systems for reasoning about points of topological spaces. Recent work has been devoted to model checking of spatial logics on discrete spatial structures, such as finite graphs and digital images, with applications in various case studies including medical image analysis. These recent developments required a generalisation step, from topological spaces to closure spaces. In this work we initiate the study of bisimilarity and minimisation algorithms that are consistent with the closure spaces semantics. For this purpose we employ coalgebraic models. We present a coalgebraic definition of bisimilarity for quasi-discrete models, which is adequate with respect to a spatial logic with reachability operators, complemented by a free and open-source minimisation tool for finite models. We also discuss the non-quasi-discrete case, by providing a generalisation of the well-known set-theoretical notion of topo-bisimilarity, and a categorical definition, in the same spirit as the coalgebraic rendition of neighbourhood frames, but employing the covariant power set functor, instead of the contravariant one. We prove its adequacy with respect to infinitary modal logic.
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With the previous notions of bisimulation presented in literature, to check if two quantum processes are bisimilar, we have to instantiate the free quantum variables of them with arbitrary quantum states, and verify the bisimilarity of resultant configurations. This makes checking bisimilarity infeasible from an algorithmic point of view because quantum states constitute a continuum. In this paper, we introduce a symbolic operational semantics for quantum processes directly at the quantum operation level, which allows us to describe the bisimulation between quantum processes without resorting to quantum states. We show that the symbolic bisimulation defined here is equivalent to the open bisimulation for quantum processes in the previous work, when strong bisimulations are considered. An algorithm for checking symbolic ground bisimilarity is presented. We also give a modal logical characterisation for quantum bisimilarity based on an extension of Hennessy-Milner logic to quantum processes.
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