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DOI: 10.4230/OASIcs.AUTOMATA.2021.5
Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets

Authors: Laurent David, Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough

Published in: OASIcs, Volume 90, 27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)

Abstract
Though iterated maps and dynamical systems are not new to combinatorics, they have enjoyed a renewed prominence over the past decade through the elevation of the subfield that has become known as dynamical algebraic combinatorics. Some of the problems that have gained popularity can also be cast and analyzed as finite asynchronous cellular automata (CA). However, these two fields are fairly separate, and while there are some individuals who work in both, that is the exception rather than the norm. In this article, we will describe our ongoing work on toggling independent sets on graphs. This will be preceded by an overview of how this project arose from new combinatorial problems involving homomesy, toggling, and resonance. Though the techniques that we explore are directly applicable to ECA rule 1, many of them can be generalized to other cellular automata. Moreover, some of the ideas that we borrow from cellular automata can be adapted to problems in dynamical algebraic combinatorics. It is our hope that this article will inspire new problems in both fields and connections between them.
Cite as

Laurent David, Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough. Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets. 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. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{david_et_al:OASIcs.AUTOMATA.2021.5,
  author =	{David, Laurent and Defant, Colin and Joseph, Michael and Macauley, Matthew and McDonough, Alex},
  title =	{{Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets}},
  booktitle =	{27th IFIP WG 1.5 International Workshop on Cellular Automata and Discrete Complex Systems (AUTOMATA 2021)},
  pages =	{5:1--5:16},
  series =	{Open Access Series in Informatics (OASIcs)},
  ISBN =	{978-3-95977-189-4},
  ISSN =	{2190-6807},
  year =	{2021},
  volume =	{90},
  editor =	{Castillo-Ramirez, Alonso and Guillon, Pierre and Perrot, K\'{e}vin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.AUTOMATA.2021.5},
  URN =		{urn:nbn:de:0030-drops-140145},
  doi =		{10.4230/OASIcs.AUTOMATA.2021.5},
  annote =	{Keywords: Asynchronous cellular automata, Covering space, Coxeter element, Dynamical algebraic combinatorics, Group action, Homomesy, Independent set, Resonance, Toggling, Toric equivalence}
}
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