27 research outputs found
Structural aspects of tilings
In this paper, we study the structure of the set of tilings produced by any
given tile-set. For better understanding this structure, we address the set of
finite patterns that each tiling contains. This set of patterns can be analyzed
in two different contexts: the first one is combinatorial and the other
topological. These two approaches have independent merits and, once combined,
provide somehow surprising results. The particular case where the set of
produced tilings is countable is deeply investigated while we prove that the
uncountable case may have a completely different structure. We introduce a
pattern preorder and also make use of Cantor-Bendixson rank. Our first main
result is that a tile-set that produces only periodic tilings produces only a
finite number of them. Our second main result exhibits a tiling with exactly
one vector of periodicity in the countable case.Comment: 11 page
Turing degrees of limit sets of cellular automata
Cellular automata are discrete dynamical systems and a model of computation.
The limit set of a cellular automaton consists of the configurations having an
infinite sequence of preimages. It is well known that these always contain a
computable point and that any non-trivial property on them is undecidable. We
go one step further in this article by giving a full characterization of the
sets of Turing degrees of cellular automata: they are the same as the sets of
Turing degrees of effectively closed sets containing a computable point
On Derivatives and Subpattern Orders of Countable Subshifts
We study the computational and structural aspects of countable
two-dimensional SFTs and other subshifts. Our main focus is on the topological
derivatives and subpattern posets of these objects, and our main results are
constructions of two-dimensional countable subshifts with interesting
properties. We present an SFT whose iterated derivatives are maximally complex
from the computational point of view, a sofic shift whose subpattern poset
contains an infinite descending chain, a family of SFTs whose finite subpattern
posets contain arbitrary finite posets, and a natural example of an SFT with
infinite Cantor-Bendixon rank.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Pi01 sets and tilings
In this paper, we prove that given any \Pi^0_1 subset of \{0,1\}^\NN
there is a tileset with a set of configurations such that
P\times\ZZ^2 is recursively homeomorphic to where is a
computable set of configurations. As a consequence, if is countable, this
tileset has the exact same set of Turing degrees
Tilings and model theory
ISBN 978-5-94057-377-7International audienceIn this paper we emphasize the links between model theory and tilings. More precisely, after giving the definitions of what tilings are, we give a natural way to have an interpretation of the tiling rules in first order logics. This opens the way to map some model theoretical properties onto some properties of sets of tilings, or tilings themselves
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
Conjugacy of transitive SFTs minus periodic points
It is a question of Hochman whether any two one-dimensional topologically
mixing subshifts of finite type (SFTs) with the same entropy are topologically
conjugate when their periodic points are removed. We give a negative answer, in
fact we prove the stronger result that there is a canonical correspondence
between topological conjugacies of transitive SFTs and topological conjugacies
between the systems obtained by removing the periodic points.Comment: 10 pages; v3 adds an introduction, and is accepted in PLM