In systems involving quantitative data, such as probabilistic, fuzzy, or
metric systems, behavioural distances provide a more fine-grained comparison of
states than two-valued notions of behavioural equivalence or behaviour
inclusion. Like in the two-valued case, 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 (bi-)simulations that need not
themselves be hemi- or pseudometrics; this is analogous to classical
simulations and bisimulations, which need not be preorders or equivalence
relations, respectively. The known 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 nonexpansive 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. All our results explicitly
hold also for asymmetric distances (hemimetrics), i.e. notions of quantitative
simulation
