Mathieu Sablik

Fundamental study: Directional dynamics for cellular automata: A sensitivity to initial condition approach

A cellular automaton is a continuous function F defined on a full-shift A^Z which commutes with the shift @s. Often, to study the dynamics of F one only considers implicitly @s. However, it is possible to emphasize the spatio-temporal structure produced by considering the dynamics of the ZxN-action induced by (@s,F). In this purpose we study the notion of directional dynamics. In particular, we are interested in directions of equicontinuity and expansivity, which generalize the concepts introduced by Gilman [Robert H. Gilman, Classes of linear automata, Ergodic Theory Dynam. Systems 7 (1) (1987) 105-118] and P. Kurka [Petr Kurka, Languages, equicontinuity and attractors in cellular automata, Ergodic Theory Dynam. Systems 17 (2) (1997) 417-433]. We study the sets of directions which exhibit this special kind of dynamics showing that they induce a discrete geometry in space-time diagrams.

Topological Dynamics of 2D Cellular Automata

Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on 2D CA and aims at showing that the situation is different and more complex. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants and the existence of CA having only non-recursive equicontinuous points. They all show a difference between the 1D and the 2D case. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case.

Measure rigidity for algebraic bipermutative cellular automata

Let be a bipermutative algebraic cellular automaton. We present conditions that force a probability measure, which is invariant for the -action of F and the shift map σ , to be the Haar measure on Σ, a closed shift-invariant subgroup of the abelian compact group . This generalizes simultaneously results of Host et al (B. Host, A. Maass and S. Martinez. Uniform Bernoulli measure in dynamics of permutative cellular automata with algebraic local rules. Discrete Contin. Dyn. Syst. 9 (6) (2003), 1423–1446) and Pivato (M. Pivato. Invariant measures for bipermutative cellular automata. Discrete Contin. Dyn. Syst. 12 (4) (2005), 723–736). This result is applied to give conditions which also force an ( F , σ )-invariant probability measure to be the uniform Bernoulli measure when F is a particular invertible affine expansive cellular automaton on .

Directional dynamics along arbitrary curves in cellular automata

This paper studies directional dynamics on one-dimensional cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behavior of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviors 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.

A notion of effectiveness for subshifts on finitely generated groups

Abstract We generalize the classical definition of effectively closed subshift to finitely generated groups. We study classical stability properties of this class and then extend this notion by allowing the usage of an oracle to the word problem of a group. This new class of subshifts forms a conjugacy class that contains all sofic subshifts. Motivated by the question of whether there exists a group where the class of sofic subshifts coincides with that of effective subshifts, we show that the inclusion is strict for several groups, including recursively presented groups with undecidable word problem, amenable groups and groups with more than two ends. We also provide an extended model of Turing machine which uses the group itself as a tape and characterizes our extended notion of effectiveness. As applications of these machines we prove that the origin constrained domino problem is undecidable for any group of the form G × Z subject to a technical condition on G and we present a simulation theorem which is valid in any finitely generated group.

AN ORDER ON SETS OF TILINGS CORRESPONDING TO AN ORDER ON LANGUAGES

Traditionally a tiling is defined with a finite number of finite forbidden patterns. We can generalize this notion considering any set of patterns. Generalized tilings defined in this way can be studied with a dynamical point of view, leading to the notion of subshift. In this article we establish a correspondence between an order on subshifts based on dynamical transformations on them and an order on languages of forbidden patterns based on computability properties.

Local Rules for Computable Planar Tilings

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question is to characterize, among a class of non-periodic tilings, the aperiodic ones. In this paper, we answer this question for the well-studied class of non-periodic tilings obtained by digitizing irrational vector spaces. Namely, we prove that such tilings are aperiodic if and only if the digitized vector spaces are computable.

Positive circuits and d-dimensional spatial differentiation : Application to the formation of sense organs in Drosophila

We discuss a rule proposed by the biologist Thomas according to which the possibility for a genetic network (represented by a signed directed graph called a regulatory graph) to have several stable states implies the existence of a positive circuit. This result is already known for different models, differential or discrete formalism, but always with a network of genes contained in a single cell. Thus, we can ask about the validity of this rule for a system containing several cells and with intercellular genetic interactions. In this paper, we consider the genetic interactions between several cells located on a d-dimensional lattice, i.e., each point of lattice represents a cell to which we associate the expression level of n genes contained in this cell. With this configuration, we show that the existence of a positive circuit is a necessary condition for a specific form of multistationarity, which naturally corresponds to spatial differentiation. We then illustrate this theorem through the example of the formation of sense organs in Drosophila.

Self-organization in cellular automata: a particle-based approach

For some classes of cellular automata, we observe empirically a phenomenon of self-organization: starting from a random configuration, regular strips separated by defects appear in the space-time diagram. When there is no creation of defects, all defects have the same direction after some time. In this article, we propose to formalise this phenomenon. Starting from the notion of propagation of defect by a cellular automaton formalized in [Piv07b, Piv07a], we show that, when iterating the automaton on a random configuration, defects in one direction only remain asymptotically.

µ-Limit sets of cellular automata from a computational complexity perspective

We characterize the set of µ-limit sets of cellular automata.We prove that the language of these limit sets can be Σ 3 -complete.We prove a Rice theorem for µ-limit sets of cellular automata. This paper concerns µ-limit sets of cellular automata: sets of configurations made of words whose probability to appear does not vanish with time, starting from an initial µ-random configuration. More precisely, we investigate the computational complexity of these sets and of related decision problems. Main results: first, µ-limit sets can have a Σ 3 0 -hard language, second, they can contain only α-complex configurations, third, any non-trivial property concerning them is at least ? 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.