Universal Gauge-Invariant Cellular Automata

Authors Pablo Arrighi , Marin Costes , Nathanaël Eon

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Pablo Arrighi
  • Université Paris-Saclay, CNRS, LMF, 91190 Gif-sur-Yvette, France
  • IXXI, Lyon, France
Marin Costes
  • ENS Paris-Saclay, CNRS, LMF, 91190 Gif-sur-Yvette, France
Nathanaël Eon
  • Aix-Marseille Université, Université de Toulon, CNRS, LIS, Marseille, France


The authors would like to thank Guillaume Theyssier for asking us the question whether any CA admits a gauge extension.

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Pablo Arrighi, Marin Costes, and Nathanaël Eon. Universal Gauge-Invariant Cellular Automata. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Gauge symmetries play a fundamental role in Physics, as they provide a mathematical justification for the fundamental forces. Usually, one starts from a non-interactive theory which governs "matter", and features a global symmetry. One then extends the theory so as make the global symmetry into a local one (a.k.a gauge-invariance). We formalise a discrete counterpart of this process, known as gauge extension, within the Computer Science framework of Cellular Automata (CA). We prove that the CA which admit a relative gauge extension are exactly the globally symmetric ones (a.k.a the colour-blind). We prove that any CA admits a non-relative gauge extension. Both constructions yield universal gauge-invariant CA, but the latter allows for a first example where the gauge extension mediates interactions within the initial CA.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Cellular automata
  • Gauge-invariance
  • Universality


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