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All binary number-conserving cellular automata based on adjacent cells are intrinsically one-dimensional

(2019) Physical Review E. 100(022126).
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Abstract
A binary number-conserving cellular automaton is a discrete dynamical system that models the movement of particles in a d-dimensional grid. Each cell of the grid is either empty or contains a particle. In subsequent time steps the particles move between the cells, but in one cell there can be at most one particle at a time. In this paper, the von Neumann neighborhood is considered, which means that in each time step a particle can move to an adjacent cell only. It is proven that regardless of the dimension d, all of these cellular automata are trivial, as they are intrinsically one-dimensional. Thus, for given d, there are only 4d+1 binary number-conserving cellular automata with the von Neumann neighborhood: the identity rule and the shift and traffic rules in each of the 2d possible directions.

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Citation

Please use this url to cite or link to this publication:

Chicago
Wolnik, Barbara, and Bernard De Baets. 2019. “All Binary Number-conserving Cellular Automata Based on Adjacent Cells Are Intrinsically One-dimensional.” Physical Review E 100 (022126).
APA
Wolnik, B., & De Baets, B. (2019). All binary number-conserving cellular automata based on adjacent cells are intrinsically one-dimensional. Physical Review E, 100(022126).
Vancouver
1.
Wolnik B, De Baets B. All binary number-conserving cellular automata based on adjacent cells are intrinsically one-dimensional. Physical Review E. 2019;100(022126).
MLA
Wolnik, Barbara, and Bernard De Baets. “All Binary Number-conserving Cellular Automata Based on Adjacent Cells Are Intrinsically One-dimensional.” Physical Review E 100.022126 (2019): n. pag. Print.
@article{8626853,
  abstract     = {A binary number-conserving cellular automaton is a discrete dynamical system that models the movement of particles in a d-dimensional grid. Each cell of the grid is either empty or contains a particle. In subsequent time steps the particles move between the cells, but in one cell there can be at most one particle at a time. In this paper, the von Neumann neighborhood is considered, which means that in each time step a particle can move to an adjacent cell only. It is proven that regardless of the dimension d, all of these cellular automata are trivial, as they are intrinsically one-dimensional. Thus, for given d, there are only 4d+1 binary number-conserving cellular automata with the von Neumann neighborhood: the identity rule and the shift and traffic rules in each of the 2d possible directions.},
  author       = {Wolnik, Barbara and De Baets, Bernard},
  issn         = {2470-0045},
  journal      = {Physical Review E},
  language     = {eng},
  number       = {022126},
  title        = {All binary number-conserving cellular automata based on adjacent cells are intrinsically one-dimensional},
  url          = {http://dx.doi.org/10.1103/physreve.100.022126},
  volume       = {100},
  year         = {2019},
}

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