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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
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Example: the Cantor set
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Example: the Cantor set
X⊆[0,1]︷ ︸︸ ︷
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Example: the Cantor set
X⊆[0,1]︷ ︸︸ ︷︸ ︷︷ ︸13 X
︸ ︷︷ ︸13 X+ 2
3
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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
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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
![Page 7: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/7.jpg)
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
![Page 8: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/8.jpg)
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
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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
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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
![Page 11: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/11.jpg)
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
![Page 12: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/12.jpg)
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
![Page 13: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/13.jpg)
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
![Page 14: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/14.jpg)
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
![Page 15: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/15.jpg)
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
![Page 16: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/16.jpg)
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
![Page 17: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/17.jpg)
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
![Page 18: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/18.jpg)
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
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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
![Page 20: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/20.jpg)
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
![Page 21: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/21.jpg)
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
![Page 22: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/22.jpg)
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
![Page 23: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/23.jpg)
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
![Page 24: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/24.jpg)
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
![Page 25: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/25.jpg)
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
![Page 26: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/26.jpg)
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
![Page 27: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/27.jpg)
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
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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
![Page 29: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/29.jpg)
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
![Page 30: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/30.jpg)
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
}
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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
}
![Page 32: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/32.jpg)
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
}
![Page 33: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/33.jpg)
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
}
![Page 34: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/34.jpg)
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))
![Page 35: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/35.jpg)
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))
![Page 36: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/36.jpg)
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
![Page 37: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/37.jpg)
Sierpiński variant 1I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 1
1)}
![Page 38: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/38.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 39: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/39.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 40: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/40.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 41: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/41.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 42: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/42.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 43: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/43.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 44: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/44.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 45: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/45.jpg)
Sierpiński variant 2I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),(−1−1)}
![Page 46: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/46.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 47: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/47.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 48: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/48.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 49: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/49.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 50: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/50.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 51: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/51.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 52: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/52.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 53: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/53.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 54: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/54.jpg)
Sierpiński variant 3I A =
( 2 00 2)
I D ={( 0
0),( 1
0),( 0
1),( 3
1)}
![Page 55: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/55.jpg)
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
![Page 56: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/56.jpg)
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
![Page 57: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/57.jpg)
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
![Page 58: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/58.jpg)
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
![Page 59: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/59.jpg)
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior (why?)
![Page 60: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/60.jpg)
Nonstandard D
A =( 2 0
0 2)
D ={( 0
0),( 1
0),( 0
1),( 1
2)}
X has empty interior (why?)
![Page 61: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/61.jpg)
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)}
![Page 62: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/62.jpg)
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)}
![Page 63: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/63.jpg)
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)}
![Page 64: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/64.jpg)
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)}
![Page 65: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/65.jpg)
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)
![Page 66: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/66.jpg)
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
![Page 67: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/67.jpg)
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
![Page 68: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/68.jpg)
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
![Page 69: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/69.jpg)
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
![Page 70: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/70.jpg)
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?
![Page 71: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/71.jpg)
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?
![Page 72: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/72.jpg)
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?
![Page 73: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/73.jpg)
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?
![Page 74: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/74.jpg)
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?
![Page 75: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/75.jpg)
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?
![Page 76: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/76.jpg)
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?
![Page 77: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/77.jpg)
Beyond self-affine tiles: affine IFSI Several contractions A1, . . . , Ai instead of just AI Arbitrary affine mappings fi(x) = Aix+ vi
![Page 78: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/78.jpg)
Beyond self-affine tiles: affine IFS
![Page 79: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/79.jpg)
Beyond self-affine tiles: affine IFS
![Page 80: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/80.jpg)
Beyond self-affine tiles: affine IFS
![Page 81: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/81.jpg)
Beyond self-affine tiles: affine IFS
![Page 82: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/82.jpg)
Beyond self-affine tiles: affine IFS
![Page 83: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/83.jpg)
Beyond self-affine tiles: affine IFS
![Page 84: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/84.jpg)
Beyond self-affine tiles: affine IFS
![Page 85: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/85.jpg)
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)
![Page 86: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/86.jpg)
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)
![Page 87: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/87.jpg)
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)
![Page 88: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/88.jpg)
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)
![Page 89: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/89.jpg)
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)
![Page 90: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/90.jpg)
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)
![Page 91: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/91.jpg)
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
![Page 92: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/92.jpg)
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
![Page 93: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/93.jpg)
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
![Page 94: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/94.jpg)
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
![Page 95: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/95.jpg)
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
![Page 96: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/96.jpg)
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
![Page 97: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/97.jpg)
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
![Page 98: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/98.jpg)
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
![Page 99: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/99.jpg)
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
![Page 100: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/100.jpg)
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
![Page 101: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/101.jpg)
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
![Page 102: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/102.jpg)
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
![Page 103: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/103.jpg)
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
![Page 104: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/104.jpg)
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
![Page 105: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/105.jpg)
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
![Page 106: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/106.jpg)
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
![Page 107: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/107.jpg)
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
![Page 108: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/108.jpg)
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
)
![Page 109: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/109.jpg)
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
)
![Page 110: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/110.jpg)
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
)
![Page 111: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/111.jpg)
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
)
![Page 112: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/112.jpg)
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
![Page 113: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/113.jpg)
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
}
![Page 114: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/114.jpg)
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
}
![Page 115: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/115.jpg)
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
}
![Page 116: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/116.jpg)
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
}
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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
}
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Multi-tape automaton 7−→ GIFSExample:
X Y
0|00|11|0
10|1110|10
11|11
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 135: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/135.jpg)
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
![Page 136: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/136.jpg)
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
![Page 137: Undecidable properties of self-affine sets and multitape ...jolivet.org/timo/docs/slides_mfcs.pdf · Wang, Undecidablepropertiesofself-affinesets](https://reader034.vdocuments.net/reader034/viewer/2022052009/601e4d9f94f409607b042d8f/html5/thumbnails/137.jpg)
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