Behavioral Metrics via Functor Lifting

TL;DR

This paper develops a coalgebraic framework for deriving behavioral pseudometrics on state spaces, generalizing Kantorovich and Wasserstein metrics, and establishing conditions for behavioral equivalence.

Contribution

It introduces two functor lifting approaches for pseudometrics in a coalgebraic setting, unifies them, and applies fixed-point techniques to define behavioral distances.

Findings

The two metric approaches coincide in natural examples.

Behavioral distance can be characterized as a fixed point.

Zero distance implies behavioral equivalence under certain conditions.

Abstract

We study behavioral metrics in an abstract coalgebraic setting. Given a coalgebra alpha: X -> FX in Set, where the functor F specifies the branching type, we define a framework for deriving pseudometrics on X which measure the behavioral distance of states. A first crucial step is the lifting of the functor F on Set to a functor in the category 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. Then a final coalgebra for F in Set can be endowed with a behavioral distance resulting as the smallest solution of a fixed-point equation, yielding the final coalgebra in PMet. The same technique, applied to an arbitrary coalgebra alpha: X -> FX in Set, provides the behavioral distance on X. Under some constraints we can prove that two states are at distance 0 if and only if they are behaviorally equivalent.