The Lattice structure of Chip Firing Games and Related Models

In this paper, we study a famous discrete dynamical system, the Chip Firing Game, used as a model in physics, economics and computer science. We use order theory and show that the set of reachable states (i.e. the configuration space) of such a system started in any configuration is a lattice, which implies strong structural properties. The lattice structure of the configuration space of a dynamical system is of great interest since it implies convergence (and more) if the configuration space is finite. If it is infinite, this property implies another kind of convergence: all the configurations reachable from two given configurations are reachable from their infimum. In other words, there is a unique first configuration which is reachable from two given configurations. Moreover, the Chip Firing Game is a very general model, and we show how known models can be encoded as Chip Firing Games, and how some results about them can be deduced from this paper. Finally, we define a new model, which is a generalization of the Chip Firing Game, and about which many interesting questions arise.Comment: See http://www.liafa.jussieu.fr/~latap

Block-sequential update schedules and Boolean automata circuits

International audienceOur work is set in the framework of complex dynamical systems and, more precisely, that of Boolean automata networks modeling regulation networks. We study how the choice of an update schedule impacts on the dynamics of such a network. To do this, we explain how studying the dynamics of any network updated with an arbitrary block-sequential update schedule can be reduced to the study of the dynamics of a different network updated in parallel. We give special attention to networks whose underlying structure is a circuit, that is, Boolean automata circuits. These particular and simple networks are known to serve as the "engines'' of the dynamics of arbitrary regulation networks containing them as sub-networks in that they are responsible for their variety of dynamical behaviours. We give both the number of attractors of period $p$, $\forall p\in \mathbb{N}$ and the total number of attractors in the dynamics of Boolean automata circuits updated with any block-sequential update schedule. We also detail the variety of dynamical behaviours that such networks may exhibit according to the update schedule

On the effects of firing memory in the dynamics of conjunctive networks

Boolean networks are one of the most studied discrete models in the context of the study of gene expression. In order to define the dynamics associated to a Boolean network, there are several \emph{update schemes} that range from parallel or \emph{synchronous} to \emph{asynchronous.} However, studying each possible dynamics defined by different update schemes might not be efficient. In this context, considering some type of temporal delay in the dynamics of Boolean networks emerges as an alternative approach. In this paper, we focus in studying the effect of a particular type of delay called \emph{firing memory} in the dynamics of Boolean networks. Particularly, we focus in symmetric (non-directed) conjunctive networks and we show that there exist examples that exhibit attractors of non-polynomial period. In addition, we study the prediction problem consisting in determinate if some vertex will eventually change its state, given an initial condition. We prove that this problem is {\bf PSPACE}-complete

Boolean networks synchronism sensitivity and XOR circulant networks convergence time

In this paper are presented first results of a theoretical study on the role of non-monotone interactions in Boolean automata networks. We propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours according to two axes. The first one consists in supporting the idea that non-monotony has a peculiar influence on the sensitivity to synchronism of such networks. It leads us to the second axis that presents preliminary results and builds an understanding of the dynamical behaviours, in particular concerning convergence times, of specific non-monotone Boolean automata networks called XOR circulant networks.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249

Computational Complexity of Avalanches in the Kadanoff two-dimensional Sandpile Model

15 pagesIn this paper we prove that the avalanche problem for Kadanoff sandpile model (KSPM) is P-complete for two-dimensions. Our proof is based on a reduction from the monotone circuit value problem by building logic gates and wires which work with configurations in KSPM. The proof is also related to the known prediction problem for sandpile which is in NC for one-dimensional sandpiles and is P-complete for dimension 3 or greater. The computational complexity of the prediction problem remains open for two-dimensional sandpiles

A New Class of Automata Networks

A new class of automata networks is defined. Their evolution rules are determined by a probability measure p on the set of all integers Z and an indicator function I_A on the interval [0,1]. It is shown that any cellular automaton rule can be represented by a (nonunique) rule formulated in terms of a pair (p,I_A). This new class of automata networks contains discrete systems which are not cellular automata. Some of their properties are discussed.Comment: Uses elsart.cls document class. Five figures (two in EPS format

Hidden structure in the randomness of the prime number sequence?

We report a rigorous theory to show the origin of the unexpected periodic behavior seen in the consecutive differences between prime numbers. We also check numerically our findings to ensure that they hold for finite sequences of primes, that would eventually appear in applications. Finally, our theory allows us to link with three different but important topics: the Hardy-Littlewood conjecture, the statistical mechanics of spin systems, and the celebrated Sierpinski fractal.Comment: 13 pages, 5 figures. New section establishing connection with the Hardy-Littlewood theory. Published in the journal where the solved problem was first describe