## Universality and decidability of number-conserving cellular automata (2003)

Venue: | THEORETICAL COMPUTER SCIENCE 292 |

Citations: | 16 - 1 self |

### BibTeX

@TECHREPORT{Moreira03universalityand,

author = {Andrés Moreira},

title = {Universality and decidability of number-conserving cellular automata},

institution = {THEORETICAL COMPUTER SCIENCE 292},

year = {2003}

}

### OpenURL

### Abstract

Number-conserving cellular automata (NCCA) are particularly interesting, both because of their natural appearance as models of real systems, and because of the strong restrictions that number-conservation implies. Here we extend the definition of the property to include cellular automata with any set of states n Z, and show that they can be always extended to “usual ” NCCA with contiguous states. We show a way to simulate any one dimensional CA through a one dimensional NCCA, proving the existence of intrinsically universal NCCA. Finally, we give an algorithm to decide, given a CA, if its states can be labeled with integers to produce a NCCA, and to find this relabeling if the answer is positive.

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Citation Context ... of some quantity Email address: anmoreir@dim.uchile.cl (Andrés Moreira). Preprint submitted to Theoretical Computer Science 8 February 2008in the system that is being modeled. In CA traffic models (=-=[13]-=-), for instance, states are interpreted as the number of indestructible particles located in a cell. In fact, an interpretation in terms of particles can be given to any NCCA [10]. In [3], Boccara and... |

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Citation Context ...us proved (provided that the numeric value of the states is given). Durand et al. also give some examples of NCCA in several classes of one-dimensional CA, intersecting the classifications of K˚urka (=-=[9]-=-) and Braga et al. ([5,6]), and prove the emptiness of the remaining classes. In [11] Morita and Imai prove the universality of the class of number-conserving reversible partitioned CA; in [12] Morita... |

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Citation Context ...jk) = ⋃ V ∩ Ejk j̸=k j̸=k ⇐⇒ ∃j, k : V = V ∩ Ejk ⇐⇒ ∃j, k : V ⊆ Ejk j̸=k where we use the fact that each V ∩Ejk is a subspace of V , and a linear space 10cannot be a finite union of proper subspaces =-=[15]-=-. The last condition can be easily checked by the algorithm, by adding the equation φ(xj) = φ(xk) to the equation system and seeing if the space of solutions is the same. If V \ ( ⋃ ) j̸=k Ejk ̸= ∅, t... |

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Citation Context ...n [3], Boccara and Fuk´s give a necessary and sufficient condition for a onedimensional CA of two states to be number-conserving, and study all the NCCA with neighborhoods {l, . . .,r}, l + r ≤ 4. In =-=[4]-=-, they give a necessary and sufficient condition that holds for one-dimensional CA of any number of states, and use it to study all the three-state NCCA for l+r ≤ 2. In [7] Durand et al. formalize thr... |

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Citation Context ...fic models ([13]), for instance, states are interpreted as the number of indestructible particles located in a cell. In fact, an interpretation in terms of particles can be given to any NCCA [10]. In =-=[3]-=-, Boccara and Fuk´s give a necessary and sufficient condition for a onedimensional CA of two states to be number-conserving, and study all the NCCA with neighborhoods {l, . . .,r}, l + r ≤ 4. In [4], ... |

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Citation Context ...so give some examples of NCCA in several classes of one-dimensional CA, intersecting the classifications of K˚urka ([9]) and Braga et al. ([5,6]), and prove the emptiness of the remaining classes. In =-=[11]-=- Morita and Imai prove the universality of the class of number-conserving reversible partitioned CA; in [12] Morita et al. embed a simple general computer in a reversible, number-conserving two-dimens... |

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Citation Context ...t the numeric value of the states is given). Durand et al. also give some examples of NCCA in several classes of one-dimensional CA, intersecting the classifications of K˚urka ([9]) and Braga et al. (=-=[5,6]-=-), and prove the emptiness of the remaining classes. In [11] Morita and Imai prove the universality of the class of number-conserving reversible partitioned CA; in [12] Morita et al. embed a simple ge... |

5 | A Simple Computer Embedded in a Reversible and NumberConserving Two-Dimensional Cellular Space, Multiple-Valued Logic, vol.6
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Citation Context ...K˚urka ([9]) and Braga et al. ([5,6]), and prove the emptiness of the remaining classes. In [11] Morita and Imai prove the universality of the class of number-conserving reversible partitioned CA; in =-=[12]-=- Morita et al. embed a simple general computer in a reversible, number-conserving two-dimensional partitioned CA. We must remark that, despite the title of [11], these articles do not settle the unive... |

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Citation Context ...{l, . . .,r}, l + r ≤ 4. In [4], they give a necessary and sufficient condition that holds for one-dimensional CA of any number of states, and use it to study all the three-state NCCA for l+r ≤ 2. In =-=[7]-=- Durand et al. formalize three different definitions of number-conservation and show their equivalence. They write the generalization of Boccara’s condition to two dimensions, and hint on the d-dimens... |

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Citation Context ...t the numeric value of the states is given). Durand et al. also give some examples of NCCA in several classes of one-dimensional CA, intersecting the classifications of K˚urka ([9]) and Braga et al. (=-=[5,6]-=-), and prove the emptiness of the remaining classes. In [11] Morita and Imai prove the universality of the class of number-conserving reversible partitioned CA; in [12] Morita et al. embed a simple ge... |

1 |
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(Show Context)
Citation Context ...n CA traffic models ([13]), for instance, states are interpreted as the number of indestructible particles located in a cell. In fact, an interpretation in terms of particles can be given to any NCCA =-=[10]-=-. In [3], Boccara and Fuk´s give a necessary and sufficient condition for a onedimensional CA of two states to be number-conserving, and study all the NCCA with neighborhoods {l, . . .,r}, l + r ≤ 4. ... |