Logical Characterization of Bisimulation Metrics

Abstract

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

A van Benthem Theorem for Fuzzy Modal Logic

“…Probabilistic systems can be compared under standard two-valued (crisp) notions of bisimilarity [25,4] under which two states are either bisimilar or not, but it has been observed previously [17] that in many respects, quantitative measures of process equivalence are more suitable in this setting: Probabilistic systems may, e.g., differ slightly in the values of individual probabilities or contain mutually deviating but very unlikely transitions, and in such cases one would like to have the possibility of saying that two processes are almost the same, or in fact quantifying their degree of distinctness. This has motivated the introduction of behavioural metrics measuring the behavioural distance between states in probabilistic systems [17,10,42,12,2,6]. More precisely, these distance functions are pseudometrics, i.e.…”

Quantitative Aspects of Programming Languages and Systems over the past 2^4 years and beyond

“…In this setting, a typical approach borrowed from pure mathematics relies on the use of metrics, or pseudo-metrics (see, e.g., [58] for a survey related to the Kantorovich metric), which provide a measure of the distance between non-equivalent processes. In particular, starting from the first developments [72], several definitions converging to bisimilarity have been proposed for various models encompassing in different ways probabilities and nondeterminism [56,149,108], sometimes enriched with characterizing logics [136,43], while others rely on alternative semantics, as in the case of linear/branching distances [67,4,66,44], as well as game-based simulation preorders [45].…”

Logical Characterization of Trace Metrics

“…the strong trace metric and its weak version, as well as the equivalences constituting their kernels. The logic L (and consequently L w ) can be seen either as a simplified version of the modal logic L from [13], which has been successfully employed in [9] to characterize the bisimilarity metric [7,12,14], or more simply as a probabilistic version of the logic characterizing the trace semantics in the fully nondeterministic case [5]. More precisely, L consists of two classes of formulae.…”

“…To this aim we follow the approach of [9] in which a logical characterization of the bisimilarity metric is provided. We introduce two boolean logics L and L w , providing a probabilistic choice operator capturing the probability weights that a process assigns to arbitrary traces, which we prove to characterize resp.…”