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