Ultimate Traces of Cellular Automata

A cellular automaton (CA) is a parallel synchronous computing model, which consists in a juxtaposition of finite automata (cells) whose state evolves according to that of their neighbors. Its trace is the set of infinite words representing the sequence of states taken by some particular cell. In this paper we study the ultimate trace of CA and partial CA (a CA restricted to a particular subshift). The ultimate trace is the trace observed after a long time run of the CA. We give sufficient conditions for a set of infinite words to be the trace of some CA and prove the undecidability of all properties over traces that are stable by ultimate coincidence.Comment: 12 pages + 5 of appendix conference STACS'1

Sofic Trace of a Cellular Automaton

The trace subshift of a cellular automaton is the subshift of all possible columns that may appear in a space-time diagram, ie the infinite sequence of states of a particular cell of a configuration; in the language of symbolic dynamics one says that it is a factor system. In this paper we study conditions for a sofic subshift to be the trace of a cellular automaton.Comment: 10 pages + 6 for included proof

Complexité dynamique et algorithmique des automates cellulaires

Ce mémoire présente une synthèse de mes travaux sur les automates cellulaires, étude de leur dynamique, dont le but est de mieux comprendre la nature des comportements chaotiques observés et de donner des définitions satisfaisantes d'automate cellulaire complexe. Le premier chapitre donne l'ensemble des définitions nécessaires. Le chapitre suivant est dédié à la complexité de Kolmogorov: les premières sections définissent la complexité de Kolmogorov et donnent des propriétés classiques et leurs implications dans le cadre des automates cellulaires; ensuite nous présentons deux preuves pour illustrer son utilisation comme outil d'aide à la démonstration; enfin, nous montrons comment cette notion permet de donner des définitions originales qui apportent une vision nouvelle des automates cellulaires. Le troisième chapitre décrit d'autres orientations concernant l'étude du chaos: un nouveau modèle dynamique de tas de sables et l'étude des traces des automates cellulaires

The Reverse Mathematics of CAC for trees

CAC for trees is the statement asserting that any infinite subtree of $\mathbb{N}^{<\mathbb{N}}$ has an infinite path or an infinite antichain. In this paper, we study the computational strength of this theorem from a reverse mathematical viewpoint. We prove that TAC for trees is robust, that is, there exist several characterizations, some of which already appear in the literature, namely, the tree antichain theorem (TCAC) introduced by Conidis, and the statement SHER introduced by Dorais et al. We show that CAC for trees is computationally very weak, in that it admits probabilistic solutions.Comment: 28 page

Basic properties for sand automata

Presented at MFCS 2005 (Gdansk, POLAND). Long version with complete proofs published in Theoretical Computer Science, 2006, under the title "From Sandpiles to Sand Automata".International audienceWe prove several results about the relations between injectivity and surjectivity for sand automata. Moreover, we begin the exploration of the dynamical behavior of sand automata proving that the property of nilpotency is undecidable. We believe that the proof technique used for this last result might reveal useful for many other results in this context

Entropy Games and Matrix Multiplication Games

Two intimately related new classes of games are introduced and studied: entropy games (EGs) and matrix multiplication games (MMGs). An EG is played on a finite arena by two-and-a-half players: Despot, Tribune and the non-deterministic People. Despot wants to make the set of possible People's behaviors as small as possible, while Tribune wants to make it as large as possible.An MMG is played by two players that alternately write matrices from some predefined finite sets. One wants to maximize the growth rate of the product, and the other to minimize it. We show that in general MMGs are undecidable in quite a strong sense.On the positive side, EGs correspond to a subclass of MMGs, and we prove that such MMGs and EGs are determined, and that the optimal strategies are simple. The complexity of solving such games is in NP\&coNP.Comment: Accepted to STACS 201

Complexité structurelle et algorithmique des pavages et des automates cellulaires

Jury: Alain Colmerauer, Marianne Delorme, Bruno Durand2, Enrico Formenti, Jean-yves Marion et Michel MorvanThis thesis examines the complexity of tilings and cellular automata. The analysis begins by structural considerations: quasiperiodic tilings. To any set of tiles that tiles the plane, we associate a quasiperiodicity function that quantifies complexity. Firstly, it is shown that any "reasonable" function may be captured by a set of tiles and that there are tilings whose quasiperiodicity function grows faster than any computable function. Then we prove a Rice theorem for tilings: the set of all tile sets that recognize the same tilings as another tile set is recursively enumerable and undecidable. Finally, we transpose our results in the context of cellular automata. The second part of our work concerns the study of cellular automata in terms of dynamical systems, particularly chaotic controllers. The usual definitions classifying chaotic controllers are not satisfactory. To overcome this problem, we use two new topologies. The first is called Besicovitch and removes the dominance of the central pattern in the study of the evolution of the automaton. It brings several results, the first indicating that our new workspace is acceptable to the study of cellular automata as dynamical systems; the latter shows that the notion of chaos remains, through the definition of sensitivity to initial conditions, but the most chaotic classes are empty. The second topology employed is defined using algorithmic complexity. The purpose of this approach is to have a distance that reflects the ease calculating a member from the other. Our results complement the earlier results. They attest formally that cellular automata can not continuously change the information in a configuration, and especially that they are incapable to create information.Ce travail de thèse étudie la complexité des pavages et des automates cellulaires. L'analyse débute par des considérations structurelles : la quasipériodicité des pavages. À tout ensemble de tuiles qui pave le plan, on associe une fonction de quasipériodicité qui quantifie sa complexité. Tout d'abord, on montre que toute fonction ``raisonnable'' peut être capturée par un ensemble de tuiles et qu'il existe des pavages dont la fonction de quasipériodicité croît plus rapidement que n'importe quelle fonction récursive. Ensuite, on démontre un théorème de Rice pour les pavages : l'ensemble des ensembles de tuiles qui admettent les même pavages qu'un autre fixé est indécidable et récursivement énumérable. Enfin, on transpose notre résultat dans le contexte des automates cellulaires. La seconde partie de notre travail concerne l'étude des automates cellulaires sous l'angle des systèmes dynamiques, et plus particulièrement des automates chaotiques. Les définitions usuelles classifiant les automates chaotiques ne sont pas satisfaisantes. Pour palier ce problème, on utilise deux nouvelles topologies. La première est dite de Besicovitch, et permet de supprimer la prédominance du motif central lors de l'étude de l'évolution de l'automate. On apporte plusieurs résultats, les premiers indiquant que notre nouvel espace de travail est acceptable à l'étude des automates cellulaires, en tant que systèmes dynamiques ; les seconds montrent que la notion de chaos subsiste, grâce à la définition de sensibilité aux conditions initiales, mais que les classes plus chaotiques sont vides. La seconde topologie employée est définie à l'aide de la complexité algorithmique. Le but de cette approche est d'avoir une distance qui traduit la facilité à calculer un élément à l'aide de l'autre. Nos résultats complètent les précédents. Ils attestent de manière formelle que les automates cellulaires ne peuvent pas modifier continûment l'information contenue dans une configuration, et surtout qu'ils sont incapables d'en créer

Covering space in the Weyl and Besicovitch topologies

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Constructing Continuous Systems from Discrete Cellular Automata

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