358 research outputs found
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
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.Comment: In Proceedings AUTOMATA&JAC 2012, arXiv:1208.249
Fixed Point and Aperiodic Tilings
An aperiodic tile set was first constructed by R.Berger while proving the
undecidability of the domino problem. It turned out that aperiodic tile sets
appear in many topics ranging from logic (the Entscheidungsproblem) to physics
(quasicrystals) We present a new construction of an aperiodic tile set that is
based on Kleene's fixed-point construction instead of geometric arguments. This
construction is similar to J. von Neumann self-reproducing automata; similar
ideas were also used by P. Gacs in the context of error-correcting
computations. The flexibility of this construction allows us to construct a
"robust" aperiodic tile set that does not have periodic (or close to periodic)
tilings even if we allow some (sparse enough) tiling errors. This property was
not known for any of the existing aperiodic tile sets.Comment: v5: technical revision (positions of figures are shifted
Aperiodic Subshifts of Finite Type on Groups
In this note we prove the following results:
If a finitely presented group admits a strongly aperiodic SFT,
then has decidable word problem. More generally, for f.g. groups that are
not recursively presented, there exists a computable obstruction for them to
admit strongly aperiodic SFTs.
On the positive side, we build strongly aperiodic SFTs on some new
classes of groups. We show in particular that some particular monster groups
admits strongly aperiodic SFTs for trivial reasons. Then, for a large class of
group , we show how to build strongly aperiodic SFTs over . In particular, this is true for the free group with 2 generators,
Thompson's groups and , and any f.g. group of
rational matrices which is bounded.Comment: New version. Adding results about monster group
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
Computability and Tiling Problems
In this thesis we will present and discuss various results pertaining to tiling problems and mathematical logic, specifically computability theory.
We focus on Wang prototiles, as defined in [32]. We begin by studying Domino Problems, and do not restrict ourselves to the usual problems concerning finite sets of prototiles.
We first consider two domino problems: whether a given set of prototiles S has total planar tilings, which we denote TILE, or whether it has infinite connected but not necessarily total tilings, WTILE (short for ‘weakly tile’). We show that both TILE ≡m ILL ≡m WTILE, and thereby both TILE and WTILE are Σ11-complete.
We also show that the opposite problems, ¬TILE and SNT (short for ‘Strongly Not Tile’) are such that ¬TILE ≡m WELL ≡m SNT and so both ¬TILE and SNT are both Π11-complete.
Next we give some consideration to the problem of whether a given (infinite) set of prototiles is periodic or aperiodic. We study the sets PTile of periodic tilings, and ATile of aperiodic tilings. We then show that both of these sets are complete for the class of problems of the form (Σ1 1 ∧Π1 1). We also present results for finite versions of these tiling problems.
We then move on to consider the Weihrauch reducibility for a general total tiling principle CT as well as weaker principles of tiling, and show that there exist Weihrauch equivalences to closed choice on Baire space, Cωω. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to Cωω.
Finally, we give a prototile set of 15 prototiles that can encode any Elementary CellularAutomaton(ECA). We make use of an unusual tileset, based on hexagons and lozenges that we have not seen in the literature before, in order to achieve this
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