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Undecidable properties of self-affine setsand multitape automata
Timo JolivetInstitut de Mathématiques de Toulouse
Université Paul Sabatier
In collaboration with Jarkko Kari
Verification seminarUniversity of OxfordThursday 18 June
Example: the Cantor set
Example: the Cantor set
X⊆[0,1]︷ ︸︸ ︷
Example: the Cantor set
X⊆[0,1]︷ ︸︸ ︷︸ ︷︷ ︸13 X
︸ ︷︷ ︸13 X+ 2
3
Example: the Cantor set
X⊆[0,1]︷ ︸︸ ︷︸ ︷︷ ︸13 X
︸ ︷︷ ︸13 X+ 2
3
I X = f1(X) ∪ f2(X) ⊆ RI f1(x) = 1
3x
I f2(x) = 13x+ 2
3
ExampleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Iterated function systems (IFS)Let f1, . . . , fn : Rd → Rd be contracting maps:∃c < 1 such that |f(x)− f(−y)| < c|x− y| for all x, y ∈ Rd.
Theorem (Hutchinson 1981)There is a unique, nonemtpy, compact set X ⊆ Rd such that
X = f1(X) ∪ · · · ∪ fn(X).
I Definition of a fractal
Iterated function systems (IFS)Let f1, . . . , fn : Rd → Rd be contracting maps:∃c < 1 such that |f(x)− f(−y)| < c|x− y| for all x, y ∈ Rd.
Theorem (Hutchinson 1981)There is a unique, nonemtpy, compact set X ⊆ Rd such that
X = f1(X) ∪ · · · ∪ fn(X).
I Definition of a fractal
Iterated function systems (IFS)Let f1, . . . , fn : Rd → Rd be contracting maps:∃c < 1 such that |f(x)− f(−y)| < c|x− y| for all x, y ∈ Rd.
Theorem (Hutchinson 1981)There is a unique, nonemtpy, compact set X ⊆ Rd such that
X = f1(X) ∪ · · · ∪ fn(X).
I Definition of a fractal
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
Example: Sierpiński’s triangleI f1(x) = x/2I f2(x) = x/2 +
(1/20
)I f3(x) = x/2 +
(01/2
)X = f1(X) ∪ f2(X) ∪ f3(X) ⊆ R2
A particular family of IFS: self-affine setsI Let A ∈ Zd×d be an expanding matrixI Let D ⊆ Zd (“digits”)
Self-affine set: IFS given by the contracting maps maps
fd(x) = A−1(x+ d)
for d ∈ D.
Very particular kind of IFS:I Affine mapsI Integer coefficientsI Same matrix A for each fd
A particular family of IFS: self-affine setsI Let A ∈ Zd×d be an expanding matrixI Let D ⊆ Zd (“digits”)
Self-affine set: IFS given by the contracting maps maps
fd(x) = A−1(x+ d)
for d ∈ D.
Very particular kind of IFS:I Affine mapsI Integer coefficientsI Same matrix A for each fd
A particular family of IFS: self-affine setsI Let A ∈ Zd×d be an expanding matrixI Let D ⊆ Zd (“digits”)
Self-affine set: IFS given by the contracting maps maps
fd(x) = A−1(x+ d)
for d ∈ D.
Very particular kind of IFS:I Affine mapsI Integer coefficientsI Same matrix A for each fd
Example 0I A = (10)I D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
I X =9⋃
d=0
110(X + d)
I X = [0, 1] (base 10 representation)
Meaning of “digits”:
X ={ ∞∑
k=1A−kdk : (dk)k>1 ∈ DN
}
Example 0I A = (10)I D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
I X =9⋃
d=0
110(X + d)
I X = [0, 1] (base 10 representation)
Meaning of “digits”:
X ={ ∞∑
k=1A−kdk : (dk)k>1 ∈ DN
}
Example 0I A = (10)I D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
I X =9⋃
d=0
110(X + d)
I X = [0, 1] (base 10 representation)
Meaning of “digits”:
X ={ ∞∑
k=1A−kdk : (dk)k>1 ∈ DN
}
Example 0I A = (10)I D = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
I X =9⋃
d=0
110(X + d)
I X = [0, 1] (base 10 representation)
Meaning of “digits”:
X ={ ∞∑
k=1A−kdk : (dk)k>1 ∈ DN
}
Example: Sierpiński’s triangle againI A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Example: Sierpiński’s triangle againI A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1)}
( 2 00 2)X = X ∪
(X +
( 10))∪(X +
( 01))
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
Self-affine tiles
DefinitionD is a standard set of digits if:
I |D| = |det(A)|I D is a complete set of representatives in Zd/AZd.
In this case, X is a self-affine tile.
Theorem [Bandt, Lagarias-Wang]Under these conditions:
I X has nonempty interiorI X is the closure of its interiorI µ(∂X) = 0I X tiles Rd by translation
I Many good properties and algorithms
Self-affine tiles
DefinitionD is a standard set of digits if:
I |D| = |det(A)|I D is a complete set of representatives in Zd/AZd.
In this case, X is a self-affine tile.
Theorem [Bandt, Lagarias-Wang]Under these conditions:
I X has nonempty interiorI X is the closure of its interiorI µ(∂X) = 0I X tiles Rd by translation
I Many good properties and algorithms
Self-affine tiles
DefinitionD is a standard set of digits if:
I |D| = |det(A)|I D is a complete set of representatives in Zd/AZd.
In this case, X is a self-affine tile.
Theorem [Bandt, Lagarias-Wang]Under these conditions:
I X has nonempty interiorI X is the closure of its interiorI µ(∂X) = 0I X tiles Rd by translation
I Many good properties and algorithms
Self-affine tiles
DefinitionD is a standard set of digits if:
I |D| = |det(A)|I D is a complete set of representatives in Zd/AZd.
In this case, X is a self-affine tile.
Theorem [Bandt, Lagarias-Wang]Under these conditions:
I X has nonempty interiorI X is the closure of its interiorI µ(∂X) = 0I X tiles Rd by translation
I Many good properties and algorithms
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior (why?)
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior (why?)
Nonstandard DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)}
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
Nonstandard DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)}
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
Nonstandard DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)}
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
Nonstandard DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)}
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2),( 2
0),( 3
0),( 2
1),( 3
2),( 0
2),( 1
2),( 0
3),( 1
4),( 2
4),( 3
4),( 2
5),( 3
6)}
Nonstandard DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)}
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2),( 2
0),( 3
0),( 2
1),( 3
2),( 0
2),( 1
2),( 0
3),( 1
4),( 2
4),( 3
4),( 2
5),( 3
6)}
so 16µ(X) 6 15µ(X)
Nonstandard DIndeed, ( 2 0
0 2)X = X +
{( 00),( 1
0),( 0
1),( 1
2)}
so ( 4 00 4)X =
( 2 00 2)X +
{( 00),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2)}
+{( 0
0),( 2
0),( 0
2),( 2
4)}
= X +{( 0
0),( 1
0),( 0
1),( 1
2),( 2
0),( 3
0),( 2
1),( 3
2),( 0
2),( 1
2),( 0
3),( 1
4),( 2
4),( 3
4),( 2
5),( 3
6)}
so 16µ(X) 6 15µ(X)so µ(X) = 0
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
2),( 1
2)}
X has nonempty interior
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
2),( 1
2)}
X has nonempty interior
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
2),( 1
2)}
X has nonempty interior
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .I Is it decidable?
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .I Is it decidable?
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .I Is it decidable?
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .I Is it decidable?
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]
I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .I Is it decidable?
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .
I Is it decidable?
Nonstandard DWe enjoy the good properties of tiles (case of standard D) :
What conditions on D for nonempty interior?
I Necessary condition |D| > |det(A)|
I Nonempty interior ⇔ µ(X) > 0 ⇔{X = closure(Xo)µ(∂(X)) = 0
⇔ #{∑N
k=1A−kdk : (dk) ∈ DN
}= |D|N for all N > 1
⇔ |D| = |det(A)| if A = (p) with p prime [Kenyon 1992]I Related works [Lagarias-Wang 1996], [Lau-Rao 2003], . . .I Is it decidable?
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ai instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFS
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ak instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
I Interesting properties:I compute fractal dimensions (Hausdorff, packing, etc.),I topological properties (connectedness, nonempty interior, etc.)I tiling properties
I Notorious difficulty of study for specific examples/familiesI Some “almost always”-type resultsI Application in number theory, dynamical systems, Fourier
analysis
I GIFS: several sets X1, . . . , Xn instead of just X (Graph-IFS)
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ak instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
I Interesting properties:I compute fractal dimensions (Hausdorff, packing, etc.),I topological properties (connectedness, nonempty interior, etc.)I tiling properties
I Notorious difficulty of study for specific examples/familiesI Some “almost always”-type resultsI Application in number theory, dynamical systems, Fourier
analysis
I GIFS: several sets X1, . . . , Xn instead of just X (Graph-IFS)
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ak instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
I Interesting properties:I compute fractal dimensions (Hausdorff, packing, etc.),I topological properties (connectedness, nonempty interior, etc.)I tiling properties
I Notorious difficulty of study for specific examples/families
I Some “almost always”-type resultsI Application in number theory, dynamical systems, Fourier
analysis
I GIFS: several sets X1, . . . , Xn instead of just X (Graph-IFS)
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ak instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
I Interesting properties:I compute fractal dimensions (Hausdorff, packing, etc.),I topological properties (connectedness, nonempty interior, etc.)I tiling properties
I Notorious difficulty of study for specific examples/familiesI Some “almost always”-type results
I Application in number theory, dynamical systems, Fourieranalysis
I GIFS: several sets X1, . . . , Xn instead of just X (Graph-IFS)
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ak instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
I Interesting properties:I compute fractal dimensions (Hausdorff, packing, etc.),I topological properties (connectedness, nonempty interior, etc.)I tiling properties
I Notorious difficulty of study for specific examples/familiesI Some “almost always”-type resultsI Application in number theory, dynamical systems, Fourier
analysis
I GIFS: several sets X1, . . . , Xn instead of just X (Graph-IFS)
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ak instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
I Interesting properties:I compute fractal dimensions (Hausdorff, packing, etc.),I topological properties (connectedness, nonempty interior, etc.)I tiling properties
I Notorious difficulty of study for specific examples/familiesI Some “almost always”-type resultsI Application in number theory, dynamical systems, Fourier
analysis
I GIFS: several sets X1, . . . , Xn instead of just X (Graph-IFS)
Graph-IFS (GIFS), example 1f1(x) = x/2
f4(x) = x/2 +(1/2
1/2
)
f2(x) = x/2 +(1/2
0
)
f5(x) = x/2 +(1/2
3/4
)
f3(x) = x/2 +(0
1/2
)
f6(x) = x/2 +(3/4
3/4
){
X = f1(X) ∪ f2(X) ∪ f3(X)
∪ f4(Y )Y = f5(Y ) ∪ f6(X)
X
Y
f1f2f3
f5
f4
f6
Graph-IFS (GIFS), example 1f1(x) = x/2 f4(x) = x/2 +
(1/21/2
)f2(x) = x/2 +
(1/20
)f5(x) = x/2 +
(1/23/4
)f3(x) = x/2 +
(01/2
)f6(x) = x/2 +
(3/43/4
){X = f1(X) ∪ f2(X) ∪ f3(X) ∪ f4(Y )Y = f5(Y ) ∪ f6(X)
X Y
f1f2f3
f5
f4
f6
Graph-IFS (GIFS), example 1f1(x) = x/2 f4(x) = x/2 +
(1/21/2
)f2(x) = x/2 +
(1/20
)f5(x) = x/2 +
(1/23/4
)f3(x) = x/2 +
(01/2
)f6(x) = x/2 +
(3/43/4
){X = f1(X) ∪ f2(X) ∪ f3(X) ∪ f4(Y )Y = f5(Y ) ∪ f6(X)
X Y
f1f2f3
f5
f4
f6
Graph-IFS (GIFS), example 2: Rauzy fractalsX1 = f(X1) ∪ f(X2) ∪ f(X3)X2 = g(X1)X3 = g(X2)
X1
X3
X2
f
f
f
g g
f(x) = αx,g(x) = αx+ 1,withα = −0.419 + 0.606i ∈ Croot of x3 − x2 − x+ 1
GIFS with 3 states
Graph-IFS (GIFS), example 2: Rauzy fractalsX1 = f(X1) ∪ f(X2) ∪ f(X3)X2 = g(X1)X3 = g(X2)
X1
X3
X2
f
f
f
g g
f(x) = αx,g(x) = αx+ 1,withα = −0.419 + 0.606i ∈ Croot of x3 − x2 − x+ 1
GIFS with 3 states
Undecidability result
Theorem [J-Kari 2013]For 2D affine GIFS with 3 states (with coefficients in Q):
I = [0, 1]2 is undecidableI empty interior is undecidable
I Also true with diagonal Ai
I Computational tools: multi-tape automata
Undecidability result
Theorem [J-Kari 2013]For 2D affine GIFS with 3 states (with coefficients in Q):
I = [0, 1]2 is undecidableI empty interior is undecidable
I Also true with diagonal Ai
I Computational tools: multi-tape automata
Undecidability result
Theorem [J-Kari 2013]For 2D affine GIFS with 3 states (with coefficients in Q):
I = [0, 1]2 is undecidableI empty interior is undecidable
I Also true with diagonal Ai
I Computational tools: multi-tape automata
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
0001010001000 . . .1100011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
XX Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
0001010001000 . . .1100011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
X YY01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
000
1010001000 . . .
11
00011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
XX Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
0001
010001000 . . .
110
0011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
XX Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
000101
0001000 . . .
11000
11011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
X YY01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
000101000
1000 . . .
1100011
011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
XX Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
0001010001
000 . . .
11000110
11 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
X YY01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
0001010001000
. . .
1100011011
. . .∈ AN = ({0, 1} × {0, 1})N
Multi-tape automata
d-tape automaton:I alphabet A = A1 × · · · ×Ad
I states QI transitions Q× (A+
1 × · · · ×A+d )→ Q
I acceptance: infinite run starting from any state
X Y01 | 00 1 | 001
000 | 11
1 | 0
A = {0, 1} × {0, 1}Q = {X,Y }
Accepted infinite word starting from X:
0001010001000 . . .1100011011 . . . ∈ AN = ({0, 1} × {0, 1})N
Multi-tape automaton 7−→ GIFS
Transitionu | v
(tape alphabets A1, A2)
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated GIFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
Multi-tape automaton 7−→ GIFS
Transitionu | v
(tape alphabets A1, A2)
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)
Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated GIFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
Multi-tape automaton 7−→ GIFS
Transitionu | v
(tape alphabets A1, A2)
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated GIFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
Multi-tape automaton 7−→ GIFS
Transitionu | v
(tape alphabets A1, A2)
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)Automaton:
X Y01 | 00 1 | 001
000 | 11
1 | 0
Associated GIFS:
X Y(1/4 0
0 1/4
)(xy
)+(0.01
0.00
) (1/2 00 1/8
)(xy
)+(0.1
0.001
)(1/8 0
0 1/4
)(xy
)+(0.000
0.11
)(1/2 0
0 1/2
)(xy
)+(0.1
0.0
)
Multi-tape automaton 7−→ GIFS
Transitionu | v
(tape alphabets A1, A2)
7−→ Mapping f(xy
)=( |A1|−|u| 0
0 |A2|−|v|)(xy
)+(0.u1 . . . u|u|
0.v1 . . . v|v|
)
Automaton GIFSstates fractal setsedges contracting mappings#tapes dimensiontape i base-|Ai| representation of ith coordinate
Multi-tape automaton 7−→ GIFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)Key correspondence
GIFS fractal associated with state X
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM starting at X
}
Multi-tape automaton 7−→ GIFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)Key correspondence
GIFS fractal associated with state X
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM starting at X
}
Multi-tape automaton 7−→ GIFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)
Key correspondenceGIFS fractal associated with state X
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM starting at X
}
Multi-tape automaton 7−→ GIFS
I
1011 | 11
I f(xy
)=
(1/16 00 1/4
)(0.x1x2 . . .0.y1y2 . . .
)+(0.1011
0.11)
=(0.0000x1x2 . . .
0.00y1y2 . . .
)+(0.1011
0.11)
=(0.1011x1x2 . . .
0.11y1y2 . . .
)Key correspondence
GIFS fractal associated with state X
={(0.x1x2 . . .0.y1y2 . . .
)∈ R2 :
(x1x2 . . .y1y2 . . .
)accepted byM starting at X
}
Multi-tape automaton 7−→ GIFSExample:
0|0
1|0 0|1
12(
xy
)+(0
0
)12(
xy
)+(0
1/2
) 12(
xy
)+(1/2
0
)Automaton GIFS
={(0.x1x2 . . .
0.y1y2 . . .
): (xn, yn) 6= (1, 1), ∀n > 1
}
Multi-tape automaton 7−→ GIFSExample:
X Y
0|00|11|0
10|1110|10
11|11
Multi-tape automaton ←→ GIFS
Automaton GIFSstates fractal setsedges contracting mappings#tapes dimensionalphabet Ai base-|Ai| representation on tape i
Languages properties GIFS properties∃ configurations with = tapes Intersects the diagonal [Dube]Is universal Is equal to [0, 1]d
Has universal prefixes Has nonempty interior? Is connected? Is totally disconnectedCompute language entropy Compute fractal dimension
Multi-tape automaton ←→ GIFS
Automaton GIFSstates fractal setsedges contracting mappings#tapes dimensionalphabet Ai base-|Ai| representation on tape i
Languages properties GIFS properties∃ configurations with = tapes Intersects the diagonal [Dube]Is universal Is equal to [0, 1]d
Has universal prefixes Has nonempty interior? Is connected? Is totally disconnectedCompute language entropy Compute fractal dimension
Multi-tape automaton ←→ GIFS
Theorem [Dube 1993]For 1-state, 2D GIFS, it is undecidable if X intersects the diagonal.
Proof idea:I X ∩ {(x, x) : x ∈ [0, 1]} 6= ∅
⇐⇒ Automaton accepts a word of the form(0.x1x2 . . .
0.x1x2 . . .
)I Reduce the infinite Post Correspondence Problem:
I PCP instance (u1, v1), . . . , (un, vn)
I Automaton ui|vi
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Example (universal): 0|0, 0|1, 1|0, 1|1I Example (not universal): 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Example (universal with prefix 1): 1, 10, 00(one-dimensional; not universal)
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Example (universal): 0|0, 0|1, 1|0, 1|1
I Example (not universal): 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Example (universal with prefix 1): 1, 10, 00(one-dimensional; not universal)
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Example (universal): 0|0, 0|1, 1|0, 1|1I Example (not universal): 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Example (universal with prefix 1): 1, 10, 00(one-dimensional; not universal)
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Example (universal): 0|0, 0|1, 1|0, 1|1I Example (not universal): 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Example (universal with prefix 1): 1, 10, 00(one-dimensional; not universal)
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Language universality ⇐⇒ nonempty interior
Fact 1: M is universal ⇐⇒ X = [0, 1]d
I Example (universal): 0|0, 0|1, 1|0, 1|1I Example (not universal): 0|0, 0|1, 1|0
Fact 2: M is prefix-universal ⇐⇒ X has nonempty interior
I Example (universal with prefix 1): 1, 10, 00(one-dimensional; not universal)
f1(x) = x/2 + 1/2f2(x) = x/4f3(x) = x/4 + 1/2
Undecidability results
Theorem [J-Kari 2013]For 3-state, 2-tape automata:
I universality is undecidableI prefix-universality is undecidable
CorollaryFor 2D affine GIFS with 3 states (with coefficients in Q):
I = [0, 1]2 is undecidableI empty interior is undecidable
Proof
Prefix-PCPInput: (u1, v1), . . . , (un, vn)
Question: ∃{i1, . . . , iki1, . . . , i`
such that ui1 · · ·uik= vi1 · · · vi`
and one of the words i1 · · · ik and i1 · · · i` is a prefix of the other
Example:I (u1, v1) = (a, abb), (u2, v2) = (bb, aa)I Prefix-PCP solution: u1u2u1u1 = v1v2 = abbaa
I No PCP solution (no (ui, vi) ends with the same letter)
Reduction to (prefix-)universality of a 3-state automaton
Proof
Prefix-PCPInput: (u1, v1), . . . , (un, vn)
Question: ∃{i1, . . . , iki1, . . . , i`
such that ui1 · · ·uik= vi1 · · · vi`
and one of the words i1 · · · ik and i1 · · · i` is a prefix of the other
Example:I (u1, v1) = (a, abb), (u2, v2) = (bb, aa)I Prefix-PCP solution: u1u2u1u1 = v1v2 = abbaa
I No PCP solution (no (ui, vi) ends with the same letter)
Reduction to (prefix-)universality of a 3-state automaton
Proof
Prefix-PCPInput: (u1, v1), . . . , (un, vn)
Question: ∃{i1, . . . , iki1, . . . , i`
such that ui1 · · ·uik= vi1 · · · vi`
and one of the words i1 · · · ik and i1 · · · i` is a prefix of the other
Example:I (u1, v1) = (a, abb), (u2, v2) = (bb, aa)I Prefix-PCP solution: u1u2u1u1 = v1v2 = abbaa
I No PCP solution (no (ui, vi) ends with the same letter)
Reduction to (prefix-)universality of a 3-state automaton
Conclusion & perspectivesDecidability of nonempty interior:
IFS 2-state GIFS > 3-state GIFSdimension 1 ? ? ?
dimension > 2 ? ? Undecidable
I dimension 1: we need new toolsI Case A = kId: “Presburger brute-forcing”,
links with automatic sequences [Charlier-Leroy-Rigo]I Reachability problems: can we reinterpret these problems in
terms of matrix/vector products?I Other topological properties:
I Connectedness, ∼= disk, . . .I int(X) 6= ∅ ⇒ µ(X) > 0 ⇒ dimH(x) = d
Thank you for your attention
Conclusion & perspectivesDecidability of nonempty interior:
IFS 2-state GIFS > 3-state GIFSdimension 1 ? ? ?
dimension > 2 ? ? Undecidable
I dimension 1: we need new tools
I Case A = kId: “Presburger brute-forcing”,links with automatic sequences [Charlier-Leroy-Rigo]
I Reachability problems: can we reinterpret these problems interms of matrix/vector products?
I Other topological properties:I Connectedness, ∼= disk, . . .I int(X) 6= ∅ ⇒ µ(X) > 0 ⇒ dimH(x) = d
Thank you for your attention
Conclusion & perspectivesDecidability of nonempty interior:
IFS 2-state GIFS > 3-state GIFSdimension 1 ? ? ?
dimension > 2 ? ? Undecidable
I dimension 1: we need new toolsI Case A = kId: “Presburger brute-forcing”,
links with automatic sequences [Charlier-Leroy-Rigo]
I Reachability problems: can we reinterpret these problems interms of matrix/vector products?
I Other topological properties:I Connectedness, ∼= disk, . . .I int(X) 6= ∅ ⇒ µ(X) > 0 ⇒ dimH(x) = d
Thank you for your attention
Conclusion & perspectivesDecidability of nonempty interior:
IFS 2-state GIFS > 3-state GIFSdimension 1 ? ? ?
dimension > 2 ? ? Undecidable
I dimension 1: we need new toolsI Case A = kId: “Presburger brute-forcing”,
links with automatic sequences [Charlier-Leroy-Rigo]I Reachability problems: can we reinterpret these problems in
terms of matrix/vector products?
I Other topological properties:I Connectedness, ∼= disk, . . .I int(X) 6= ∅ ⇒ µ(X) > 0 ⇒ dimH(x) = d
Thank you for your attention
Conclusion & perspectivesDecidability of nonempty interior:
IFS 2-state GIFS > 3-state GIFSdimension 1 ? ? ?
dimension > 2 ? ? Undecidable
I dimension 1: we need new toolsI Case A = kId: “Presburger brute-forcing”,
links with automatic sequences [Charlier-Leroy-Rigo]I Reachability problems: can we reinterpret these problems in
terms of matrix/vector products?I Other topological properties:
I Connectedness, ∼= disk, . . .I int(X) 6= ∅ ⇒ µ(X) > 0 ⇒ dimH(x) = d
Thank you for your attention
Conclusion & perspectivesDecidability of nonempty interior:
IFS 2-state GIFS > 3-state GIFSdimension 1 ? ? ?
dimension > 2 ? ? Undecidable
I dimension 1: we need new toolsI Case A = kId: “Presburger brute-forcing”,
links with automatic sequences [Charlier-Leroy-Rigo]I Reachability problems: can we reinterpret these problems in
terms of matrix/vector products?I Other topological properties:
I Connectedness, ∼= disk, . . .I int(X) 6= ∅ ⇒ µ(X) > 0 ⇒ dimH(x) = d
Thank you for your attention
ReferencesI Lagarias, Wang, Self-affine tiles in Rn, Adv. Math., 1996I Wang, Self-affine tiles, in book Advances in wavelets, 1999
(great survey, available online)I Lai, Lau, Rao, Spectral structure of digit sets ofself-similar tiles on R1, Trans. Amer. Math. Soc., 2013
I Dube, Undecidable problems in fractal geometry, ComplexSystems, 1993
I Jolivet, Kari, Undecidable properties of self-affine setsand multi-tape automata, MFCS 2014, arXiv:1401.0705