Substitutions and Strongly Deterministic Tilesets - 1ex-3ex · Substitutions and Strongly...

SubstitutionsandStronglyDeterministicTilesets

Bastien Le Gloannec and Nicolas Ollinger

LIFO, Université d’Orléans

Aperiodic 2017, LyonSeptember 26, 2017

Recognizing families of colorings

Family of Σ-colorings(subshift)

Σ = {□,■}

Set of local rules (Wangtiles)

Family of tilings

projection

We recognize colorings of the discrete plane via local constraints.Theorem[Mozes89]. Colorings “generated by” (expansive) substitu-tions are “recognizable”.

2/54

Recognizing families of colorings

Family of Σ-colorings(subshift)

Σ = {□,■}

Set of deterministic local rules (Wangtiles)

Family of tilings

projection

We recognize colorings of the discrete plane via local constraints.Theorem[inCiE’12]. Colorings “generated by” 2×2 substitutions are“deterministically recognizable”.

2/54

1. Tilings

Tilings by Wang tiles

A Wangtile is an oriented (norotations allowed) unit square tilecarrying a coloroneachside.

A tileset τ is a finite set of Wangtiles.

A configuration c ∈ τZ2

associates a tile to each cell of thediscrete plane Z2.

A tiling is a configuration wherethe colors of the common sides ofneighboring tiles match.

4/54

Historical context

In the early 60s, H. Wang reduces the decidability problem of the∀∃∀ class of firstorderlogicformulas to a question of discretetiling.

DominoProblem, DP [Wang, 1961]. Given a tileset, is it possible totile the plane?

Theorem[Berger, 1964]. The DominoProblem is undecidable.

Proof by construction of an aperiodic tileset describing a self-similarstructure able to contain some Turingcomputations.

Aperiodicity. A tileset is aperiodic if it tiles the plane, but never in aperiodic way.

1. Tilings 6/54

Deterministic tilesetsIntroduced by J. Kari in 1991 to prove the undecidability of thenilpotency problem for 1D cellular automata.

Notations: NW for North-West, SE pour South-East…

Deterministictileset. A tileset τ is NE-deterministic if for any pairof tiles (tW, tS) ∈ τ2, there exists atmostone tile t compatible to thewest with tW and to the south with tS.

? Partiallocalmap.

f : τ2 → τ

We symmetrically define {NW,SE,SW}-determinism.1. Tilings 7/54

Deterministic tilesetsIntroduced by J. Kari in 1991 to prove the undecidability of thenilpotency problem for 1D cellular automata.

Notations: NW for North-West, SE pour South-East…

Deterministictileset. A tileset τ is NE-deterministic if for any pairof tiles (tW, tS) ∈ τ2, there exists atmostone tile t compatible to thewest with tW and to the south with tS.

?

Partiallocalmap.

f : τ2 → τ

We symmetrically define {NW,SE,SW}-determinism.1. Tilings 7/54

Deterministic tilesets

Bideterminism. A tileset is bideterministic if it is simultaneously de-terministic in twooppositedirections: NE & SW, or NW & SE.

?

Strongdeterminism. A tileset is stronglydeterministic (4-waydeter-ministic) if it is simultaneously deterministic in the fourdirections NE,NW, SW and SE.

1. Tilings 8/54

Deterministic tilesets: a short history[Kari, 1991] introduced a (bi)determinization of [Robinson, 1971] totreat the nilpotency problem for cellular automata in dimension 1(Nil1D)…Already proven in [Aanderaa-Lewis1974]?!

Theorem[Kari, 1991; Aanderaa-Lewis, 1974]. Nil1D is undecid-able.

Theorem [Kari, 1991; Aanderaa-Lewis, 1974]. There exist some(bi)deterministic aperiodic tilesets.

Rmk The 16 Wang tiles derived from Ammann’s geometric tiles arebideterministic.

Theorem[Kari, 1991; Aanderaa-Lewis, 1974]. DP remains unde-cidable for deterministic tilesets.

1. Tilings 9/54

Deterministic tilesets: a short history

[Kari-Papasoglu, 1999] builds a strong determinization of[Robinson, 1971].

Theorem[Kari-Papasoglu, 1999]. There exist some stronglydeter-ministic aperiodic tilesets.

[Lukkarila, 2009] introduces a strong determinization of [Robinson,1971]+Turingcomputation.

Theorem[Lukkarila, 2009]. DP remains undecidable for stronglydeterministic tilesets.

1. Tilings 11/54

2. Colorings, subshifts and (directional) soficity

Colorings of the discrete planeGiven a finitealphabet Σ, a Σ-coloring of Z2 is a map

c : Z2 −→ Σ

Σ ={

, ,}

0

2. Colorings, subshifts and (directional) soficity 13/54

Topology

The set ΣZ2of Σ-colorings is endowed with the producttopology

over Z2 of the discretetopology over Σ.

Theorem. ΣZ2is a compact space.

Translation. ∀c ∈ ΣZ2, ∀z, x ∈ Z2, σz(c)(x) = c(x− z)

Subshift. A subshift Y ⊆ ΣZ2is a close and translationinvariant

set of colorings.

Tilings. The set of tilings of a tileset τ is a subshift.


2D SoficitySoficity. A subshift Y ⊆ ΣZ2

is sofic if it can be obtained as thealphabeticprojection of the set of tilings Xτ by a tileset τ: Y =π(Xτ).

π7−→

π7−→2. Colorings, subshifts and (directional) soficity 15/54

Directional soficity

Directionalsoficity. A subshift Y ⊆ ΣZ2is NW/NE/SW/SE-sofic if

it can be obtained as the alphabetic projection of the set of tiling Xτ

by a NW/NE/SW/SE-deterministic tileset τ.

N.B. Those colorings are generated by partialcellularautomata ofdimension 1.

Motivation. Opens the doors for tools from dimension1.

The coloringsgeneratedbysubstitutions are sofic.

Today… What about the directional case?


3. Substitutions and soficity

SubstitutionsSubstitution. A (deterministic) substitution is a set of rules replacingletters from a finite alphabet Σ by rectangles of letters over Σ.

s7−→ + rotations

3. Substitutions and soficity 18/54

Limit set

Whatarethecoloringsgeneratedbyasubstitution?

Limitset. Given a substitution s, we define its limitset:

Λs =∩n≥0

⟨sn(ΣZ2

)⟩σ

The limit set is a subshift.

Proposition. The limit set exactly is the set of colorings c admittinga history: (cn) ∈ ΣZ2

verifying c0 = c and ∀n ≥ 0, ∃σn translation,σn ◦ s(cn+1) = cn.


Soficity of Λs

We want to force the hierarchicalstructure imposed by thesubstitution using localrules.

Idea. Code the history of a coloring into the tilings.

Theorem[Mozes, 1989]. Limit sets of (expansive) substitutions aresofic.

Seminal construction method for [Goodman-Strauss, 1998] and[Fernique-O,2010].

Theorem[104]. Limit sets of 2×2 substitutions are sofic.


v1,1R1,1

(1, 1)

v2,1R1,1

(2, 1)

v1,2R1,1

(1, 2)

v2,2R1,1

(2, 2)

v1,3R1,1

(1, 3)

v2,3R1,1

(2, 3)

b1,1S1,1

(1, 1)

c1,1R2,1

(1, 1)

c2,1R2,1

(2, 1)

c3,1R2,1

(3, 1)

c1,2R2,1

(1, 2)

c2,2R2,1

(2, 2)

c3,2R2,1

(3, 2)

c1,3R2,1

(1, 3)

c2,3R2,1

(2, 3)

c3,3R2,1

(3, 3)

b2,1S1,1

(2, 1)

j1,1R3,1

(1, 1)

j2,1R3,1

(2, 1)

j1,2R3,1

(1, 2)

j2,2R3,1

(2, 2)

j1,3R3,1

(1, 3)

j2,3R3,1

(2, 3)

b3,1S1,1

(3, 1)

q1,1R4,1

(1, 1)

q2,1R4,1

(2, 1)

q3,1R4,1

(3, 1)

q1,2R4,1

(1, 2)

q2,2R4,1

(2, 2)

q3,2R4,1

(3, 2)

q1,3R4,1

(1, 3)

q2,3R4,1

(2, 3)

q3,3R4,1

(3, 3)

b1,1S2,1

(1, 1)

x1,1

R5,1

(1, 1)

x2,1

R5,1

(2, 1)

x3,1

R5,1

(3, 1)

x1,2

R5,1

(1, 2)

x2,2

R5,1

(2, 2)

x3,2

R5,1

(3, 2)

x1,3

R5,1

(1, 3)

x2,3

R5,1

(2, 3)

x3,3

R5,1

(3, 3)

b2,1S2,1

(2, 1)

e1,1R1,2

(1, 1)

e2,1R1,2

(2, 1)

e1,2R1,2

(1, 2)

e2,2R1,2

(2, 2)

b1,2S1,1

(1, 2)

l1,1R2,2

(1, 1)

l2,1R2,2

(2, 1)

l3,1R2,2

(3, 1)

l1,2R2,2

(1, 2)

l2,2R2,2

(2, 2)

l3,2R2,2

(3, 2)

b2,2S1,1

(2, 2)

s1,1R3,2

(1, 1)

s2,1R3,2

(2, 1)

s1,2R3,2

(1, 2)

s2,2R3,2

(2, 2)

b3,2S1,1

(3, 2)

z1,1R4,2

(1, 1)

z2,1R4,2

(2, 1)

z3,1R4,2

(3, 1)

z1,2R4,2

(1, 2)

z2,2R4,2

(2, 2)

z3,2R4,2

(3, 2)

b1,2S2,1

(1, 2)

g1,1R5,2

(1, 1)

g2,1R5,2

(2, 1)

g3,1R5,2

(3, 1)

g1,2R5,2

(1, 2)

g2,2R5,2

(2, 2)

g3,2R5,2

(3, 2)

b2,2S2,1

(2, 2)

n1,1

R1,3

(1, 1)

n2,1

R1,3

(2, 1)

n1,2

R1,3

(1, 2)

n2,2

R1,3

(2, 2)

b1,1S1,2

(1, 1)

u1,1

R2,3

(1, 1)

u2,1

R2,3

(2, 1)

u3,1

R2,3

(3, 1)

u1,2

R2,3

(1, 2)

u2,2

R2,3

(2, 2)

u3,2

R2,3

(3, 2)

b2,1S1,2

(2, 1)

b1,1R3,3

(1, 1)

b2,1R3,3

(2, 1)

b1,2R3,3

(1, 2)

b2,2R3,3

(2, 2)

b3,1S1,2

(3, 1)

i1,1R4,3

(1, 1)

i2,1R4,3

(2, 1)

i3,1R4,3

(3, 1)

i1,2R4,3

(1, 2)

i2,2R4,3

(2, 2)

i3,2R4,3

(3, 2)

b1,1S2,2

(1, 1)

p1,1R5,3

(1, 1)

p2,1R5,3

(2, 1)

p3,1R5,3

(3, 1)

p1,2R5,3

(1, 2)

p2,2R5,3

(2, 2)

p3,2R5,3

(3, 2)

b2,1S2,2

(2, 1)

w1,1

R1,4

(1, 1)

w2,1

R1,4

(2, 1)

w1,2

R1,4

(1, 2)

w2,2

R1,4

(2, 2)

w1,3

R1,4

(1, 3)

w2,3

R1,4

(2, 3)

b1,2S1,2

(1, 2)

d1,1R2,4

(1, 1)

d2,1R2,4

(2, 1)

d3,1R2,4

(3, 1)

d1,2R2,4

(1, 2)

d2,2R2,4

(2, 2)

d3,2R2,4

(3, 2)

d1,3R2,4

(1, 3)

d2,3R2,4

(2, 3)

d3,3R2,4

(3, 3)

b2,2S1,2

(2, 2)

k1,1R3,4

(1, 1)

k2,1R3,4

(2, 1)

k1,2R3,4

(1, 2)

k2,2R3,4

(2, 2)

k1,3R3,4

(1, 3)

k2,3R3,4

(2, 3)

b3,2S1,2

(3, 2)

r1,1R4,4

(1, 1)

r2,1R4,4

(2, 1)

r3,1R4,4

(3, 1)

r1,2R4,4

(1, 2)

r2,2R4,4

(2, 2)

r3,2R4,4

(3, 2)

r1,3R4,4

(1, 3)

r2,3R4,4

(2, 3)

r3,3R4,4

(3, 3)

b1,2S2,2

(1, 2)

y1,1R5,4

(1, 1)

y2,1R5,4

(2, 1)

y3,1R5,4

(3, 1)

y1,2R5,4

(1, 2)

y2,2R5,4

(2, 2)

y3,2R5,4

(3, 2)

y1,3R5,4

(1, 3)

y2,3R5,4

(2, 3)

y3,3R5,4

(3, 3)

b2,2S2,2

(2, 2)

c1,1T

(1, 1)

c2,1T

(2, 1)

c1,2T

(1, 2)

c2,2T

(2, 2)

dU

(2, 6)

Goodman-Strauss 1998

Theorem[Goodman-Strauss 1998]. The limit set of homotheticsubstitution (+somehypothesis) is sofic.


Fernique-O 2010

c

db

a

c

db

a

b

b

b

b

b

d

d

dd

d

a

a

a

ab

c d

b b b

d d d

c

c

a

a

b

b

b

b

b

d

d

dd

d

c c caa a

c c c

c

db

a

b

b

b

b

b

aa a

d

d

dd

d

c

c

c

a

b

cd

d

d

dd

d

c

c

b

b

b

b

b

a

a

Theorem[Fernique-O 2010]. The limit set of a combinatorialsub-stitution (+somehypothesis) is sofic.


a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

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a b

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a b

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a b

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a b

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a b

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a b

c d

a b

c d

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c d

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c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

a b

c d

Is this self-encoding?Iterating the coding rule one obtains 56 tiles.

⊢

codingrule

Unfortunately, this tile set is notself-coding.

Idea Add a synchronizingsubstitution as a thirdlayer.3. Substitutions and soficity 25/54

àla Robinson

` `

` `

`

Proposition. The associated tile set of 104tiles admits a tiling andcodes an unambiguous substitution.


104 in brief: 3 layersTileset τ introduced in [104].

Layer1: Parity.

Layer2: Cables.

Layer3.


104 in brief

Smallest fixed point of a 2×2substitutionscheme.

Each macro-tile codes a tile.

104 is self-simulating for a 2×2substitution s on tiles.

The limit set Λs of s is aperiodicand the set of tilings verifiesXτ ⊂ Λs.

Theorem[104].104 is aperiodic.

7−→

7−→

7−→29/54

SoficityEvery tilings contains an infinitequaternarytree hierarchicalstructure.

For any 2×2 substitution s ′ overan alphabet Σ, we enrich τ into atileset τ(s ′) such that:

⋄ cables of the tree structurecarry letters of Σ ;

⋄ new rules enforce the structureto code the whole history of acoloring of Λs ′ .

Theorem[104].

π(Xτ(s ′)

)= Λs ′

b = s ′(a)(0, 0) b = s ′(a)(1, 0)

b = s ′(a)(0, 1) b = s ′(a)(1, 1)

b = s ′(a)(0, 0) b = s ′(a)(1, 0)

b = s ′(a)(0, 1) b = s ′(a)(1, 1)

31/54

4. Historical interlude

“(…)In1966R. Bergerdiscoveredthefirstaperiodictileset. Itcontains20,426Wangtiles, (…)Bergerhimselfmanagedtoreducethenumberoftilesto104andhedescribedtheseinhisthesis, thoughtheywereomittedfromthepublishedversion(Berger[1966]). (…)” [GrSh, p.584]

Berger’s skeleton substitution

` ` `

` ` `

`

` ` `

`

4. Historical interlude 34/54

Berger’s forgotten aperiodic tile set

Proposition. The associated tile set of 103tiles admits a tiling andcodes an unambiguous substitution.

Remark. The number of tiles doesnotgrowmonotonically in thenumber of letters of the synchronizing layer.

5 letters → 104 tiles11 letters → 103 tiles

4. Historical interlude 36/54

5. Substitutions and directional soficity

Directional soficity of Λs

We want to force the hierarchical structure imposed by thesubstitution using 4-waydeterministic local rules.

Inpractice. Let’s adapt some (rather technical) existing constructionscoding the history of colorings into tilings to make themdeterministic.

Inthefollowingofthistalk…

Theorem. Limit sets of 2×2 substitutions are 4-way sofic.

5. Substitutions and directional soficity 38/54

Directional soficity of Λs

Battleplan.1. We determinize 104 in the fourdirections simultaneously.2. We determinize 104+substitutions in one direction.3. We bideterminize 104+substitutions.4. We strongly determinize 104+substitutions.


104-way104 is not deterministic in anydirection.

Problem. How to chose betweenH and V tiles in ■□□□ positions?

Idea. Go & search for informationon the mother tile back in the his-tory.

Needfor:

⋄ new constraints at radius 1 forlevel 0 ;

⋄ new wires (carrying H/V/Xlabels) to inferior levels.

40/54

104 1-way + Substitutions

We combine the stronglydeterministic version of 104 with theencodingofasubstitution s ′ on the quaternarytree.

Again, the obtained tileset is not deterministic in any direction!

Problem. For the NE direction, we do not know how to “predict” theletter carried by cables of color ■□□□ in NE position on X tiles.

Idea. In our construction, hereditary information is translated in theSW direction. We could setupsomewires to go & find it.

ByJove! We have already done something similar at the previousstep.


104 2-way + SubstitutionsWe consider the cartesian product of the two 1041-way+substitution tilesets obtained from the following two symmetricalsubstitutionschemes.

7−→ 7−→We synchronize the parity layer in order to codethesamecoloringon the level 0 of both components.

The component 1 (resp. 2) is NE-deterministic (resp.SW-deterministic).

We need to synchronize the wholehistory on both components.5. Substitutions and directional soficity 44/54

104 4-way + Substitutions

We consider this time the cartesian product of the four 1041-way+substitution tilesets obtained from the following four symmetricalsubstitutionschemes.

7−→ 7−→7−→ 7−→

Similar analysis and same solution (3 × 3 grouping) as before.


6. Going further

Conclusion and perspectivesWe have considered the case of colorings generated bydeterministic 2 × 2 (and 2n × 2n) substitutions.

This might be adapted for regular n× n substitutions.

Generalopenquestion: What subshifts can be recognized in adeterministic way?

Bi-deterministic version of [Durand-Romashchenko-Shen 08]6. Going further 53/54

That’s all folks!

Thankyouforyourattention.

54/54

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