A Topological Study of Tilings -...

IntroductionBesicovitch topology

Cantor topologyTopological games

A Topological Study of Tilings

Grégory Lafitte1 Michaël Weiss2

1Laboratoire d’Informatique Fondamentale de Marseille (LIF), CNRS –Aix-Marseille Université, 39, rue Joliot-Curie, F-13453 Marseille Cedex 13, France

2Centre Universitaire d’Informatique, Université de Genève, Battelle bâtiment A, 7route de Drize, 1227 Carouge, Switzerland

Theory and Applications of Models of Computation, 2008

Lafitte, Weiss A Topological Study of Tilings

Outline

1 IntroductionWang TilingsMotivation

2 Besicovitch topologyDefinitionThe topological space

3 Cantor topologyDefinitionThe topological space

4 Topological gamesIntroductionGames on tilings

Wang TilingsMotivation

Outline

Wang Tilings

A Wang tile is a unit size square with colored edges.A tile set "τ " is a finite set of Wang tiles.To tile consists in placing the tiles in the grid Z2, respectingthe color matching. A tiling P is associated to a tilingfunction fP : Z2 → τ .

Wang Tilings

Domino ProblemDoes a given tile set tile the plane?

Wang Tilings

Domino ProblemDoes a given tile set tile the plane?

Berger’s proofThe problem is undecidable. Tilings areequivalent to Turing machine.

Different kinds of tile sets

A tile set can be:FinitePeriodicAperiodicNon-recursiveComplete. . .

Outline

Motivation

Two commonly-used topologies on cellular automataProximity between tilings and cellular automataAdapt these topologies to tilingsUnderstand the behavior of the space of tilingsStudy computation from a geometrical point of view

Motivation

What we have, what we expect

ResultsHow does the tiling space behave inside the mappingspaceWhat both topologies can say on tilingsTopological games to study meagerness

ExpectationsGames to solve problems as:Which kinds of structures are more likely to appear?What is most common ? Periodic or aperiodic?

DefinitionThe topological space

Outline

Definition

Traditional topology on cellular automataLimsup of the different bits of two configurationsHow to compare tile sets?

Definition

1 111 100 0 0

1 11 0 1 00 1

Definition

1 111 100 0 0

1 11 0 1 00 1

Definition

1 111 100 0 0

1 11 0 1 00 1

S = 0 , 0

Definition

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1 11 0 1 00 1

S = 0 , 0 , 1 / 5

Definition

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1 11 0 1 00 1

S = 0 , 0 , 1 / 5 , 2 / 7 , 2 / 9 , . . .

Definition

1 111 100 0 0

1 11 0 1 00 1

S = 0 , 0 , 1 / 5 , 2 / 7 , 2 / 9 , . . .

The metric

S = 0 , 2 / 9

The metric

S = 0 , 2 / 9 , 3 / 2 5

The metric

S = 0 , 2 / 9 , 3 / 2 5 , 1 0 / 4 9

The metric

S = 0 , 2 / 9 , 3 / 2 5 , 1 0 / 4 9 , 1 5 / 8 1 . . .

Outline

Properties

MB: {{1, . . . , s}Z2}s>1 with the Besicovitch metricTB: Tilings with the Besicovitch metric

MotivationUnderstand the behavior of TB in MB

PropertyThe classes of equivalent tilings in TB are still uncountablymany

Properties

Results

Metric spacesThere is a countable infinite set A of Wang tilings such thatany two tilings of A are almost at distance 1There is no countable set of Wang tilings dense in TB

TB is path-connected

Topological spacesTB is meager in MB

TB is a Baire space

Results

TB is a Baire space

Results

TB is a Baire space

Results

TB is a Baire space

Results

TB is a Baire space

Outline

Introduction

Another traditional cellular automata metricConsider the biggest common sequence of twoconfigurationsFor tilings, the biggest common pattern

Introduction

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1 11 0 1 00 1

Introduction

1 111 100 0 0

1 11 0 1 00 1

Introduction

1 111 100 0 0

1 11 0 1 00 1

Introduction

1 111 100 0 0

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nd = 2- n

Introduction

1 111 100 0 0

1 11 0 1 00 1

nd = 2- n

Definition

Outline

Properties

MC : {{1, . . . , s}Z2}s>1 with the Cantor metricTC : Tilings with the Cantor metric

MotivationHave a better understanding of the local structure

PropertyTC is a 0-dimensional space

Properties

Results

Metric spacesThere exists an open ball of MC that does not contain anyWang tilingThere exists a countable set of Wang tilings dense in TC

Topological spacesTC is a Baire space

Results

IntroductionGames on tilings

Outline

Motivation

Banach-MazurGames with perfect information with two playersMakes possible the study of meagerness of subsets of atopological space

Motivation

Banach-Mazur Games

A topological space X and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 ⊂ U1, and so on. . .If

⋂Ui ⊂ C, Player II wins

If Player II has a winning strategy, X \ C is meager in X

Banach-Mazur Games

L i m U

Banach-Mazur Games

L i m U

Outline

Games on the Cantor space

An open set X of tilings, and a subset C of XPlayer I chooses a pattern mPlayer II chooses a pattern m′ such that m ∈ m′, and soon. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy

TheoremTC is meager in MC

Games on the Besicovitch space

An open set X of tilings, and a subset C of XPlayer I chooses an open set U1 of XPlayer II chooses an open set U2 such that U2 ⊂ U1, andso on. . .Player II wins if the final tiling is in CX \ C is meager if Player II has a winning strategy

TheoremTB is meager in MB

Further works

Games seems a promising tools for tilingsStudy the meagerness of subsets of tilingsObtain strong results on the probability of a certainstructure to appear

Further works

Shen’s question

Tilings with errorsFinite holeAperiodic tile set stabilityConsider any aperiodic tile set, can we always remove afinite border of the hole to complete the tiling?

A Topological Study of Tilings -...

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