45 research outputs found
1D Effectively Closed Subshifts and 2D Tilings
Michael Hochman showed that every 1D effectively closed subshift can be
simulated by a 3D subshift of finite type and asked whether the same can be
done in 2D. It turned out that the answer is positive and necessary tools were
already developed in tilings theory. We discuss two alternative approaches:
first, developed by N. Aubrun and M. Sablik, goes back to Leonid Levin; the
second one, developed by the authors, goes back to Peter Gacs.Comment: Journ\'ees Automates Cellulaires, Turku : Finland (2010
Quasiperiodicity and non-computability in tilings
We study tilings of the plane that combine strong properties of different
nature: combinatorial and algorithmic. We prove existence of a tile set that
accepts only quasiperiodic and non-recursive tilings. Our construction is based
on the fixed point construction; we improve this general technique and make it
enforce the property of local regularity of tilings needed for
quasiperiodicity. We prove also a stronger result: any effectively closed set
can be recursively transformed into a tile set so that the Turing degrees of
the resulted tilings consists exactly of the upper cone based on the Turing
degrees of the later.Comment: v3: the version accepted to MFCS 201
About the domino problem in the hyperbolic plane from an algorithmic point of view
In this paper, we prove that the general problem of tiling the hyperbolic
plane with \`a la Wang tiles is undecidable.Comment: 11 pages, 6 figure
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
Aperiodic Tilings: Breaking Translational Symmetry
Classical results on aperiodic tilings are rather complicated and not widely
understood. Below, an alternative approach is discussed in hope to provide
additional intuition not apparent in classical works.Comment: 4 pages, 2 figures, minor change
Computing (or not) Quasi-Periodicity Functions of Tilings
We know that tilesets that can tile the plane always admit a quasi-periodic
tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The
quasi-periodicity function is one way to measure the regularity of a
quasi-periodic tiling. We prove that the tilings by a tileset that admits only
quasi-periodic tilings have a recursively (and uniformly) bounded
quasi-periodicity function. This corrects an error from [6, theorem 9] which
stated the contrary. Instead we construct a tileset for which any
quasi-periodic tiling has a quasi-periodicity function that cannot be
recursively bounded. We provide such a construction for 1-dimensional effective
subshifts and obtain as a corollary the result for tilings of the plane via
recent links between these objects [1, 10].Comment: Journ\'ees Automates Cellulaires 2010, Turku : Finland (2010
Enumeration Reducibility in Closure Spaces with Applications to Logic and Algebra
In many instances in first order logic or computable algebra, classical
theorems show that many problems are undecidable for general structures, but
become decidable if some rigidity is imposed on the structure. For example, the
set of theorems in many finitely axiomatisable theories is nonrecursive, but
the set of theorems for any finitely axiomatisable complete theory is
recursive. Finitely presented groups might have an nonrecursive word problem,
but finitely presented simple groups have a recursive word problem. In this
article we introduce a topological framework based on closure spaces to show
that many of these proofs can be obtained in a similar setting. We will show in
particular that these statements can be generalized to cover arbitrary
structures, with no finite or recursive presentation/axiomatization. This
generalizes in particular work by Kuznetsov and others. Examples from first
order logic and symbolic dynamics will be discussed at length