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

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

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ACM Subject Classification
  • Mathematics of computing → Trees
  • Theory of computation → Categorical semantics
Keywords
  • Trees
  • Coalgebra
  • Factorization Systems

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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) https://doi.org/10.4230/LIPIcs.CALCO.2025.15

Author Details

Thorsten Wißmann
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Bálint Kocsis
  • Radboud University, Nijmegen, The Netherlands
Jurriaan Rot
  • Radboud University, Nijmegen, The Netherlands
Ruben Turkenburg
  • Radboud University, Nijmegen, The Netherlands

Funding

This research is supported by the NWO grants VI.Vidi.223.096 and OCENW.M20.053.

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