Abstract
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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This work has been supported by the French National Research Agency project EMC (ANR-09-BLAN-0164) and by the Italian MIUR PRIN 2010-2011 grant “Automata and Formal Languages: Mathematical and Applicative Aspects” H41J12000190001.
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References
Brylawski, T.: The lattice of integer partitions. Discrete Mathematics 6, 201–219 (1973)
Cattaneo, G., Chiaselotti, G., Gentile, T., Oliverio, P.A.: The lattice structure of equally extended signed partitions (2013), preprint, submitted for publication
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 publication
Cattaneo, G., Comito, M., Bianucci, D.: Sand piles: from physics to cellular automata models. Theor. Comput. Sci. 436, 35–53 (2012)
Formenti, E., Masson, B.: Fixed points of generalized ice pile models. Poster Proceedings of ECCS 2005, Paris (November 2005)
Formenti, E., Masson, B.: A note on fixed points of generalized ice piles models, vol. 2, pp. 183–191 (2006)
Formenti, E., Goles, E., Martin, B.: Computational complexity of avalanches in the kadanoff sandpile model. Fundam. Inform. 115, 107–124 (2012)
Goles, E., Kiwi, M.A.: Games on line graphs and sand piles. Theoret. Comput. Sci. 115, 321–349 (1993)
Goles, E., Morvan, M., Phan, H.D.: Sandpiles and order structure of integer partitions. Discrete Applied Mathematics 117, 51–64 (2002)
Goles, E., Morvan, M., Phan, H.D.: The structure of linear chip firing game and related models. Theoret. Comput. Sci. 270, 827–841 (2002)
Green, C., Kleitman, D.J.: Longest chains in the lattice of integere partitions ordered by majorization. Europ. J. Combinatoric 7, 1–10 (1986)
Latapy, M., Mantaci, R., Morvan, M., Phan, H.D.: Structure of some sand piles models. Theoret. Comput. Sci. 262, 525–556 (2001)
Perrot, K., Rémila, E.: Kadanoff sand pile model. avalanche structure and wave shape. Theor. Comput. Sci. 504, 52–72 (2013)
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)
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Cattaneo, G., Chiaselotti, G., Dennunzio, A., Formenti, E., Manzoni, L. (2014). Non Uniform Cellular Automata Description of Signed Partition Versions of Ice and Sand Pile Models. In: Wąs, J., Sirakoulis, G.C., Bandini, S. (eds) Cellular Automata. ACRI 2014. Lecture Notes in Computer Science, vol 8751. Springer, Cham. https://doi.org/10.1007/978-3-319-11520-7_13
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DOI: https://doi.org/10.1007/978-3-319-11520-7_13
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