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The pseudometric based on the Kantorovich lifting is one of the most popular notion of distance between probabilistic processes proposed in the literature. However, its application in verification is limited to linear properties. We propose a generalization which allows to deal with a wider class of properties, such as those used in security and privacy. More precisely, we propose a family of pseudometrics, parametrized on a notion of distance which depends on the property we want to verify. Furthermore, we show that the members of this family still characterize bisimilarity in terms of their kernel, and provide a bound on the corresponding distance between trace distributions. Finally, we study the instance corresponding to differential privacy, and we show that it has a dual form, easier to compute. We also prove that the typical processalgebra constructs are non-expansive, thus paving the way to a modular approach to verification.

Unlike in land plants, photosynthesis in many aquatic plants relies on bicarbonate in addition to carbon dioxide (CO2) to compensate for the low diffusivity and potential depletion of CO2 in water. Concentrations of bicarbonate and CO2 vary greatly with catchment geology. In this study, we investigate whether there is a link between these concentrations and the frequency of freshwater plants possessing the bicarbonate use trait. We show, globally, that the frequency of plant species with this trait increases with bicarbonate concentration. Regionally, however, the frequency of bicarbonate use is reduced at sites where the CO2 concentration is substantially above the air equilibrium, consistent with this trait being an adaptation to carbon limitation. Future anthropogenic changes of bicarbonate and CO2 concentrations may alter the species compositions of freshwater plant communities.

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.Comment: In Proceedings QAPL'16, arXiv:1610.0769

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