Non Uniform Cellular Automata Description of Signed Partition Versions of Ice and Sand Pile Models

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.

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.