Graded Monads and Graded Logics for the Linear Time - Branching Time Spectrum

Authors Ulrich Dorsch, Stefan Milius, Lutz Schröder

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Ulrich Dorsch
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Stefan Milius
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Lutz Schröder
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

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Ulrich Dorsch, Stefan Milius, and Lutz Schröder. Graded Monads and Graded Logics for the Linear Time - Branching Time Spectrum. In 30th International Conference on Concurrency Theory (CONCUR 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 140, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time - branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in the present paper that this framework of graded semantics does subsume the most important equivalences from the linear time - branching time spectrum. An important feature of graded semantics is that it allows for the principled extraction of characteristic modal logics. We have established invariance of these graded logics under the given graded semantics in earlier work; in the present paper, we extend the logical framework with an explicit propositional layer and provide a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract graded logics for a range of graded semantics on labelled transition systems and probabilistic systems, and give exemplary proofs of their expressiveness based on our generic criterion.

Subject Classification

ACM Subject Classification
  • Theory of computation → Concurrency
  • Theory of computation → Modal and temporal logics
  • Linear Time
  • Branching Time
  • Monads
  • System Equivalences
  • Modal Logics
  • Expressiveness


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