Towards Trace Metrics via Functor Lifting

TL;DR

This paper develops a coalgebraic framework for deriving trace metrics by lifting functors from Set to pseudometric spaces, enabling the computation of behavioral distances in automata models.

Contribution

It introduces a systematic method for lifting functors and related structures to pseudometric spaces, facilitating trace metric derivation in coalgebraic systems.

Findings

Framework applies to nondeterministic automata

Framework applies to probabilistic automata

Enables behavioral distance computation in coalgebraic models

Abstract

We investigate the possibility of deriving metric trace semantics in a coalgebraic framework. First, we generalize a technique for systematically lifting functors from the category Set of sets to the category PMet of pseudometric spaces, showing under which conditions also natural transformations, monads and distributive laws can be lifted. By exploiting some recent work on an abstract determinization, these results enable the derivation of trace metrics starting from coalgebras in Set. More precisely, for a coalgebra on Set we determinize it, thus obtaining a coalgebra in the Eilenberg-Moore category of a monad. When the monad can be lifted to PMet, we can equip the final coalgebra with a behavioral distance. The trace distance between two states of the original coalgebra is the distance between their images in the determinized coalgebra through the unit of the monad. We show how our framework applies to nondeterministic automata and probabilistic automata.