Towards Trace Metrics via Functor Lifting
Résumé
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, by identifying conditions under which
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.