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In probabilistic transition systems, behavioural metrics provide a more fine-grained and stable measure of system equivalence than crisp notions of bisimilarity. They correlate strongly to quantitative probabilistic logics, and in fact the distance induced by a probabilistic modal logic taking values in the real unit interval has been shown to coincide with behavioural distance. For probabilistic systems, probabilistic modal logic thus plays an analogous role to that of Hennessy-Milner logic on classical labelled transition systems. In the quantitative setting, invariance of modal logic under bisimilarity becomes non-expansivity of formula evaluation w.r.t. behavioural distance. In the present paper, we provide a characterization of the expressive power of probabilistic modal logic based on this observation: We prove a probabilistic analogue of the classical van Benthem theorem, which states that modal logic is precisely the bisimulation-invariant fragment of first-order logic. Specifically, we show that quantitative probabilistic modal logic lies dense in the bisimulation-invariant fragment, in the indicated sense of non-expansive formula evaluation, of quantitative probabilistic first-order logic; more precisely, bisimulation-invariant first-order formulas are approximable by modal formulas of bounded rank.For a description logic perspective on the same result, see [46].

Behavioural distances provide a fine-grained measure of equivalence in systems involving quantitative data, such as probabilistic, fuzzy, or metric systems. Like in the classical setting of crisp bisimulationtype equivalences, the wide variation found in system types creates a need for generic methods that apply to many system types at once. Approaches of this kind are emerging within the paradigm of universal coalgebra, based either on lifting pseudometrics along set functors or on lifting general real-valued (fuzzy) relations along functors by means of fuzzy lax extensions. An immediate benefit of the latter is that they allow bounding behavioural distance by means of fuzzy bisimulations that need not themselves be (pseudo-)metrics, in analogy to classical bisimulations (which need not be equivalence relations). We show that the known instances of generic pseudometric liftings, specifically the generic Kantorovich and Wasserstein liftings, both can be extended to yield fuzzy lax extensions, using the fact that both are effectively given by a choice of quantitative modalities. Our central result then shows that in fact all fuzzy lax extensions are Kantorovich extensions for a suitable set of quantitative modalities, the so-called Moss modalities. For non-expansive fuzzy lax extensions, this allows for the extraction of quantitative modal logics that characterize behavioural distance, i.e. satisfy a quantitative version of the Hennessy-Milner theorem; equivalently, we obtain expressiveness of a quantitative version of Moss' coalgebraic logic.

[1] Energetic electrons (tens to hundreds of keV) deposit significant energy into the D layer of the ionosphere. Riometers provide a means of monitoring this electron precipitation by measuring the associated cosmic noise absorption (CNA), but individually they are incapable of resolving the associated energy. However, the combination of two imaging riometers with overlapping beams allows an estimate of the height of peak CNA and so the associated energy to be made. We examine two methods for estimating the height of CNA using data from two imaging riometers in northern Fennoscandia; a 3-D reconstruction of CNA using Occam's inversion and a technique based upon the triangulation of discrete absorption structures are developed. We compare these two methods with the results from a previously published technique. It is found that for the case studies and test phantoms the height triangulation and 3-D reconstruction offer improvement over previous methods. These techniques are tested by comparison with data from the EISCAT incoherent scatter radar. Observations show good correlation between the estimates of peak height of CNA from EISCAT and from the triangulation and 3-D reconstruction methods for this case. Three case studies are examined in detail, a slowly varying absorption, afternoon spike, and evening absorption spike event. Estimates of the characteristic energy are made. The substorm event had a characteristic energy of ∼5 keV, whereas the characteristic energy for the morning event was 17-20 keV. Analyses indicate the afternoon spike event having characteristic energy greater than 100 keV.

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