Damien Regnault

Progresses in the analysis of stochastic 2D cellular automata: A study of asynchronous 2D minority

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changes under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on 1D probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from 1D to 2D systems is a long lasting challenging problem. In this paper, we address the problem of analyzing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our simulations reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.

About non-monotony in Boolean automata networks

This paper aims at presenting motivations and first results of a prospective theoretical study on the role of non-monotone interactions in the modelling process of biological regulation networks. Focusing on discrete models of these networks, namely, Boolean automata networks, we propose to analyse the contribution of non-monotony to the diversity and complexity in their dynamical behaviours. More precisely, in this paper, we start by detailing some motivations, both mathematical and biological, for our interest in non-monotony, and we discuss how it may account for phenomena that cannot be produced by monotony only. Then, to build some understanding in this direction, we show some preliminary results on the dynamical behaviours of some specific non-monotone Boolean automata networks called xor circulant networks.

Progresses in the analysis of stochastic 2D cellular automata: a study of asynchronous 2D minority

Cellular automata are often used to model systems in physics, social sciences, biology that are inherently asynchronous. Over the past 20 years, studies have demonstrated that the behavior of cellular automata drastically changed under asynchronous updates. Still, the few mathematical analyses of asynchronism focus on one-dimensional probabilistic cellular automata, either on single examples or on specific classes. As for other classic dynamical systems in physics, extending known methods from one- to two-dimensional systems is a long lasting challenging problem. In this paper, we address the problem of analysing an apparently simple 2D asynchronous cellular automaton: 2D Minority where each cell, when fired, updates to the minority state of its neighborhood. Our experiments reveal that in spite of its simplicity, the minority rule exhibits a quite complex response to asynchronism. By focusing on the fully asynchronous regime, we are however able to describe completely the asymptotic behavior of this dynamics as long as the initial configuration satisfies some natural constraints. Besides these technical results, we have strong reasons to believe that our techniques relying on defining an energy function from the transition table of the automaton may be extended to the wider class of threshold automata.

On the Convergence of Boolean Automata Networks without Negative Cycles

Since the 1980’s, automata networks have been at the centre of numerous studies, from both theoretical (around the computational abilities) and applied (around the modelling power of real phenomena) standpoints. In this paper, basing ourselves on the seminal works of Robert and Thomas, we focus on a specific family of Boolean automata networks, those without negative cycles. For these networks, subjected to both asynchronous and elementary updating modes, we give new answers to well known problems (some of them having already been solved) about their convergence towards stable configurations. For the already solved ones, the proofs given are much simpler and neater than the existing ones. For the others, in any case, the proofs presented are constructive.

Improved local search for universal facility location

We consider the universal facility location problem in which the goal is to assign clients to facilities in order to minimize the sum of connection and facility costs. The connection cost is proportional to the distance each client has to travel to its assigned facility, whereas the cost of a facility is a non-decreasing function depending on the number of clients assigned to the facility. This model generalizes several variants of facility location problems. We present a

Stochastic minority on graphs

(5.83 + \epsilon )

Lost in Self-Stabilization

(5.83+ϵ) approximation algorithm for this problem based on local search technique.

Asynchronous Dynamics of Boolean Automata Double-Cycles

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 speeds, of specific non-monotone Boolean automata networks called XOR circulant networks.