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(Co)algebraic pearl

DOI: 10.4230/LIPIcs.CALCO.2025.15

Trees in Coalgebra from Generalized Reachability ((Co)algebraic pearl)

Authors: Thorsten Wißmann, Bálint Kocsis, Jurriaan Rot, and Ruben Turkenburg

Published in: LIPIcs, Volume 342, 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)

Abstract

An automaton is called reachable if every state is reachable from the initial state. This notion has been generalized coalgebraically in two ways: first, via a universal property on pointed coalgebras, namely, that a reachable coalgebra has no proper subcoalgebra; and second, a coalgebra is reachable if it arises as the union of an iterative computation of successor states, starting from the initial state. In the current paper, we present corresponding universal properties and iterative constructions for trees. The universal property captures when a coalgebra is a tree, namely, when it has no proper tree unravelling. The iterative construction unravels an arbitrary coalgebra to a tree. We show that this yields the expected notion of tree for a variety of standard examples. We obtain our characterization of trees by first generalizing the previous theory of reachable coalgebras. Surprisingly, both the universal property and the iterative construction for trees arise as an instance of this generalized notion of reachability.

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Thorsten Wißmann, Bálint Kocsis, Jurriaan Rot, and Ruben Turkenburg. Trees in Coalgebra from Generalized Reachability ((Co)algebraic pearl). In 11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 342, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{wimann_et_al:LIPIcs.CALCO.2025.15,
  author =	{Wi{\ss}mann, Thorsten and Kocsis, B\'{a}lint and Rot, Jurriaan and Turkenburg, Ruben},
  title =	{{Trees in Coalgebra from Generalized Reachability}},
  booktitle =	{11th Conference on Algebra and Coalgebra in Computer Science (CALCO 2025)},
  pages =	{15:1--15:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-383-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{342},
  editor =	{C\^{i}rstea, Corina and Knapp, Alexander},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2025.15},
  URN =		{urn:nbn:de:0030-drops-235740},
  doi =		{10.4230/LIPIcs.CALCO.2025.15},
  annote =	{Keywords: Trees, Coalgebra, Factorization Systems}
}

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DOI: 10.4230/LIPIcs.CALCO.2023.6

Forward and Backward Steps in a Fibration

Authors: Ruben Turkenburg, Harsh Beohar, Clemens Kupke, and Jurriaan Rot

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

Abstract

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.

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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.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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Documents authored by Turkenburg, Ruben

Trees in Coalgebra from Generalized Reachability ((Co)algebraic pearl)

Abstract

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Forward and Backward Steps in a Fibration

Abstract

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