Dynamical Algebraic Combinatorics, Asynchronous Cellular Automata, and Toggling Independent Sets
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.
Asynchronous cellular automata
Covering space
Coxeter element
Dynamical algebraic combinatorics
Group action
Homomesy
Independent set
Resonance
Toggling
Toric equivalence
Mathematics of computing~Combinatoric problems
Theory of computation~Formal languages and automata theory
Hardware~Cellular neural networks
5:1-5:16
Regular Paper
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.
Laurent
David
Laurent David
University of Texas, Dallas, TX, USA
Colin
Defant
Colin Defant
Princeton University, NJ, USA
https://cdefant.wixsite.com/theoremspace
National Science Foundation Graduate Research Fellowship (grant no. DGE-1656466); Fannie and John Hertz Foundation Fellowship.
Michael
Joseph
Michael Joseph
Dalton State College, GA, USA
Matthew
Macauley
Matthew Macauley
Clemson University, SC, USA
http://www.math.clemson.edu/~macaule/
https://orcid.org/0000-0002-4409-2248
Simons Foundation Collaboration Grant #358242.
Alex
McDonough
Alex McDonough
Brown University, Providence, RI, USA
https://www.math.brown.edu/amcdono1/
https://orcid.org/0000-0002-3816-7805
10.4230/OASIcs.AUTOMATA.2021.5
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Laurent David, Colin Defant, Michael Joseph, Matthew Macauley, and Alex McDonough
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