On the structure of covers of sofic shifts Rune Johansen Department - PowerPoint PPT Presentation

On the structure of covers of sofic shifts Rune Johansen Department of Mathematical Sciences, University of Copenhagen November 26 2010

Overview Presentations of sofic shifts Irreducible sofic shifts Generalizing the Fischer cover Sofic shifts

Sofic shifts and presentations On the structure of covers of sofic shifts E = ( E 0 , E 1 , r , s ) is a finite directed graph. a is a finite set (alphabet). Presentations L : E 1 → a labels the edges. Irred. sofic shifts Layers Range result Generalized LFC Definition Properties Sofic shifts Layers

Sofic shifts and presentations On the structure of covers of sofic shifts E = ( E 0 , E 1 , r , s ) is a finite directed graph. a is a finite set (alphabet). Presentations L : E 1 → a labels the edges. Irred. sofic shifts Layers Range result The labelled graph ( E , L ) defines a sofic shift : Generalized LFC Definition Properties ( L ( x i )) i ∈ a Z | x i ∈ E 1 , r ( x i ) = s ( x i + 1 ) � � X ( E , L ) = . Sofic shifts Layers Example: The even shift 0 u v 1 0

Sofic shifts and presentations On the structure of covers of sofic shifts E = ( E 0 , E 1 , r , s ) is a finite directed graph. a is a finite set (alphabet). Presentations L : E 1 → a labels the edges. Irred. sofic shifts Layers Range result The labelled graph ( E , L ) defines a sofic shift : Generalized LFC Definition Properties ( L ( x i )) i ∈ a Z | x i ∈ E 1 , r ( x i ) = s ( x i + 1 ) � � X ( E , L ) = . Sofic shifts Layers Example: The even shift 0 0 u v u v 1 1 0 0 1 w 0

A nice presentation: The Krieger cover On the structure of covers of sofic shifts X sofic shift. Presentations X + = { x 0 x 1 x 2 . . . | x ∈ X } (right rays) Irred. sofic shifts X − = { . . . x − 3 x − 2 x − 1 | x ∈ X } Layers (left rays) Range result Generalized LFC For x + ∈ X + , define the predecessor set of x + to be Definition Properties P ∞ ( x + ) = { y − ∈ X − | y − x + ∈ X } . Sofic shifts Layers

A nice presentation: The Krieger cover On the structure of covers of sofic shifts X sofic shift. Presentations X + = { x 0 x 1 x 2 . . . | x ∈ X } (right rays) Irred. sofic shifts X − = { . . . x − 3 x − 2 x − 1 | x ∈ X } Layers (left rays) Range result Generalized LFC For x + ∈ X + , define the predecessor set of x + to be Definition Properties P ∞ ( x + ) = { y − ∈ X − | y − x + ∈ X } . Sofic shifts Layers The left Krieger cover of X is a labelled graph ( E K , L K ) K = { P ∞ ( x + ) | x + ∈ X + } , Vertices: E 0 Edges: Draw an edge labelled a ∈ a from P ∈ E 0 K to P ′ ∈ E 0 K if and only if there exists x + ∈ X + such that P = P ∞ ( ax + ) and P ′ = P ∞ ( x + ) .

A nice presentation: The Krieger cover On the structure of covers of sofic shifts X sofic shift. Presentations X + = { x 0 x 1 x 2 . . . | x ∈ X } (right rays) Irred. sofic shifts X − = { . . . x − 3 x − 2 x − 1 | x ∈ X } Layers (left rays) Range result Generalized LFC For x + ∈ X + , define the predecessor set of x + to be Definition Properties P ∞ ( x + ) = { y − ∈ X − | y − x + ∈ X } . Sofic shifts Layers The left Krieger cover of X is a labelled graph ( E K , L K ) K = { P ∞ ( x + ) | x + ∈ X + } , Vertices: E 0 Edges: Draw an edge labelled a ∈ a from P ∈ E 0 K to P ′ ∈ E 0 K if and only if there exists x + ∈ X + such that P = P ∞ ( ax + ) and P ′ = P ∞ ( x + ) . Past set cover : Use predecessor sets of words (finite factors) instead of predecessor sets of rays.

Example: Krieger cover of the even shift On the structure of covers of sofic shifts P ∞ ( 0 2 n 1 x + ) = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Presentations Irred. sofic shifts Layers Range result Generalized LFC Definition Properties Sofic shifts Layers

Example: Krieger cover of the even shift On the structure of covers of sofic shifts P 1 = P ∞ ( 0 2 n 1 x + ) = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Presentations Irred. sofic shifts Layers Range result Generalized LFC Definition Properties Sofic shifts Layers P 1

Example: Krieger cover of the even shift On the structure of covers of sofic shifts P 1 = P ∞ ( 0 2 n 1 x + ) = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Presentations P 2 = P ∞ ( 0 2 n + 1 1 x + ) = { y − 10 2 k + 1 ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Irred. sofic shifts Layers Range result Generalized LFC Definition Properties Sofic shifts Layers P 1 P 2

Example: Krieger cover of the even shift On the structure of covers of sofic shifts P 1 = P ∞ ( 0 2 n 1 x + ) = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Presentations P 2 = P ∞ ( 0 2 n + 1 1 x + ) = { y − 10 2 k + 1 ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Irred. sofic shifts Layers P 3 = P ∞ ( 0 ∞ ) = X − Range result Generalized LFC Definition Properties Sofic shifts Layers P 1 P 2 P 3

Example: Krieger cover of the even shift On the structure of covers of sofic shifts P 1 = P ∞ ( 0 2 n 1 x + ) = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Presentations P 2 = P ∞ ( 0 2 n + 1 1 x + ) = { y − 10 2 k + 1 ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Irred. sofic shifts Layers P 3 = P ∞ ( 0 ∞ ) = X − Range result Generalized LFC Definition Properties Sofic shifts Layers P 1 P 2 0 P 3 Edge: P ∞ ( 10 ∞ ) = P 1 P ∞ ( 010 ∞ ) = P 2

Example: Krieger cover of the even shift On the structure of covers of sofic shifts P 1 = P ∞ ( 0 2 n 1 x + ) = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Presentations P 2 = P ∞ ( 0 2 n + 1 1 x + ) = { y − 10 2 k + 1 ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } Irred. sofic shifts Layers P 3 = P ∞ ( 0 ∞ ) = X − Range result Generalized LFC Definition 0 Properties Sofic shifts Layers 1 P 1 P 2 0 1 P 3 0 Edge: P ∞ ( 10 ∞ ) = P 1 P ∞ ( 010 ∞ ) = P 2

Irred. sofic shifts and the Fischer cover On the structure of covers of sofic shifts Presentations For now : Irred. sofic shifts Assume that X is irreducible , i.e. there exists an irreducible Layers Range result (transitive, strongly connected) presentation of X. Generalized LFC Definition Properties A presentation ( E , L ) of X is left-resolving if no vertex in Sofic shifts E 0 receives two edges with the same label. Layers

Irred. sofic shifts and the Fischer cover On the structure of covers of sofic shifts Presentations For now : Irred. sofic shifts Assume that X is irreducible , i.e. there exists an irreducible Layers Range result (transitive, strongly connected) presentation of X. Generalized LFC Definition Properties A presentation ( E , L ) of X is left-resolving if no vertex in Sofic shifts E 0 receives two edges with the same label. Layers Theorem (Fischer) There is a unique minimal left-resolving presentation ( E K , L K ) of X when X is irreducible. This presentation is the left Fischer cover of X.

Layers in the Krieger cover On the structure of covers of sofic shifts Presentations For v ∈ E 0 F define P ∞ ( v ) to be the set of left rays which Irred. sofic shifts have a presentation terminating at v . Layers Range result For x + ∈ X + define S ( x + ) to be the set of vertices in E 0 Generalized LFC F Definition that are sources of presentations of x + . Properties Sofic shifts Note: P ∞ ( x + ) = ∪ v ∈ S ( x + ) P ∞ ( v ) Layers

Layers in the Krieger cover On the structure of covers of sofic shifts Presentations For v ∈ E 0 F define P ∞ ( v ) to be the set of left rays which Irred. sofic shifts have a presentation terminating at v . Layers Range result For x + ∈ X + define S ( x + ) to be the set of vertices in E 0 Generalized LFC F Definition that are sources of presentations of x + . Properties Sofic shifts Note: P ∞ ( x + ) = ∪ v ∈ S ( x + ) P ∞ ( v ) Layers A vertex P ∞ ( x + ) ∈ E 0 K is in the n th layer of the left Krieger cover if n is the smallest number such that there exist v 1 , . . . , v n ∈ E 0 F with P ∞ ( x + ) = P ∞ ( v 1 ) ∪ · · · ∪ P ∞ ( v n ) . x + is said to be 1 / n-synchronizing . Same definition can be used for the past set cover .

Example: The even shift On the structure of covers of sofic shifts P 1 = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } = P ∞ ( u ) P 2 = { y − 10 2 k + 1 ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } = P ∞ ( v ) Presentations P 3 = X − = P ∞ ( u ) ∪ P ∞ ( v ) Irred. sofic shifts Layers Range result Left Fischer cover and left Krieger cover: Generalized LFC Definition Properties Sofic shifts ( E F , L F ) ( E K , L K ) Layers 0 0 u v 1 1 P 1 P 2 0 0 1 P 3 0

Example: The even shift On the structure of covers of sofic shifts P 1 = { y − 10 2 k ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } = P ∞ ( u ) P 2 = { y − 10 2 k + 1 ∈ X − | k ∈ N 0 } ∪ { 0 ∞ } = P ∞ ( v ) Presentations P 3 = X − = P ∞ ( u ) ∪ P ∞ ( v ) Irred. sofic shifts Layers Range result Left Fischer cover and left Krieger cover: Generalized LFC Definition Properties Sofic shifts ( E F , L F ) ( E K , L K ) Layers 0 0 u v 1 1 P ∞ ( u ) P ∞ ( v ) 0 0 1 P ∞ ( u ) ∪ P ∞ ( v ) 0

Theorem (Krieger) On the structure of covers of sofic shifts The left Fischer cover is (isomorphic as a labelled graph to) the first layer of the left Krieger cover. Presentations Irred. sofic shifts Proof. Layers Range result Identify u ∈ E 0 F with P ∞ ( u ) ∈ E 0 K . Generalized LFC Definition Properties Sofic shifts Layers