# Publication: Response Curves and Preimage Sequences of Two-Dimensional Cellular Automata

All || By Area || By YearTitle | Response Curves and Preimage Sequences of Two-Dimensional Cellular Automata |
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Authors/Editors* | H. FukÅ and A. Skelton |

Where published* | Proceedings of the 2011 International Conference on Scientific Computing: CSC-2011 |

How published* | Proceedings |

Year* | 2011 |

Volume | |

Number | |

Pages | 165--171 |

Publisher | CSREA Press |

Keywords | |

Link | http://arxiv.org/abs/1108.1559 |

Abstract |
We consider the problem of finding response curves for a class of binary two-dimensional cellular automata with $L$-shaped neighbourhood. We show that the dependence of the density o f ones after an arbitrary number of iterations, on the initial density of ones, can be calculated for a fairly large number of rules by considering preimage sets. We provide several examples and a summary of all known results. We consider a special case of initial density equal to $0.5$ for other rules and compute explicitly the density of ones after $n$ iterations of the rule. This analysis includes surjective rules, which in the case of $L$-shaped neighbourhood are all found to be permutive. We conclude with the observation that all rules for which preimage curves can be computed explicitly are either finite or asymptotic emulators of identity or shift. |

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