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Expressive Quantale-Valued Logics for Coalgebras: An Adjunction-Based Approach

Authors Harsh Beohar , Sebastian Gurke , Barbara König , Karla Messing , Jonas Forster , Lutz Schröder , Paul Wild

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ACM Subject Classification
  • Theory of computation → Modal and temporal logics
Keywords
  • modal logics
  • coalgebras
  • behavioural equivalences
  • behavioural metrics
  • linear-time semantics
  • Eilenberg-Moore categories

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Abstract

We address the task of deriving fixpoint equations from modal logics characterizing behavioural equivalences and metrics (summarized under the term conformances). We rely on an earlier work that obtains Hennessy-Milner theorems as corollaries to a fixpoint preservation property along Galois connections between suitable lattices. We instantiate this to the setting of coalgebras, in which we spell out the compatibility property ensuring that we can derive a behaviour function whose greatest fixpoint coincides with the logical conformance. We then concentrate on the linear-time case, for which we study coalgebras based on the machine functor living in Eilenberg-Moore categories, a scenario for which we obtain a particularly simple logic and fixpoint equation. The theory is instantiated to concrete examples, both in the branching-time case (bisimilarity and behavioural metrics) and in the linear-time case (trace equivalences and trace distances).

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Harsh Beohar, Sebastian Gurke, Barbara König, Karla Messing, Jonas Forster, Lutz Schröder, and Paul Wild. Expressive Quantale-Valued Logics for Coalgebras: An Adjunction-Based Approach. In 41st International Symposium on Theoretical Aspects of Computer Science (STACS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 289, pp. 10:1-10:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.STACS.2024.10

Author Details

Harsh Beohar
  • University of Sheffield, UK
Sebastian Gurke
  • Universität Duisburg-Essen, Germany
Barbara König
  • Universität Duisburg-Essen, Germany
Karla Messing
  • Universität Duisburg-Essen, Germany
Jonas Forster
  • Friedrich-Alexander-Universität Erlangen-, Nürnberg, Germany
Lutz Schröder
  • Friedrich-Alexander-Universität Erlangen-, Nürnberg, Germany
Paul Wild
  • Friedrich-Alexander-Universität Erlangen-, Nürnberg, Germany

Funding

The first author was supported by the EPSRC NIA Grant EP/X019373/1, while the remaining authors were supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 434050016 (SpeQt).

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