Up-To Techniques for Behavioural Metrics via Fibrations

Authors Filippo Bonchi, Barbara König, Daniela Petrisan

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Filippo Bonchi
  • Universitá di Pisa, Italy
Barbara König
  • Universität Duisburg-Essen, Germany
Daniela Petrisan
  • CNRS, IRIF, Université Paris Diderot, France

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Filippo Bonchi, Barbara König, and Daniela Petrisan. Up-To Techniques for Behavioural Metrics via Fibrations. In 29th International Conference on Concurrency Theory (CONCUR 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 118, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Up-to techniques are a well-known method for enhancing coinductive proofs of behavioural equivalences. We introduce up-to techniques for behavioural metrics between systems modelled as coalgebras and we provide abstract results to prove their soundness in a compositional way. In order to obtain a general framework, we need a systematic way to lift functors: we show that the Wasserstein lifting of a functor, introduced in a previous work, corresponds to a change of base in a fibrational sense. This observation enables us to reuse existing results about soundness of up-to techniques in a fibrational setting. We focus on the fibrations of predicates and relations valued in a quantale, for which pseudo-metric spaces are an example. To illustrate our approach we provide an example on distances between regular languages.

Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Logic and verification
  • behavioural metrics
  • bisimilarity
  • up-to techniques
  • coalgebras
  • fibrations


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