Multidimensional morphic words, embedding types and some decidability

problems Ivan Mitrofanov

Moscow State University

January 17, 2013, CIRM

Preliminaries on multidimensional words

Let A be a finite alphabet.An array in (or a d-dimensional infinite word) is a map

a b ab b c

c a a

a

b

a

cc

a

a

... ...

......

a b c a bcb ……

d = 1 d = 2

b b b

bb

b

can be viewed as a tiling of .

1/10

Preliminaries on multidimensional words

a b ab b c

c a a

a

b

a

cc

a

a

... ...

......

a b c a bcb ……

d = 1 d = 2

b b b

bb

b

A d-dimensional picture (or a rectangular word) is a map d-tuple ) is called size of x.We denote the set of all d-dimensional pictures .

c a aa b a = ; size() = (3; 2)

size() = 3 size() = 2

Let A be a finite alphabet.An array in (or a d-dimensional infinite word) is a map

can be viewed as a tiling of .

1/10

Rectangular words and morphisms

A d-dimensional morphism is a map . If it is called a substitution

2/10

Rectangular words and morphisms

A d-dimensional morphism is a map . If it is called a substitution Suppose x to be a rectangular word. Image is said to be well-defined if images of adjacent letters can be arranged in the same order side-to-side without overlaps and holes.More precisely, if two letters and have the same i-th position in , and should have the same i-th component of size.

Example (from Charlier, Karki, and Rigo article)

Consider the map given by →a ab da ; →

cbb ; →a ac ; →dd .

If 𝑥=¿a bc d , then is well-defined and given by 𝜑 (𝑥)=¿

a a cb d ca a d

.

2/10

Rectangular words and morphisms

Suppose x to be a rectangular word. Image is said to be well-defined if images of adjacent letters can be arranged in the same order side-to-side without overlaps and holes.More precisely, if two letters and have the same i-th position in , and should have the same i-th component of size.

Example (from Charlier, Karki, and Rigo article)

Consider the map given by →a ab da ; →

cbb ; →a ac ; →dd .

If 𝑥=¿a bc d , then is well-defined and given by 𝜑 (𝑥)=¿

a a cb d ca a d

.

But is not well-defined. If is well-defined for any integer k and letter , is well-defined for alphabet . This property of a substitution can be algorithmically checked.

A d-dimensional morphism is a map . If it is called a substitution

2/10

Primitive morphic systems

In further: is well defined for is such that is well defined for and all integer k. The set of all rectangle subwords of words is called the language of morphic system Morphic system is said to be primitive if such that iteration contains all letters of .NB If we consider language of a primitive system, we may consider h is non-erasing.

There exists an infinite d-dimensional word T) such that the set of finite subwords of is language of .Problem To find the translation group (subgroup in ) for given and .

0 0 1 1 1 1 0

1 1 1 1 0 0 1

1 1 0 0 1 1 0

0 0 1 1 0 0 1

1 1 0 0 1 1 1

3/10

Primitive morphic systems – some properties

Lemma Morphic system is primitive there exist d-tuple of reals and such that holds true for all integer k. In further we call the d-tuple the growth type of system and denote , ,…, ) as .We suppose for all

4/10

Primitive morphic systems – some properties

Lemma Morphic system is primitive there exist d-tuple of reals and such that holds true for all integer k. In further we call the d-tuple the growth type of system and denote , ,…, ) as .We suppose for all

3 41 21→

4 32 12→

1 23→ 2 14→

𝑒=(2 ; 1+√52

)

4/10

Primitive morphic systems – some properties

Lemma Suppose and language of primitive morphic system and there exist such that , . Then is a subword of.

is a number depending on and and it can be algorithmically estimated)

Lemma Morphic system is primitive there exist d-tuple of reals and such that holds true for all integer k. In further we call the d-tuple the growth type of system and denote , ,…, ) as .We suppose for all

3 41 21→

4 32 12→

1 23→ 2 14→

𝑒=(2 ; 1+√52

)

4/10

Embedding types

0 0 1 0 1 0 01 1 1 1 1 0 11 1 0 1 1 1 01 1 0 0 1 1 00 1 1 0 0 1 00 1 1 1 0 0 11 1 1 1 0 1 0

1 1 10 1 10 0 1

1 11 11 1

Suppose we have d-dimensional picture and set of pictures ..,

𝑋

𝒙𝟏 𝒙𝟐

1. Find all the occurrences of in

5/10

Embedding types

1 1 10 1 10 0 1

1 11 11 1

Suppose we have d-dimensional picture and set of pictures ..,

𝑋

𝒙𝟏 𝒙𝟐

1. Find all the occurrences of in .2. For each occurrence find vertices.

0 0 1 0 1 0 01 1 1 1 1 0 11 1 0 1 1 1 01 1 0 0 1 1 00 1 1 0 0 1 00 1 1 1 0 0 11 1 1 1 0 1 0

5/10

Embedding types

1 1 10 1 10 0 1

1 11 11 1

Suppose we have d-dimensional picture and set of pictures ..,

𝑋

𝒙𝟏 𝒙𝟐

1. Find all the occurrences of in .2. For each occurrence find vertices.3. Numerate these vertices (way of numeration is

not important).

0 0 1 0 1 0 01 1 1 1 1 0 11 1 0 1 1 1 01 1 0 0 1 1 00 1 1 0 0 1 00 1 1 1 0 0 11 1 1 1 0 1 0

5/10

Embedding types

1 1 10 1 10 0 1

1 11 11 1

Suppose we have d-dimensional picture and set of pictures ..,

𝑋

𝒙𝟏 𝒙𝟐

1. Find all the occurrences of in .2. For each occurrence find vertices.3. Numerate these vertices (way of numeration is

not important).4. For each write down its occurrences encoded by

vertices.

0 0 1 0 1 0 01 1 1 1 1 0 11 1 0 1 1 1 01 1 0 0 1 1 00 1 1 0 0 1 00 1 1 1 0 0 11 1 1 1 0 1 0

1

13

7 9

14

1211

54

32

86 10

5/10

Embedding types

1 1 10 1 10 0 1

1 11 11 1

Suppose we have d-dimensional picture and set of pictures ..,

𝑋

𝒙𝟏 𝒙𝟐

1. Find all the occurrences of in .2. For each occurrence find vertices.3. Numerate these vertices (way of numeration is

not important).4. For each right down its occurrences encoded by

vertices.

0 0 1 0 1 0 01 1 1 1 1 0 11 1 0 1 1 1 01 1 0 0 1 1 00 1 1 0 0 1 00 1 1 1 0 0 11 1 1 1 0 1 0

1

13

7 9

14

1211

54

32

86 10

This is embedding type .Two embedding types are equal iff numbers of vertices are permuted.

:(2,3,10,8), (4,5,12,11):(1,2,8,6), (7,9,14,13)

5/10

Embedding types

1 1 10 1 10 0 1

1 11 11 1

Suppose we have d-dimensional picture and set of pictures ..,

𝑋

𝒙𝟏 𝒙𝟐

1. Find all the occurrences of in .2. For each occurrence find vertices.3. Numerate these vertices (way of numeration is

not important).4. For each right down its occurrences encoded by

vertices.

0 0 1 0 1 0 01 1 1 1 1 0 11 1 0 1 1 1 01 1 0 0 1 1 00 1 1 0 0 1 00 1 1 1 0 0 11 1 1 1 0 1 0

1

13

7 9

14

1211

54

32

86 10

This is embedding type .Two embedding types are equal iff numbers of vertices are permuted.

:(2,3,10,8), (4,5,12,11):(1,2,8,6), (7,9,14,13)

:(1,3,10,8), (4,5,12,11):(2,1,8,6), (7,9,14,13)

5/10

Periodicity of embedding types

Let and be two (ordered) sets of d-dimensional pictures.The embedding type , ) is the ordered set of embedding types, , ), , , ), .., , , ).We will call pictures big and pictures small. Theorem Suppose is a primitive morphic system. If is trivial, then there exists integer N such that sequence is ultimately periodic.

6/10

Periodicity of embedding types

Let and be two (ordered) sets of d-dimensional pictures.The embedding type , ) is the ordered set of embedding types, , ), , , ), .., , , ).We will call pictures big and pictures small.

Algorithm Input: Primitive morphic system .Output: 1. Answer for the question “Does there exist a vector of periodicity for T()”2. Vector of periodicity (in case of positive answer).

Theorem Suppose is a primitive morphic system. If is trivial, then there exists integer N such that sequence is ultimately periodic.

6/10

Extra embedding types

Suppose is a d-dimensional picture. We will denote as . is the set of words in language of , which size is .

In two-dimensional case: ,

,

,

In order to construct 1. Find

7/10

Extra embedding types

Suppose is a d-dimensional picture. We will denote as . is the set of words in language of , which size is .

In two-dimensional case: ,

,

,

In order to construct 1. Find

2. Picture has two parts:

and

ab

a

b

𝑘1(𝑔¿¿2)¿

𝑘1(𝑔¿¿1)¿

𝑘1(𝑔¿¿3)¿

7/10

Extra embedding types

Suppose is a d-dimensional picture. We will denote as . is the set of words in language of , which size is .

In two-dimensional case: ,

,

,

In order to construct 1. Find

2. Picture has two parts:

and

3. Correspondence between points.

ab

a

b

𝑘1(𝑔¿¿2)¿

𝑘1(𝑔¿¿1)¿

𝑘1(𝑔¿¿3)¿

7/10

Extra embedding types

Suppose is a d-dimensional picture. We will denote as . is the set of words in language of , which size is .

In two-dimensional case: ,

,

,

In order to construct 1. Find

2. Picture has two parts:

and

3. Correspondence between points. Example: point 5 in is situated in bottom part and corresponds . to point 1 in

ab

a

b1

5

𝑘1(𝑔¿¿2)¿

𝑘1(𝑔¿¿1)¿

𝑘1(𝑔¿¿3)¿

7/10

Extra embedding types (2)

Lemma is defined by Lemma If then is defined by

8/10

Extra embedding types (2)

Lemma is defined by Lemma If then is defined by

𝑘1(𝑔¿¿3)¿

8/10

Extra embedding types (2)

Lemma is defined by Lemma If then is defined by

𝑘1(𝑔¿¿3)¿

Each block is for some

8/10

Extra embedding types (2)

Lemma is defined by Lemma If then is defined by

𝑘1(𝑔¿¿3)¿

Each block is for some

8/10

Extra embedding types (2)

Lemma is defined by Lemma If then is defined by

𝑘1(𝑔¿¿3)¿

Each block is for some

Find and so on.

Lemma We can find (algorithmically) such that if (for some and in this sequence) contains more than points then is not trivial.

8/10

Extra embedding types (3)

… and so on wait until

Fall into circle (periodicity of types, is

trivial)

has more than points for some has two close occurrences and their shift vector is a vector of periodicity.

9/10

Translation group

Induction on dimension.standard basis: Suppose translation group, are linearly independent.WLOG is basis in .

Question: does the translation group intersect ?

We construct a (d-k)-dimensional primitive morphic system such that .

10/10

Translation group

Induction on dimension.standard basis: Suppose translation group, are linearly independent.WLOG is basis in .

Question: does the translation group intersect ?

We construct a (d-k)-dimensional primitive morphic system such that .

Thank you!

10/10