Multidimensional morphic words embedding types and some decidability

Multidimensional morphic words, embedding types and some decidability problems Ivan Mitrofanov Moscow State University January 17, 2013, CIRM

Preliminaries on multidimensional words 1/10 Let A be a finite alphabet. d=2 d=1 a b c b … . . . c a a b a c b b b c c b a a b b . . . … . . .

Preliminaries on multidimensional words 1/10 Let A be a finite alphabet. d=2 d=1 a b c a a a b c b … . . . c a a b a c b b b c c b a a b b . . . … . . .

Rectangular words and morphisms 2/10

Rectangular words and morphisms 2/10 Example (from Charlier, Karki, and Rigo article) a If a b c d a a b d ; b c b ; c a a ; d a a c b d c a a d d . .

Rectangular words and morphisms 2/10 Example (from Charlier, Karki, and Rigo article) a If a b c d a a b d ; b c b ; c a a ; d a a c b d c a a d d . .

Primitive morphic systems 0 0 1 1 1 1 0 0 0 1 1 0 0 1 1 1 3/10

Primitive morphic systems – some properties 4/10

Primitive morphic systems – some properties 4/10 1 2 3 1 2 4 3 2 1 4 2 3 4 1 2 1

Primitive morphic systems – some properties 4/10 1 2 3 1 2 4 3 2 1 4 2 3 4 1 2 1

Embedding types 5/10 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0

Embedding types 5/10 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0

Embedding types 5/10 0 0 1 1 1 1 1 0 0 1 1 0 0 0 1 1 1 0 0 1 1 1 1 1 0

Embedding types 1 6 0 0 1 1 1 7 1 2 8 1 0 1 1 0 0 4 9 3 10 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 14 1 0 1 1 13 11 5/10 1 5 12 0 1 0

Embedding types 1 6 0 0 1 1 1 7 1 2 8 1 0 1 1 0 0 4 9 3 10 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 14 1 0 1 1 13 11 5/10 1 5 12 0 1 0

Embedding types 1 6 0 0 1 1 1 7 1 2 8 1 0 1 1 0 0 4 9 3 10 0 1 1 1 1 0 0 0 1 1 1 0 0 1 1 14 1 0 1 1 13 11 5/10 1 5 12 0 1 0

Periodicity of embedding types 6/10

Periodicity of embedding types 6/10

Extra embedding types , 7/10 , ,

Extra embedding types 7/10 , , , a a b b

Extra embedding types 7/10 , , , a a b b

Extra embedding types 7/10 , , , a a b b 5 1

Extra embedding types (2) 8/10

Extra embedding types (2) 8/10

Extra embedding types (2) 8/10

Extra embedding types (2) 8/10

Extra embedding types (2) 8/10

Extra embedding types (3) 9/10

Translation group 10/10

Translation group 10/10 Thank you!