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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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References

  1. D. Armstrong, C. Stump, and H. Thomas. A uniform bijection between nonnesting and noncrossing partitions. Trans. Amer. Math. Soc., 365(8):4121-4151, 2013. Google Scholar
  2. A. Björner and F. Brenti. Combinatorics of Coxeter groups. Springer Verlag, New York, 2005. Google Scholar
  3. P. Cameron and D. Fon-Der-Flaass. Orbits of antichains revisited. European J. Combin., 16(6):545-554, 1995. Google Scholar
  4. L. David, C. Defant, M. Joseph, M. Macauley, and A. McDonough. Toggling independent sets in cycle graphs, 2021. In preparation. Google Scholar
  5. C. Defant. Flexible toggles and symmetric invertible asynchronous elementary cellular automata. Discrete Math., 341(9):2367-2379, 2018. Google Scholar
  6. K. Dilks, O. Pechenik, and J. Striker. Resonance in orbits of plane partitions and increasing tableaux. J. Comb. Theory Ser. A, 148:244-274, 2017. Google Scholar
  7. D. Einstein, M. Farber, E. Gunawan, M. Joseph, M. Macauley, J. Propp, and S. Rubinstein-Salzedo. Noncrossing partitions, toggles, and homomesies. Electron. J. Combin., 23(3), 2016. Google Scholar
  8. H. Eriksson and K. Eriksson. Conjugacy of Coxeter elements. Electron. J. Combin., 16(2):#R4, 2009. Google Scholar
  9. M. Joseph. Antichain toggling and rowmotion. Electron. J. Combin., 26(1), 2019. Google Scholar
  10. M. Joseph and T. Roby. Toggling independent sets of a path graph. Electron J. Combin., 25(1):1-18, 2018. Google Scholar
  11. G. Kreweras. Sur les partitions non croisées d'un cycle. Discrete Math., 1(4):333-350, 1972. Google Scholar
  12. M. Macauley, J. McCammond, and H. Mortveit. Dynamics groups of asynchronous cellular automata. J. Algebraic Combin., 33(1):11-35, 2011. Google Scholar
  13. M. Macauley, J. McCammond, and H.S. Mortveit. Order independence in asynchronous cellular automata. J. Cell. Autom., 3(1):37-56, 2008. Google Scholar
  14. M. Macauley and H.S. Mortveit. Cycle equivalence of graph dynamical systems. Nonlinearity, 22:421-436, 2009. Google Scholar
  15. H.S. Mortveit and C.M. Reidys. Discrete, sequential dynamical systems. Discrete Math., 226(1-3):281-295, 2001. Google Scholar
  16. H.S. Mortveit and C.M. Reidys. An Introduction to Sequential Dynamical Systems. Universitext. Springer Verlag, 2007. Google Scholar
  17. Y. Numata and Y. Yamanouchi. On the action of the toggle group of the Dynkin diagram of type A. https://arxiv.org/abs/2103.16217, 2021.
  18. O. Pretzel. On reorienting graphs by pushing down maximal vertices. Order, 3(2):135-153, 1986. Google Scholar
  19. J. Propp and T. Roby. Homomesy in products of two chains. Electron. J. Combin., 22(3), 2015. Google Scholar
  20. T. Roby. Dynamical algebraic combinatorics and the homomesy phenomenon. In Recent Trends in Combinatorics, pages 619-652. Springer, 2016. Google Scholar
  21. V. Salo. Universal gates with wires in a row. https://arxiv.org/abs/1809.08050, 2018.
  22. B. Schönfisch and A. de Roos. Synchronous and asynchronous updating in cellular automata. BioSystems, 51(3):123-143, 1999. Google Scholar
  23. M. Schützenberger. Promotion des morphismes d'ensembles ordonnés. Discrete Math., 2(1):73-94, 1972. Google Scholar
  24. J. Striker. Dynamical algebraic combinatorics: promotion, rowmotion, and resonance. Notices Amer. Math. Soc., 64(6), 2017. Google Scholar
  25. J. Striker. Rowmotion and generalized toggle groups. Discrete Math. Theor. Comput. Sci., 20, 2018. Google Scholar
  26. J. Striker and N. Williams. Promotion and rowmotion. European J. Combin., 33:1919-1942, 2012. Google Scholar
  27. M. Vielhaber. Computing by temporal order: asynchronous cellular automata. In AUTOMATA, volume 90, pages 166-176. Electron. Proc. Theor. Comput. Sci., 2012. Google Scholar
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