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Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets

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

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Author Details

Laurent David
  • University of Texas, Dallas, TX, USA
Colin Defant
  • Princeton University, NJ, USA
Michael Joseph
  • Dalton State College, GA, USA
Matthew Macauley
  • Clemson University, SC, USA
Alex McDonough
  • Brown University, Providence, RI, USA

Funding

  • Defant, Colin: National Science Foundation Graduate Research Fellowship (grant no. DGE-1656466); Fannie and John Hertz Foundation Fellowship.
  • Macauley, Matthew: Simons Foundation Collaboration Grant #358242.

Acknowledgements

The authors would like to thank the Banff International Research Station, and the organizers of the Dynamical Algebraic Combinatorics Workshop, held virtually in October 2020.

Cite AsGet BibTex

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)
https://doi.org/10.4230/OASIcs.AUTOMATA.2021.5

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
  • Theory of computation → Formal languages and automata theory
  • Hardware → Cellular neural networks
Keywords
  • Asynchronous cellular automata
  • Covering space
  • Coxeter element
  • Dynamical algebraic combinatorics
  • Group action
  • Homomesy
  • Independent set
  • Resonance
  • Toggling
  • Toric equivalence

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