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Emergence on Decreasing Sandpile Models

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Mathematical Foundations of Computer Science 2015 (MFCS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9234))

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Abstract

Sand is a proper instance for the study of natural algorithmic phenomena. Idealized square/cubic sand grains moving according to “simple” local toppling rules may exhibit surprisingly “complex” global behaviors. In this paper we explore the language made by words corresponding to fixed points reached by iterating a toppling rule starting from a finite stack of sand grains in one dimension. Using arguments from linear algebra, we give a constructive proof that for all decreasing sandpile rules the language of fixed points is accepted by a finite (Muller) automaton. The analysis is completed with a combinatorial study of cases where the emergence of precise regular patterns is formally proven. It extends earlier works presented in [15–17], and asks how far can we understand and explain emergence following this track?

This work was partially supported by IXXI (Complex System Institute, Lyon), ANR projects Dynamite and QuasiCool (ANR-12-JS02-011-01), Modmad Federation of U. St-Etienne, FONDECYT Grant 3140527 (DIM, Universidad de Chile), and Núcleo Milenio Información y Coordinación en Redes (ACGO).

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Notes

  1. 1.

    stands for \(i \le c\,f(N)\) for a suitable constant c.

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Authors and Affiliations

  1. Aix Marseille Université - CNRS - LIF UMR 7279, 13288, Marseille, France

    Kévin Perrot

  2. Université de Lyon - CNRS - GATE LSE UMR 5824, 42023, Saint-Etienne, France

    Éric Rémila

Authors
  1. Kévin Perrot
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  2. Éric Rémila
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Correspondence to Kévin Perrot .

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Editors and Affiliations

  1. Università di Roma "Tor Vergata", Rome, Italy

    Giuseppe F Italiano

  2. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  3. University of Edinburgh, Edinburgh, United Kingdom

    Donald T. Sannella

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Perrot, K., Rémila, É. (2015). Emergence on Decreasing Sandpile Models. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_33

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  • DOI: https://doi.org/10.1007/978-3-662-48057-1_33

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  • Print ISBN: 978-3-662-48056-4

  • Online ISBN: 978-3-662-48057-1

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