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

We study many-valued coalgebraic logics with primal algebras of truth-degrees. We describe a way to lift algebraic semantics of classical coalgebraic logics, given by an endofunctor on the variety of Boolean algebras, to this many-valued setting, and we show that many important properties of the original logic are inherited by its lifting. Then, we deal with the problem of obtaining a concrete axiomatic presentation of the variety of algebras for this lifted logic, given that we know one for the original one. We solve this problem for a class of presentations which behaves well with respect to a lattice structure on the algebra of truth-degrees.

Alexander Kurz and Wolfgang Poiger. Many-Valued Coalgebraic Logic: From Boolean Algebras to Primal Varieties. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kurz_et_al:LIPIcs.CALCO.2023.17, author = {Kurz, Alexander and Poiger, Wolfgang}, title = {{Many-Valued Coalgebraic Logic: From Boolean Algebras to Primal Varieties}}, booktitle = {10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023)}, pages = {17:1--17:17}, 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.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2023.17}, URN = {urn:nbn:de:0030-drops-188147}, doi = {10.4230/LIPIcs.CALCO.2023.17}, annote = {Keywords: coalgebraic modal logic, many-valued logic, primal algebras, algebraic semantics, presenting functors} }

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**Published in:** LIPIcs, Volume 139, 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)

We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application.

Samuel Balco and Alexander Kurz. Nominal String Diagrams. In 8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 139, pp. 18:1-18:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{balco_et_al:LIPIcs.CALCO.2019.18, author = {Balco, Samuel and Kurz, Alexander}, title = {{Nominal String Diagrams}}, booktitle = {8th Conference on Algebra and Coalgebra in Computer Science (CALCO 2019)}, pages = {18:1--18:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-120-7}, ISSN = {1868-8969}, year = {2019}, volume = {139}, editor = {Roggenbach, Markus and Sokolova, Ana}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2019.18}, URN = {urn:nbn:de:0030-drops-114466}, doi = {10.4230/LIPIcs.CALCO.2019.18}, annote = {Keywords: string diagrams, nominal sets, separated product, simultaneous substitutions, internal category, monoidal category, internal monoidal categories, PROP} }

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**Published in:** LIPIcs, Volume 72, 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)

We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing a endofunctor T': Pos->Pos from an endofunctor T: Set->Set, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor L': DL->DL from a syntax-building functor L: BA->BA, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case of several modal logics. We show how the semantics of a boolean coalgebraic logic can be canonically lifted to define a semantics for its positive fragment, and that weak completeness transfers from the boolean case to the positive case.

Fredrik Dahlqvist and Alexander Kurz. The Positivication of Coalgebraic Logics. In 7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 72, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{dahlqvist_et_al:LIPIcs.CALCO.2017.9, author = {Dahlqvist, Fredrik and Kurz, Alexander}, title = {{The Positivication of Coalgebraic Logics}}, booktitle = {7th Conference on Algebra and Coalgebra in Computer Science (CALCO 2017)}, pages = {9:1--9:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-033-0}, ISSN = {1868-8969}, year = {2017}, volume = {72}, editor = {Bonchi, Filippo and K\"{o}nig, Barbara}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2017.9}, URN = {urn:nbn:de:0030-drops-80425}, doi = {10.4230/LIPIcs.CALCO.2017.9}, annote = {Keywords: Coalgebraic logic, coalgebras, enriched category theory, boolean algebra, distributive lattice, positive modal logic, monotone modal logic} }

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**Published in:** LIPIcs, Volume 35, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)

We show that for a commutative quantale V every functor from Set to V-cat has an enriched left-Kan extension. As a consequence, coalgebras over Set are subsumed by coalgebras over V-cat. Moreover, one can build functors on V-cat by equipping Set-functors with a metric.

Adriana Balan, Alexander Kurz, and Jiri Velebil. Extensions of Functors From Set to V-cat. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 17-34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{balan_et_al:LIPIcs.CALCO.2015.17, author = {Balan, Adriana and Kurz, Alexander and Velebil, Jiri}, title = {{Extensions of Functors From Set to V-cat}}, booktitle = {6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)}, pages = {17--34}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-84-2}, ISSN = {1868-8969}, year = {2015}, volume = {35}, editor = {Moss, Lawrence S. and Sobocinski, Pawel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.17}, URN = {urn:nbn:de:0030-drops-55244}, doi = {10.4230/LIPIcs.CALCO.2015.17}, annote = {Keywords: enriched category, quantale, final coalgebra} }

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**Published in:** LIPIcs, Volume 35, 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)

The question addressed in this paper is how to correctly approximate infinite data given by systems of simultaneous corecursive definitions. We devise a categorical framework for reasoning about regular datatypes, that is, datatypes closed under products, coproducts and fixpoints. We argue that the right methodology is on one hand coalgebraic (to deal with possible nontermination and infinite data) and on the other hand 2-categorical (to deal with parameters in a disciplined manner). We prove a coalgebraic version of Bekic lemma that allows us to reduce simultaneous fixpoints to a single fix point. Thus a possibly infinite object of interest is regarded as a final coalgebra of a many-sorted polynomial functor and can be seen as a limit of finite approximants. As an application, we prove correctness of a generic function that calculates the approximants on a large class of data types.

Alexander Kurz, Alberto Pardo, Daniela Petrisan, Paula Severi, and Fer-Jan de Vries. Approximation of Nested Fixpoints – A Coalgebraic View of Parametric Dataypes. In 6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 35, pp. 205-220, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{kurz_et_al:LIPIcs.CALCO.2015.205, author = {Kurz, Alexander and Pardo, Alberto and Petrisan, Daniela and Severi, Paula and de Vries, Fer-Jan}, title = {{Approximation of Nested Fixpoints – A Coalgebraic View of Parametric Dataypes}}, booktitle = {6th Conference on Algebra and Coalgebra in Computer Science (CALCO 2015)}, pages = {205--220}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-84-2}, ISSN = {1868-8969}, year = {2015}, volume = {35}, editor = {Moss, Lawrence S. and Sobocinski, Pawel}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2015.205}, URN = {urn:nbn:de:0030-drops-55351}, doi = {10.4230/LIPIcs.CALCO.2015.205}, annote = {Keywords: coalgebra, Bekic lemma, infinite data, functional programming, type theory} }

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**Published in:** Dagstuhl Reports, Volume 5, Issue 1 (2015)

This report documents the program and the outcomes of Dagstuhl Seminar 15042 "Coalgebraic Semantics of Reflexive Economics".

Samson Abramsky, Alexander Kurz, Pierre Lescanne, and Viktor Winschel. Coalgebraic Semantics of Reflexive Economics (Dagstuhl Seminar 15042). In Dagstuhl Reports, Volume 5, Issue 1, pp. 197-206, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@Article{abramsky_et_al:DagRep.5.1.197, author = {Abramsky, Samson and Kurz, Alexander and Lescanne, Pierre and Winschel, Viktor}, title = {{Coalgebraic Semantics of Reflexive Economics (Dagstuhl Seminar 15042)}}, pages = {197--206}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2015}, volume = {5}, number = {1}, editor = {Abramsky, Samson and Kurz, Alexander and Lescanne, Pierre and Winschel, Viktor}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.5.1.197}, URN = {urn:nbn:de:0030-drops-50398}, doi = {10.4230/DagRep.5.1.197}, annote = {Keywords: Programming language semantics, Coalgebra, Category theory, Economics, Epistemic game theory} }

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**Published in:** Dagstuhl Reports, Volume 3, Issue 10 (2014)

This report documents the program and the outcomes of Dagstuhl Seminar 13422 "Nominal Computation Theory". The underlying theme of the seminar was nominal sets (also known as sets with atoms or Fraenkel-Mostowski sets) and they role and applications in three distinct research areas: automata over infinite alphabets, program semantics using nominal sets and nominal calculi of concurrent processes.

Mikolaj Bojanczyk, Bartek Klin, Alexander Kurz, and Andrew M. Pitts. Nominal Computation Theory (Dagstuhl Seminar 13422). In Dagstuhl Reports, Volume 3, Issue 10, pp. 58-71, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@Article{bojanczyk_et_al:DagRep.3.10.58, author = {Bojanczyk, Mikolaj and Klin, Bartek and Kurz, Alexander and Pitts, Andrew M.}, title = {{Nominal Computation Theory (Dagstuhl Seminar 13422)}}, pages = {58--71}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2014}, volume = {3}, number = {10}, editor = {Bojanczyk, Mikolaj and Klin, Bartek and Kurz, Alexander and Pitts, Andrew M.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.3.10.58}, URN = {urn:nbn:de:0030-drops-44285}, doi = {10.4230/DagRep.3.10.58}, annote = {Keywords: nominal sets, Fraenkel-Mostowski sets} }

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**Published in:** Dagstuhl Reports, Volume 2, Issue 10 (2013)

This report documents the program and the outcomes of Dagstuhl Seminar 12411 "Coalgebraic Logics". The seminar deals with recent developments in the area of coalgebraic logic, a branch of logics which combines modal logics with coalgebraic
semantics. Modal logic finds its uses when reasoning about behavioural and temporal properties of computation and communication, coalgebras have evolved into a general theory of systems. Consequently, it is natural to combine both areas for a mathematical description of system specification. Coalgebraic logics are closely related to the broader categories semantics/formal methods and verification/logic.

Ernst-Erich Doberkat and Alexander Kurz. Coalgebraic Logics (Dagstuhl Seminar 12411). In Dagstuhl Reports, Volume 2, Issue 10, pp. 38-59, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@Article{doberkat_et_al:DagRep.2.10.38, author = {Doberkat, Ernst-Erich and Kurz, Alexander}, title = {{Coalgebraic Logics (Dagstuhl Seminar 12411)}}, pages = {38--59}, journal = {Dagstuhl Reports}, ISSN = {2192-5283}, year = {2013}, volume = {2}, number = {10}, editor = {Doberkat, Ernst-Erich and Kurz, Alexander}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagRep.2.10.38}, URN = {urn:nbn:de:0030-drops-38938}, doi = {10.4230/DagRep.2.10.38}, annote = {Keywords: Modal Logic, Coalgebra, Category Theory, Stochastic Logic, Categorical Semantics} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 9502, Coalgebraic Logics (2010)

The seminar dealt with recent developments in the emerging area of coalgebraic logic and was the first Dagstuhl seminar on that topic. Coalgebraic logic is a branch of logic which studies coalgebras as models of systems and their logics. It can be seen as generalising and extending the classical theory of modal logic to more general models of systems than labelled transition systems. Traditionally, modal logics find their use when reasoning about behavioural and temporal properties of computation and communication, whereas coalgebras give a uniform account for a large class of different systems.
The seminar discussed foundational topics in a particular branch of logic, so problems which command a direct application in an industrial context were outside the seminar's scope. We expect, however, that specification methods related to coalgebraic logics will enter fields like model checking and other areas of industrial interest, once the mathematical foundations in this
area are firmer and better understood.

Ernst-Erich Doberkat and Alexander Kurz. 09502 Abstracts Collection – Coalgebraic Logics. In Coalgebraic Logics. Dagstuhl Seminar Proceedings, Volume 9502, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010)

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@InProceedings{doberkat_et_al:DagSemProc.09502.1, author = {Doberkat, Ernst-Erich and Kurz, Alexander}, title = {{09502 Abstracts Collection – Coalgebraic Logics}}, booktitle = {Coalgebraic Logics}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2010}, volume = {9502}, editor = {Ernst-Erich Doberkat and Alexander Kurz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09502.1}, URN = {urn:nbn:de:0030-drops-24203}, doi = {10.4230/DagSemProc.09502.1}, annote = {Keywords: Modal logics, coalgebras, bisimulation and behavioral equivalence, relations, Markov transition systems} }

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