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**Published in:** LIPIcs, Volume 270, 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)

Distributive laws of various kinds occur widely in the theory of coalgebra, for instance to model automata constructions and trace semantics, and to interpret coalgebraic modal logic. We study steps, which are a general type of distributive law, that allow one to map coalgebras along an adjunction. In this paper, we address the question of what such mappings do to well known notions of equivalence, e.g., bisimilarity, behavioural equivalence, and logical equivalence.
We do this using the characterisation of such notions of equivalence as (co)inductive predicates in a fibration. Our main contribution is the identification of conditions on the interaction between the steps and liftings, which guarantees preservation of fixed points by the mapping of coalgebras along the adjunction. We apply these conditions in the context of lax liftings proposed by Bonchi, Silva, Sokolova (2021), and generalise their result on preservation of bisimilarity in the construction of a belief state transformer. Further, we relate our results to properties of coalgebraic modal logics including expressivity and completeness.

Ruben Turkenburg, Harsh Beohar, Clemens Kupke, and Jurriaan Rot. Forward and Backward Steps in a Fibration. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{turkenburg_et_al:LIPIcs.CALCO.2023.6, author = {Turkenburg, Ruben and Beohar, Harsh and Kupke, Clemens and Rot, Jurriaan}, title = {{Forward and Backward Steps in a Fibration}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {6:1--6:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-287-7}, ISSN = {1868-8969}, year = {2023}, volume = {270}, editor = {Baldan, Paolo and de Paiva, Valeria}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.6}, URN = {urn:nbn:de:0030-drops-188032}, doi = {10.4230/LIPIcs.CALCO.2023.6}, annote = {Keywords: Coalgebra, Fibration, Bisimilarity} }

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**Published in:** LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Quantitative parity automata (QPAs) generalise non-deterministic parity automata (NPAs) by adding weights from a certain semiring to transitions. QPAs run on infinite word/tree-like structures, modelled as coalgebras of a polynomial functor F. They can also arise as certain products between a quantitative model (with branching modelled via the same semiring of quantities, and linear behaviour described by the functor F) and an NPA (modelling a qualitative property of F-coalgebras). We build on recent work on semiring-valued measures to define a way to measure the set of paths through a quantitative branching model which satisfy a qualitative property (captured by an unambiguous NPA running on F-coalgebras). Our main result shows that the notion of extent of a QPA (which generalises non-emptiness of an NPA, and is defined as the solution of a nested system of equations) provides an equivalent characterisation of the measure of the accepting paths through the QPA. This result makes recently-developed methods for computing nested fixpoints available for model checking qualitative, linear-time properties against quantitative branching models.

Corina Cîrstea and Clemens Kupke. Measure-Theoretic Semantics for Quantitative Parity Automata. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{cirstea_et_al:LIPIcs.CSL.2023.14, author = {C\^{i}rstea, Corina and Kupke, Clemens}, title = {{Measure-Theoretic Semantics for Quantitative Parity Automata}}, booktitle = {31st EACSL Annual Conference on Computer Science Logic (CSL 2023)}, pages = {14:1--14:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-264-8}, ISSN = {1868-8969}, year = {2023}, volume = {252}, editor = {Klin, Bartek and Pimentel, Elaine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.14}, URN = {urn:nbn:de:0030-drops-174755}, doi = {10.4230/LIPIcs.CSL.2023.14}, annote = {Keywords: parity automaton, coalgebra, measure theory} }

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**Published in:** LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Many algorithmic results on the modal mu-calculus use representations of formulas such as alternating tree automata or hierarchical equation systems. At closer inspection, these results are not always optimal, since the exact relation between the formula and its representation is not clearly understood. In particular, there has been confusion about the definition of the fundamental notion of the size of a mu-calculus formula.
We propose the notion of a parity formula as a natural way of representing a mu-calculus formula, and as a yardstick for measuring its complexity. We discuss the close connection of this concept with alternating tree automata, hierarchical equation systems and parity games. We show that well-known size measures for mu-calculus formulas correspond to a parity formula representation of the formula using its syntax tree, subformula graph or closure graph, respectively. Building on work by Bruse, Friedmann & Lange we argue that for optimal complexity results one needs to work with the closure graph, and thus define the size of a formula in terms of its Fischer-Ladner closure. As a new observation, we show that the common assumption of a formula being clean, that is, with every variable bound in at most one subformula, incurs an exponential blow-up of the size of the closure.
To realise the optimal upper complexity bound of model checking for all formulas, our main result is to provide a construction of a parity formula that (a) is based on the closure graph of a given formula, (b) preserves the alternation-depth but (c) does not assume the input formula to be clean.

Clemens Kupke, Johannes Marti, and Yde Venema. Succinct Graph Representations of μ-Calculus Formulas. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kupke_et_al:LIPIcs.CSL.2022.29, author = {Kupke, Clemens and Marti, Johannes and Venema, Yde}, title = {{Succinct Graph Representations of \mu-Calculus Formulas}}, booktitle = {30th EACSL Annual Conference on Computer Science Logic (CSL 2022)}, pages = {29:1--29:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-218-1}, ISSN = {1868-8969}, year = {2022}, volume = {216}, editor = {Manea, Florin and Simpson, Alex}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.29}, URN = {urn:nbn:de:0030-drops-157491}, doi = {10.4230/LIPIcs.CSL.2022.29}, annote = {Keywords: modal mu-calculus, model checking, alternating tree automata, hierachical equation systems} }

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**Published in:** LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

The classical Hennessy-Milner theorem says that two states of an image-finite transition system are bisimilar if and only if they satisfy the same formulas in a certain modal logic. In this paper we study this type of result in a general context, moving from transition systems to coalgebras and from bisimilarity to coinductive predicates. We formulate when a logic fully characterises a coinductive predicate on coalgebras, by providing suitable notions of adequacy and expressivity, and give sufficient conditions on the semantics. The approach is illustrated with logics characterising similarity, divergence and a behavioural metric on automata.

Clemens Kupke and Jurriaan Rot. Expressive Logics for Coinductive Predicates. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kupke_et_al:LIPIcs.CSL.2020.26, author = {Kupke, Clemens and Rot, Jurriaan}, title = {{Expressive Logics for Coinductive Predicates}}, booktitle = {28th EACSL Annual Conference on Computer Science Logic (CSL 2020)}, pages = {26:1--26:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-132-0}, ISSN = {1868-8969}, year = {2020}, volume = {152}, editor = {Fern\'{a}ndez, Maribel and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.26}, URN = {urn:nbn:de:0030-drops-116698}, doi = {10.4230/LIPIcs.CSL.2020.26}, annote = {Keywords: Coalgebra, Fibration, Modal Logic} }

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