Realizing Finitely Presented Groups as Projective Fundamental Groups of SFTs

Subshifts are sets of colourings - or tilings - of the plane, defined by local constraints. Historically introduced as discretizations of continuous dynamical systems, they are also heavily related to computability theory. In this article, we study a conjugacy invariant for subshifts, known as the projective fundamental group. It is defined via paths inside and between configurations. We show that any finitely presented group can be realized as a projective fundamental group of some SFT

Pi01 sets and tilings

In this paper, we prove that given any \Pi^0_1 subset $P$ of \{0,1\}^\NN there is a tileset $\tau$ with a set of configurations $C$ such that P\times\ZZ^2 is recursively homeomorphic to $C\setminus U$ where $U$ is a computable set of configurations. As a consequence, if $P$ is countable, this tileset has the exact same set of Turing degrees

Slopes of Tilings

We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they coincide with recursively enumerable sets of rationals.Comment: Journ\'ees Automates Cellulaires 2010, Turku : Finland (2010

Periodicity in tilings

Tilings and tiling systems are an abstract concept that arise both as a computational model and as a dynamical system. In this paper, we characterize the sets of periods that a tiling system can produce. We prove that up to a slight recoding, they correspond exactly to languages in the complexity classes \nspace{n} and \cne

Undecidable word problem in subshift automorphism groups

International audienceThis article studies the complexity of the word problem in groups of automorphisms of subshifts. We show in particular that for any Turing degree, there exists a subshift whose automorphism group contains a subgroup whose word problem has exactly this degree

Pavages : périodicité et complexité calculatoire

Cette thèse est dédiée à l'étude des pavages : des ensembles de coloriages du plan discret respectant des contraintes locales données par un jeu de tuiles. Nous nous penchons en particulier sur les liens qui unissent les pavages et la calculabilité. Les pavages étant des ensembles effectivement clos particuliers, nous étudions dans un premier temps la structure des ensembles de degrés Turing des pavages, la comparant à celle des ensembles effectivement clos en général : pour tout ensemble effectivement clos il existe un pavage qui a les même degrés Turing à 0 près, le degré des ensembles récursifs. De plus les pavages ne contenant pas de membre récursif ont une structure particulière : ils contiennent toujours un cône de degrés Turing, un degré Turing et tous les degrés qui lui sont supérieurs. Dans un second temps, nous étudions les ensembles de périodes des pavages, pour diverses notions de périodicité, parvenant à des caractérisations à l'aide de classes de complexité ou de calculabilité pour chaque notion étudiée. Enfin nous nous intéressons à la difficulté calculatoire des problèmes de la factorisation et de la conjugaison, des notions de simulation et d'équivalence adaptées aux spécificités des pavages.This thesis is dedicated to the study of subshifts of finite type (SFTs) : sets of colorings of the discrete plane which respect some local constraints given by a set of forbidden patterns. We study the links between SFTs and computation. SFTs being specific effectively closed classes, we fist study their Turing degree structure, comparing it to the one of effectively closed classes in general: for any effectively closed class, there exist an SFT having the same Turing degrees except maybe 0, the degree of recursive sets. Furthermore, SFTs containing no recursive member have a particular structure: they always contain a cone of Turing degrees, ie. a Turing degree and all degrees above it. We then study the sets of periods of SFTs, for different notions of periodicity, reaching characterizations by means of computational complexity classes or computability classes for each notion introduced. Finally we look at the computable hardness of the factorization and conjugacy problems, the right notions of simulation and equivalence for SFTs

A Characterization of Subshifts with Computable Language

International audienceSubshifts are sets of colorings of Z^d by a finite alphabet that avoid some family of forbidden patterns. We investigate here some analogies with group theory that were first noticed by the first author. In particular we prove several theorems on subshifts inspired by Higman's embedding theorems of group theory, among which, the fact that subshifts with a computable language can be obtained as restrictions of minimal subshifts of finite type