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

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

On the Expressive Power of Quasiperiodic SFT

In this paper we study the shifts, which are the shift-invariant and topologically closed sets of configurations over a finite alphabet in Z^d. The minimal shifts are those shifts in which all configurations contain exactly the same patterns. Two classes of shifts play a prominent role in symbolic dynamics, in language theory and in the theory of computability: the shifts of finite type (obtained by forbidding a finite number of finite patterns) and the effective shifts (obtained by forbidding a computably enumerable set of finite patterns). We prove that every effective minimal shift can be represented as a factor of a projective subdynamics on a minimal shift of finite type in a bigger (by 1) dimension. This result transfers to the class of minimal shifts a theorem by M.Hochman known for the class of all effective shifts and thus answers an open question by E. Jeandel. We prove a similar result for quasiperiodic shifts and also show that there exists a quasiperiodic shift of finite type for which Kolmogorov complexity of all patterns of size ntimes n is Omega(n)

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

Weak local rules for planar octagonal tilings

We provide an effective characterization of the planar octagonal tilings which admit weak local rules. As a corollary, we show that they are all based on quadratic irrationalities, as conjectured by Thang Le in the 90s.Comment: 23 pages, 6 figure

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

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 $TILE \equiv_m ILL \equiv_m WTILE$, and thereby both $TILE$ and $WTILE$ are $\Sigma^1_1$-complete. We also show that the opposite problems, $\neg TILE$ and $SNT$ (short for `Strongly Not Tile') are such that $\neg TILE \equiv_m WELL \equiv_m SNT$ and so both $\neg TILE$ and $SNT$ are both $\Pi^1_1$-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 $(\Sigma^1_1 \wedge \Pi^1_1)$. 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, $C_{\omega^\omega}$. We also show that all Domino Problems that tile some infinite connected region are Weihrauch reducible to $C_{\omega^\omega}$. Finally, we give a prototile set of 15 prototiles that can encode any Elementary Cellular Automaton (ECA). We make use of an unusual tile set, based on hexagons and lozenges that we have not see in the literature before, in order to achieve this.Comment: PhD thesis. 179 pages, 13 figure