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Cellular Automata and Kan Extensions

Authors Alexandre Fernandez, Luidnel Maignan, Antoine Spicher

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Author Details

Alexandre Fernandez
  • Université Paris Est Creteil, LACL, F-94010 Creteil, France
Luidnel Maignan
  • Université Paris Est Creteil, LACL, F-94010 Creteil, France
Antoine Spicher
  • Université Paris Est Creteil, LACL, F-94010 Creteil, France

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Alexandre Fernandez, Luidnel Maignan, and Antoine Spicher. Cellular Automata and Kan Extensions. In 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021). Open Access Series in Informatics (OASIcs), Volume 90, pp. 7:1-7:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


In this paper, we formalize precisely the sense in which the application of a cellular automaton to partial configurations is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis-Hedlund theorem of cellular automata from the extended topological point of view provided by category theory. These links also allow to relatively easily generalize concepts pioneered by cellular automata to arbitrary kinds of possibly evolving spaces. No prior knowledge of category theory is assumed.

Subject Classification

ACM Subject Classification
  • Theory of computation → Rewrite systems
  • Theory of computation → Models of computation
  • Cellular Automata
  • Kan Extension
  • Category Theory
  • Global Transformation


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