# Behavioral Metrics via Functor Lifting

## Reference

Paolo Baldan, Filippo Bonchi, Henning Kerstan, and Barbara König.

Behavioral Metrics via Functor Lifting. In Venkatesh Raman and S. P. Suresh, editors,34th International Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2014), volume 29 ofLeibniz International Proceedings in Informatics (LIPIcs), pages 403–415. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, 2014.

## 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.

## Keywords:

behavioral metric, functor lifting, pseudometric, coalgebra

## Suggested BibTeX entry:

@

inproceedings{BBKK14,

author= {Paolo Baldan and Filippo Bonchi and Henning Kerstan and Barbara K{\"o}nig},

booktitle= {34th International Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2014)},

editor= {Venkatesh Raman and S. P. Suresh},

pages= {403--415},

publisher= {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},

series= {Leibniz International Proceedings in Informatics (LIPIcs)},

title= {{Behavioral Metrics via Functor Lifting}},

volume= {29},

year= {2014}

}

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