Substitutions and
Strongly Deterministic Tilesets
Bastien Le Gloannec and Nicolas Ollinger
LIFO, University of Orléans, France
CiE 2012, Cambridge
June 22, 2012
Recognizing
families of colorings
.
Σ = {□, ■}
.
Family of Σ-colorings
(subshift)
We recognize colorings of the discrete plane via local constraints.
.
Theorem [Mozes 89]. Colorings “generated by” (expansive) substitu.tions are “recognizable”.
2/15
Recognizing
families of colorings
.
Σ = {□, ■}
.
Family of Σ-colorings
(subshift)
Set of local rules (Wang tiles)
We recognize colorings of the discrete plane via local constraints.
.
Theorem [Mozes 89]. Colorings “generated by” (expansive) substitu.tions are “recognizable”.
2/15
Recognizing
families of colorings
.
Σ = {□, ■}
projection
.
Family of tilings
Family of Σ-colorings
(subshift)
Set of local rules (Wang tiles)
We recognize colorings of the discrete plane via local constraints.
.
Theorem [Mozes 89]. Colorings “generated by” (expansive) substitu.tions are “recognizable”.
2/15
Recognizing
families of colorings
.
Σ = {□, ■}
projection
.
Family of tilings
Family of Σ-colorings
(subshift)
Set of deterministic local rules (Wang tiles)
We recognize colorings of the discrete plane via local constraints.
.
Theorem.
Colorings “generated by” 2×2 substitutions are
.“deterministically recognizable”.
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1. Tilings
.
Tilings
by Wang tiles
.
A Wang tile is an oriented (no
rotations allowed) unit square tile
carrying a color on each side.
.
A tileset τ is a finite set of Wang
tiles.
A tiling is a map t : Z2 → τ,
associating a tile to each cell of
the discrete plane Z2 , for which
the colors of the common sides
of neighboring tiles match.
.
.
3/15
.
Tilings
by Wang tiles
.
A Wang tile is an oriented (no
rotations allowed) unit square tile
carrying a color on each side.
.
A tileset τ is a finite set of Wang
tiles.
A tiling is a map t : Z2 → τ,
associating a tile to each cell of
the discrete plane Z2 , for which
the colors of the common sides
of neighboring tiles match.
.
.
3/15
Deterministic
tilesets
.
.
Deterministic tileset. A tileset τ is NE-deterministic if for any pair
of tiles (tW , tS ) ∈ τ2 , there exists at most one tile t compatible to the
west
with tW and to the south with tS .
.
?
.
1. Tilings
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Deterministic
tilesets
.
.
Deterministic tileset. A tileset τ is NE-deterministic if for any pair
of tiles (tW , tS ) ∈ τ2 , there exists at most one tile t compatible to the
west
with tW and to the south with tS .
.
.
1. Tilings
4/15
Deterministic
tilesets
.
.
Deterministic tileset. A tileset τ is NE-deterministic if for any pair
of tiles (tW , tS ) ∈ τ2 , there exists at most one tile t compatible to the
west
with tW and to the south with tS .
.
.
?
.
Strong determinism. A tileset is 4-way deterministic if it is simultaneously
deterministic in the four directions NE, NW, SW and SE.
.
1. Tilings
4/15
Deterministic
tilesets
.
.
Deterministic tileset. A tileset τ is NE-deterministic if for any pair
of tiles (tW , tS ) ∈ τ2 , there exists at most one tile t compatible to the
west
with tW and to the south with tS .
.
?
.
.
Strong determinism. A tileset is 4-way deterministic if it is simultaneously
deterministic in the four directions NE, NW, SW and SE.
.
1. Tilings
4/15
Deterministic
tilesets
.
.
Deterministic tileset. A tileset τ is NE-deterministic if for any pair
of tiles (tW , tS ) ∈ τ2 , there exists at most one tile t compatible to the
west
with tW and to the south with tS .
.
.
?
.
Strong determinism. A tileset is 4-way deterministic if it is simultaneously
deterministic in the four directions NE, NW, SW and SE.
.
1. Tilings
4/15
Deterministic
tilesets
.
.
Deterministic tileset. A tileset τ is NE-deterministic if for any pair
of tiles (tW , tS ) ∈ τ2 , there exists at most one tile t compatible to the
west
with tW and to the south with tS .
.
.
.
Strong determinism. A tileset is 4-way deterministic if it is simultaneously
deterministic in the four directions NE, NW, SW and SE.
.
1. Tilings
4/15
A
short history
. very
.
Domino Problem, DP [Wang 61]. Given a tileset, is it possible to tile
.the plane?
.
Theorem
[Berger 64]. The Domino Problem is undecidable.
.
Proof by construction of an aperiodic tileset, tiling the plane but
never in a periodic way.
[Kari 91] introduces the notion of deterministic tileset.
.
Theorem [Kari 91]. DP remains undecidable for deterministic tilesets
(and there exist some (bi)deterministic aperiodic tilesets).
.
.
Theorem [Kari-Papasoglu 99]. There exist some 4-way deterministic
aperiodic tilesets.
.
.
Theorem [Lukkarila 09]. DP remains undecidable for 4-way deterministic
tilesets.
.
1. Tilings
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2. Colorings, subshifts and (directional) soficity
Colorings
of the discrete plane
.
Given a finite alphabet Σ, a Σ-coloring of Z2 is a map c
{
}
Σ= . ,. ,.
0
: Z2 → Σ.
.
Endowed with the product of the discrete topology, ΣZ is a
compact space.
2
A subshift Y
colorings.
⊆ ΣZ is a closed and translation invariant set of
2. Colorings, subshifts and (directional) soficity
2
6/15
(Directional)
soficity
. .
Soficity. A subshift Y ⊆ ΣZ is sofic if it can be obtained as the
alphabetic projection of the set of tilings Xτ by a tileset τ: Y =
π(X
τ ).
.
2
π
7−→
.
.
π
7−→
.
2. Colorings, subshifts and (directional) soficity
.
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(Directional)
soficity
. .
Directional soficity. A subshift Y ⊆ ΣZ is NW/NE/SW/SE-sofic if it
can be obtained as the alphabetic projection of the set of tilings Xτ
.by a NW/NE/SW/SE-deterministic tileset τ: Y = π(Xτ ).
2
π
7−→
.
.
π
7−→
.
2. Colorings, subshifts and (directional) soficity
.
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3. Substitutions and directional soficity
Substitutions
. .
Substitution. A (deterministic) substitution is a set of rules replacing
.letters from a finite alphabet Σ by rectangles of letters over Σ.
s
7−→
3. Substitutions and directional soficity
+ rotations
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Limit
set
.
What are the colorings generated by a substitution?
.
Limit set. Given a substitution s, we define its limit set:
Λs =
.
∩⟨ (
sn
n ≥0
ΣZ
2
)⟩
σ
The limit set is a subshift.
.
Proposition. The limit set exactly is the set of colorings c admitting
2
a history: (cn ) ∈ ΣZ verifying c0 = c and ∀n ≥ 0, ∃σn translation,
.σn ◦ s(cn+1 ) = cn .
3. Substitutions and directional soficity
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Directional
soficity of Λs
.
Soficity consists in enforcing the hierarchical structure imposed by
the substitution using local rules.
.
Idea.
Code the history of a coloring into the tilings.
.
.
Theorem
[Mozes 89]. Limit sets of (expansive) substitutions are sofic.
.
.
.Theorem [Ollinger 08]. Limit sets of 2×2 substitutions are sofic.
3. Substitutions and directional soficity
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Directional
soficity of Λs
.
Soficity consists in enforcing the hierarchical structure imposed by
the substitution using local rules.
.
Idea.
Code the history of a coloring into the tilings.
.
.
Theorem
[Mozes 89]. Limit sets of (expansive) substitutions are sofic.
.
.
.Theorem [Ollinger 08]. Limit sets of 2×2 substitutions are sofic.
Directional soficity consists in enforcing the hierarchical structure
imposed by the substitution using deterministic local rules.
.
.Theorem. Limit sets of 2×2 substitutions are 4-way sofic.
3. Substitutions and directional soficity
10/15
.
Sketch
of the proof (1/ 4): 104 4-way
.
We start from the 104 tiles
aperiodic tileset of [Ollinger 08].
Smallest fixed point of a 2×2
substitution scheme.
Each macro-tile codes a tile.
104 is self-simulating for a 2×2
substitution on tiles.
.
7−→
.
We add some cables to obtain a
4-way deterministic structure.
This will be used as a skeleton for
the rest of the construction.
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Sketch of the proof (2/4): enforcing a substitution
.
Observe that every tiling contains an infinite quaternary tree
hierarchical structure.
We add letters of Σ on its nodes to hold the whole history of a
coloring of Λs : the 4 children of a node with label a ∈ Σ carry the
4 letters of σ(a).
3. Substitutions and directional soficity
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Sketch
of the proof (3/4): NE-soficity
.
The strongly deterministic version of 104 combined with the
encoding of a substitution s on the quaternary tree is not
deterministic any more.
.
Problem. For the NE direction, we do not know how to “predict” the
letter
carried by cables of color □□
■□ in NE position on X tiles.
.
.
Idea. In our construction, hereditary information is translated in the
SW
direction. We could set up some wires to go & find it.
.
3. Substitutions and directional soficity
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Sketch
of the proof (3/4): NE-soficity
.
The strongly deterministic version of 104 combined with the
encoding of a substitution s on the quaternary tree is not
deterministic any more.
.
Problem. For the NE direction, we do not know how to “predict” the
letter
carried by cables of color □□
■□ in NE position on X tiles.
.
.
Idea. In our construction, hereditary information is translated in the
SW
direction. We could set up some wires to go & find it.
.
That way we get:
.
T
. heorem. Limit sets of 2×2 substitutions are NE-sofic.
3. Substitutions and directional soficity
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Sketch
of the proof (4/4): 4-way soficity
.
Consider the cartesian product of the four 104 one-way tilesets
obtained from the following four symmetrical substitution
schemes.
7→
7→
.
7→
.
7→
.
.
Each component is deterministic in one of the four directions.
We synchronize the parity layer in order to code the same coloring
on the level 0 of both components.
We then synchronize the whole history on both components using a
3 × 3 grouping of tiles.
■
3. Substitutions and directional soficity
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4. Conclusion
Conclusion
and perspectives
.
We have introduced a strengthened notion of soficity and
considered the case of colorings generated by deterministic 2 × 2
(and 2n × 2n ) substitutions.
This can be adapted for regular n × n substitutions.
This can also be adapted to enforce the generated set Ω of
substitutions.
General open question: What subshifts can be recognized in a
deterministic way?
4. Conclusion
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That’s
all folks!
.
. 4. Conclusion
Thank you for your attention.
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