Construction of mu-Limit Sets of Two-dimensional Cellular Automata

We prove a characterisation of mu-limit sets of two-dimensional cellular automata, extending existing results in the one-dimensional case. This sets describe the typical asymptotic behaviour of the cellular automaton, getting rid of exceptional cases, when starting from the uniform measure

Directional Dynamics along Arbitrary Curves in Cellular Automata

This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers

μ\mu-Limit Sets of Cellular Automata from a Computational Complexity Perspective

This paper concerns $\mu$-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial $\mu$-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, $\mu$-limit sets can have a $\Sigma\_3^0$-hard language, second, they can contain only $\alpha$-complex configurations, third, any non-trivial property concerning them is at least $\Pi\_3^0$-hard. We prove complexity upper bounds, study restrictions of these questions to particular classes of CA, and different types of (non-)convergence of the measure of a word during the evolution.Comment: 41 page

Construction of µ-limit Sets

International audienceThe µ-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to infinity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as µ-limit set where µ is the uniform Bernoulli measure

Matrix Metalloproteinase Gene Polymorphisms and Bronchopulmonary Dysplasia: Identification of MMP16 as a New Player in Lung Development

International audienceBACKGOUND: Alveolarization requires coordinated extracellular matrix remodeling, a process in which matrix metalloproteinases (MMPs) play an important role. We postulated that polymorphisms in MMP genes might affect MMP function in preterm lungs and thus influence the risk of bronchopulmonary dysplasia (BPD). METHODS AND FINDINGS: Two hundred and eighty-four consecutive neonates with a gestational age of <28 weeks were included in this prospective study. Forty-five neonates developed BPD. Nine single-nucleotide polymorphisms (SNPs) were sought in the MMP2, MMP14 and MMP16 genes. After adjustment for birth weight and ethnic origin, the TT genotype of MMP16 C/T (rs2664352) and the GG genotype of MMP16 A/G (rs2664349) were found to protect from BPD. These genotypes were also associated with a smaller active fraction of MMP2 and with a 3-fold-lower MMP16 protein level in tracheal aspirates collected within 3 days after birth. Further evaluation of MMP16 expression during the course of normal human and rat lung development showed relatively low expression during the canalicular and saccular stages and a clear increase in both mRNA and protein levels during the alveolar stage. In two newborn rat models of arrested alveolarization the lung MMP16 mRNA level was less than 50% of normal. CONCLUSIONS: MMP16 may be involved in the development of lung alveoli. MMP16 polymorphisms appear to influence not only the pulmonary expression and function of MMP16 but also the risk of BPD in premature infants

Automates cellulaires : dynamique directionnelle et asymptotique typique

Les automates cellulaires sont à la fois un modèle de calcul parallèle, un système complexe et un système dynamique. Ils fonctionnent de manière synchrone et en temps discret, leur particularité est que les fonctions qu'ils définissent sont issues de l'application simultanée, en tout point de l'espace, d'une règle d'évolution locale. L'ensemble limite est un objet classique des systèmes dynamiques, c'est l'ensemble des états que le système peut atteindre arbitrairement tard. Il a été très étudié dans le cadre des automates cellulaires, et les résultats sont nombreux. Parmi ces résultats, un théorème de Rice démontré par Jarkko Kari dit que toute propriété des ensembles limites est indécidable. Dans ce mémoire, on ne s'intéresse plus à l'ensemble limite traditionnel, mais à une variante pour laquelle on utilise une mesure sur l'espace des entrées, sélectionnant ainsi les comportements susceptibles d'apparaître arbitrairement tard et souvent. Ce nouvel ensemble, que l'on nomme ensemble mu-limite, a été introduit en 2000 par Petr Kurka et Alejandro Maass. La plupart des résultats sur les ensembles limites ne se transposent pas naturellement. On étudie la famille des ensembles mu-limites d'automates cellulaires. On montre que sous certaines contraintes sur la dynamique, l'ensemble mu-limite peut être entièrement décrit. On classe ainsi les automates en fonction de ces contraintes. Dans le cas général, on montre l'existence d'automates cellulaires ayant comme ensembles mu-limites un grand nombre d'ensembles complexes. On finit par montrer un théorème de Rice pour les ensembles mu-limites d'automates cellulaires: tout propriété non triviale de ces ensembles est indécidable.Cellular automata are simultaneously a model of parallel computation, a complex system and a dynamical system. They are synchronous and time is discrete. The functions defined by their application is the result of the synchronous application of the same local rule everywhere. The limit set is a classical tool of dynamical systems theory, it is the set of states the system can reach arbitrarily late. It has been studied often in the particular case of cellular automata and there are numerous results. Amongst them, a Rice's theorem proved by Jarkko Kari states that any non-trivial property of limit sets of cellular automata is undecidable. In this thesis, we do not consider the classical limit set, as we add a measure on the space of states of the system. Thus, we get a set which contains behaviors that appear arbitrarily far and often. This set is named mu-limit set and was introduced in 2000 by Petr Kurka and Alejandro Maass. Most of the results on limit sets cannot be directly adapted for mu-limit sets. We study the family of all mu-limit sets of cellular automata. We show that under some constraints on the dynamics, the mu-limit set can be entirely described. We then produce a classification of cellular automata according to these constraints. In the general case, we prove the existence of cellular automata whose mu-limit sets are among a large set of complex sets. We finally prove Rice's theorem for mu-limit sets: any non-trivial property is undecidable

Rice's Theorem for µ-Limit Sets of Cellular Automata

38th International Colloquium, ICALP 2011, Zurich, Switzerland, July 4-8, 2011, Proceedings, Part IIInternational audienceCellular automata are a parallel and synchronous computing model, made of innitely many nite automata updating according to the same local rule. Rice's theorem states that any nontrivial property over computable functions is undecidable. It has been adapted by Kari to limit sets of cellular automata [Kar94], that is the set of congurations that can be reached arbitrarily late. This paper proves a new Rice theorem for µ-limit sets, which are sets of congurations often reached arbitrarily late