Coalgebraic Behavioral Metrics

Abstract

We study different behavioral metrics, such as those arising from bothbranching and linear-time semantics, in a coalgebraic setting. Given acoalgebra $\alpha\colon X \to HX$ for a functor $H \colon \mathrm{Set}\to\mathrm{Set}$, we define a framework for deriving pseudometrics on $X$ whichmeasure the behavioral distance of states. A crucial step is the lifting of the functor $H$ on $\mathrm{Set}$ to afunctor $\overline{H}$ on the category $\mathrm{PMet}$ of pseudometric spaces.We present two different approaches which can be viewed as generalizations ofthe Kantorovich and Wasserstein pseudometrics for probability measures. We showthat the pseudometrics provided by the two approaches coincide on severalnatural examples, but in general they differ. If $H$ has a final coalgebra, every lifting $\overline{H}$ yields in acanonical way a behavioral distance which is usually branching-time, i.e., itgeneralizes bisimilarity. In order to model linear-time metrics (generalizingtrace equivalences), we show sufficient conditions for lifting distributivelaws and monads. These results enable us to employ the generalized powersetconstruction