## Paolo Baldan ; Filippo Bonchi ; Henning Kerstan ; Barbara König - Coalgebraic Behavioral Metrics

lmcs:4715 - Logical Methods in Computer Science, September 14, 2018, Volume 14, Issue 3 - https://doi.org/10.23638/LMCS-14(3:20)2018
Coalgebraic Behavioral Metrics

Authors: Paolo Baldan ; Filippo Bonchi ; Henning Kerstan ; Barbara König

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

Volume: Volume 14, Issue 3
Published on: September 14, 2018
Accepted on: September 14, 2018
Submitted on: July 26, 2018
Keywords: Computer Science - Logic in Computer Science