Breadth-first signature of trees and rational languages Victor - PowerPoint PPT Presentation

Breadth-first signature of trees and rational languages Victor Marsault, joint work with Jacques Sakarovitch CNRS / Telecom-ParisTech, Paris, France Developments in Language Theory 2014, Ekateringburg, 2014–08–30

Breadth-first serialisation of languages and numeration systems: The rational case Victor Marsault, joint work with Jacques Sakarovitch CNRS / Telecom-ParisTech, Paris, France Developments in Language Theory 2014, Ekateringburg, 2014–08–30

Outline 1 1 Signature of trees and of languages 2 Substitutive signatures and finite automata 3 A word on numeration system

We call tree a... 1 Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.)

We call tree a... 1 Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.) � =

We call tree a... 1 Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.)

We call tree a... 1 Directed graph which is Rooted: a node is called the root (leftmost in the figures) Directed outward from the root: there is a unique path from the root to every other node. Ordered: the children of every node are ordered (In the figures, lower children are smaller.) � =

Every tree has a canonical breadth-first traversal 2 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13

Two more features 3 We consider infinite trees only. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13

Two more features 3 We consider infinite trees only. For convenience, there is loop on the root. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s =

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1 2 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1 2 2 1

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1 2 2 1 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1 2 2 1 2 2

Signature of a tree 4 Definition The signature of a tree is the sequence of the degrees of the nodes taken in breadth-first order. 12 20 4 7 19 11 18 0 1 2 17 6 10 16 3 9 15 5 14 8 13 s = 2 1 2 2 1 2 1 2 2 1 2 2 1 · · ·

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 0

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 0 1

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 2 0 1

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 2 0 1 3

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 2 0 4 1 3

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 5 2 0 4 1 3

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 5 2 0 4 1 3 6

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 5 2 0 4 1 3 7 6

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 5 2 0 4 1 8 3 7 6

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 5 2 9 0 4 1 8 3 7 6

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 10 5 2 9 0 4 1 8 3 7 6

The signature is characteristic of a tree 5 s = ( 3 2 1 ) ω 11 10 5 2 9 0 4 1 8 3 7 6

Prefix-closed languages and labelled trees 6 Alphabets are ordered hence prefix-closed languages = labelled trees. 12 0 20 1 4 0 7 19 0 1 11 0 1 18 0 1 1 0 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 13 Figure : Integer representations in the Fibonacci numeration system.

Prefix-closed languages and labelled trees 6 Alphabets are ordered hence prefix-closed languages = labelled trees. 12 0 20 1 4 0 7 19 0 1 11 0 1 18 1 0 0 1 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 5 = F 4 13 Figure : Integer representations in the Fibonacci numeration system.

Prefix-closed languages and labelled trees 6 Alphabets are ordered hence prefix-closed languages = labelled trees. 12 0 20 1 0 4 7 19 0 1 11 1 0 18 1 0 0 1 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 7 = 5 + 2 = F 4 + F 2 13 Figure : Integer representations in the Fibonacci numeration system.

Serialisation of a prefix-closed language 7 Definition The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order. 12 0 20 1 4 0 7 19 0 1 11 0 1 18 0 1 1 0 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 13 s = λ =

Serialisation of a prefix-closed language 7 Definition The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order. 12 0 20 1 4 0 7 19 0 1 11 0 1 18 1 0 1 0 2 17 1 6 0 10 0 0 16 1 0 3 9 0 15 0 1 5 14 0 1 8 0 13 s = 2 λ =01

Serialisation of a prefix-closed language 7 Definition The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order. 12 0 20 1 4 0 7 19 0 1 11 0 1 18 0 0 1 1 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 13 s = 2 1 λ =01 0

Serialisation of a prefix-closed language 7 Definition The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order. 12 0 20 1 4 0 7 19 0 1 11 1 0 18 0 1 1 0 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 13 s = 2 1 2 λ =01 0 01

Serialisation of a prefix-closed language 7 Definition The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order. 12 0 20 1 4 0 7 19 0 1 11 0 1 18 0 1 1 0 2 17 1 6 0 10 0 0 16 1 0 3 9 0 15 0 1 5 14 0 1 8 0 13 s = 2 1 2 2 λ =01 0 01 01

Serialisation of a prefix-closed language 7 Definition The labelling of a language is the sequence of arc labels of its transitions taken in breadth-first order. 12 0 20 1 0 4 7 19 0 1 11 0 1 18 0 1 1 0 2 17 1 6 0 10 0 0 16 0 1 3 9 0 15 0 1 5 14 0 1 8 0 13 s = 2 1 2 2 1 λ =01 0 01 01 0