Non Uniform Cellular Automata Description of Signed Partition Versions of Ice and Sand Pile Models
This paper reviews the well-known formalisations for ice and sand piles, based on a finite sequence of non-negative integers and its recent extension to signed partitions, i.e. sequences of a non-negative and a non-positive part of integers, both non increasing.
The ice pile model can be interpreted as a discrete time dynamical system under the action of a vertical and a horizontal evolution rule, whereas the sand pile model is characterized by the unique action of the vertical rule.
The signed partition extension, besides these two dynamical evolution rules, also takes into account an annihilation rule at the boundary region between the non-negative and the non-positive regions. We provide an original physical interpretation of this model as a p-n junction of two semiconductors.
Moreover, we show how the sand pile extension of the signed partition environment can be formalized by mean of a non-uniform cellular automaton (CA) since the vertical and the annihilation evolution rules have the formal description of two CA local rules. Finally, we provide a similar construction for the ice pile extension.
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- 2.Cattaneo, G., Chiaselotti, G., Gentile, T., Oliverio, P.A.: The lattice structure of equally extended signed partitions (2013), preprint, submitted for publicationGoogle Scholar
- 3.Cattaneo, G., Chiaselotti, G., Stumbo, F., Oliverio, P.A.: Signed integer partitions as extension of the Goles–Kiwi ice and sand pile models (2014), preprint, submitted for publicationGoogle Scholar
- 5.Formenti, E., Masson, B.: Fixed points of generalized ice pile models. Poster Proceedings of ECCS 2005, Paris (November 2005)Google Scholar
- 6.Formenti, E., Masson, B.: A note on fixed points of generalized ice piles models, vol. 2, pp. 183–191 (2006)Google Scholar
- 14.Perrot, K., Rémila, É.: Emergence of wave patterns on kadanoff sandpiles. In: Pardo, A., Viola, A. (eds.) LATIN 2014. LNCS, vol. 8392, pp. 634–647. Springer, Heidelberg (2014)Google Scholar