The profinite theory of rational languages Laure Daviaud LIP, ENS Lyon Toulouse, 22th June 2016
The 3 reasons I am here... 2/20
The 3 reasons I am here... 1 - Topology: metric space, limits of sequences of words... 2/20
The 3 reasons I am here... 1 - Topology: metric space, limits of sequences of words... 2 - Languages: classes of rational languages... 2/20
The 3 reasons I am here... 1 - Topology: metric space, limits of sequences of words... 2 - Languages: classes of rational languages... 3 - Toulouse... 2/20
Profinite theory → a topological approach for the study of rational languages. 3/20
Profinite theory → a topological approach for the study of rational languages. A : finite alphabet 3/20
Profinite theory → a topological approach for the study of rational languages. A : finite alphabet A ∗ : set of words ⊆ L : rational language 3/20
Profinite theory → a topological approach for the study of rational languages. A : finite alphabet ( A ∗ , d ) : metric space A ∗ : set of words ⊆ ⊆ L : rational language L : subset satisfying some topological properties 3/20
Profinite theory → a topological approach for the study of rational languages. A : finite alphabet ( A ∗ , d ) : metric space A ∗ : set of words ⊆ ⊆ L : rational language L : subset satisfying some topological properties � automata � logic � rational expressions � monoids 3/20
Profinite theory → a topological approach for the study of rational languages. A : finite alphabet ( A ∗ , d ) : metric space A ∗ : set of words ⊆ ⊆ L : rational language L : subset satisfying some topological properties � automata � logic � rational expressions � monoids 3/20
Finite monoids and rational languages A few things to know about monoids... Monoid: a set with an associative operation and a neutral element. Idempotent: e 2 = e bla In a finite monoid, every element has a unique idempotent power → x | M | ! is idempotent bla x ∈ M − 4/20
Finite monoids and rational languages A few things to know about monoids... Monoid: a set with an associative operation and a neutral element. Idempotent: e 2 = e bla In a finite monoid, every element has a unique idempotent power → x | M | ! is idempotent bla x ∈ M − A monoid M recognises a language L if there is a morphism ϕ : A ∗ → M and P ⊆ M s.t. L = ϕ − 1 ( P ) . ϕ A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L P 4/20
Finite monoids and rational languages A few things to know about monoids... Monoid: a set with an associative operation and a neutral element. Idempotent: e 2 = e bla In a finite monoid, every element has a unique idempotent power → x | M | ! is idempotent bla x ∈ M − A monoid M recognises a language L if there is a morphism ϕ : A ∗ → M and P ⊆ M s.t. L = ϕ − 1 ( P ) . ϕ A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L P A language is rational iff it is recognised by a finite monoid. 4/20
Examples and syntactic monoid ϕ A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L P Example: L = { w ∈ A ∗ | | w | is even } 5/20
Examples and syntactic monoid ϕ A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L P Example: L = { w ∈ A ∗ | | w | is even } ( Z / 2 Z , +) - ϕ ( A ) = { 1 } ϕ A ∗ Z / 2 Z ⊆ ⊆ ϕ − 1 ( { 0 } ) = L { 0 } 5/20
Examples and syntactic monoid ϕ A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L P Example: L = { w ∈ A ∗ | | w | is even } ( Z / 2 Z , +) - ϕ ( A ) = { 1 } ϕ A ∗ Z / 2 Z ⊆ ⊆ ϕ − 1 ( { 0 } ) = L { 0 } Syntactic monoid: the smallest monoid recognising L . = Monoid of transitions of a minimal deterministic automaton. 5/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . 6/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . Example 1: separate u � = v ? 6/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . Example 1: separate u � = v ? Syntatic monoid of { u } (or { v } ...) 6/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . Example 1: separate u � = v ? Syntatic monoid of { u } (or { v } ...) Example 2: a ∈ A - separate a 99 and a 100 ? 6/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . Example 1: separate u � = v ? Syntatic monoid of { u } (or { v } ...) Example 2: a ∈ A - separate a 99 and a 100 ? Z / 2 Z 6/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . Example 1: separate u � = v ? Syntatic monoid of { u } (or { v } ...) Example 2: a ∈ A - separate a 99 and a 100 ? Z / 2 Z Example 3: u ∈ A ∗ , n ∈ N - separate u n ! and u ( n + 1 )! ? 6/20
Separation of words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . Example 1: separate u � = v ? Syntatic monoid of { u } (or { v } ...) Example 2: a ∈ A - separate a 99 and a 100 ? Z / 2 Z Example 3: u ∈ A ∗ , n ∈ N - separate u n ! and u ( n + 1 )! ? x ∈ M then x | M | ! = x ( | M | + 1 )! = the idempotent power of x in M ⇒ ϕ ( u ) | M | ! = ϕ ( u ) ( | M | + 1 )! = u n ! and u ( n + 1 )! cannot be separated by a monoid of size less than n 6/20
Distance over words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . 7/20
Distance over words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . d ( u , v ) = 2 − n where n is the minimal size of a monoid that separates u and v . 7/20
Distance over words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . d ( u , v ) = 2 − n where n is the minimal size of a monoid that separates u and v . d is an ultrametric distance: � d ( u , v ) = 0 iff u = v � d ( u , v ) = d ( v , u ) � d ( u , v ) � max ( d ( u , w ) , d ( w , v )) 7/20
Distance over words A monoid M separates u and v if: there is a morphism ϕ : A ∗ → M such that ϕ ( u ) � = ϕ ( v ) . d ( u , v ) = 2 − n where n is the minimal size of a monoid that separates u and v . d is an ultrametric distance: � d ( u , v ) = 0 iff u = v � d ( u , v ) = d ( v , u ) � d ( u , v ) � max ( d ( u , w ) , d ( w , v )) The words u n ! and u ( n + 1 )! are closer and closer... 7/20
Profinite monoid Definition Profinite monoid � A ∗ : completion of A ∗ with respect to the distance d . � Monoid if u and v sequences of words, ( u . v ) n = u n v n � Metric space � A ∗ dense subset � Compact 8/20
V.I.P. words (very important profinite words) Idempotent power u ω = lim n →∞ u n ! 9/20
V.I.P. words (very important profinite words) Idempotent power u ω = lim n →∞ u n ! Zero (Reilly-Zhang 2000, Almeida-Volkov 2003) | A | � 2 u 0 , u 1 , . . . an enumeration of the words of A ∗ v n + 1 = ( v n u n + 1 v n ) ( n + 1 )! v 0 = u 0 , ρ A = lim n →∞ v n 9/20
Profinite monoid and rational languages Universal property M a finite monoid. Every morphism ϕ : A ∗ → M can be uniquely extended to a ϕ : � A ∗ → M . continuous morphism � ϕ � � A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L � P 10/20
Profinite monoid and rational languages Universal property M a finite monoid. Every morphism ϕ : A ∗ → M can be uniquely extended to a ϕ : � A ∗ → M . continuous morphism � ϕ � � A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L � P 10/20
Profinite monoid and rational languages Universal property M a finite monoid. Every morphism ϕ : A ∗ → M can be uniquely extended to a ϕ : � A ∗ → M . continuous morphism � ϕ � � A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L � P 10/20
Profinite monoid and rational languages Universal property M a finite monoid. Every morphism ϕ : A ∗ → M can be uniquely extended to a ϕ : � A ∗ → M . continuous morphism � ϕ � � A ∗ M ⊆ ⊆ ϕ − 1 ( P ) = L � P A language L is rational iff L is open and closed in � A ∗ . 10/20
V.I.P. words (very important profinite words) Idempotent power u ω = lim n →∞ u n ! 11/20
V.I.P. words (very important profinite words) Idempotent power u ω = lim n →∞ u n ! → For all morphisms ϕ : A ∗ → M (finite monoid): − ϕ ( u ω ) is the idempotent power of � − → � ϕ ( u ) in M . 11/20
V.I.P. words (very important profinite words) Idempotent power u ω = lim n →∞ u n ! → For all morphisms ϕ : A ∗ → M (finite monoid): − ϕ ( u ω ) is the idempotent power of � − → � ϕ ( u ) in M . Zero (Reilly-Zhang 2000, Almeida-Volkov 2003) | A | � 2 u 0 , u 1 , . . . an enumeration of the words of A ∗ v n + 1 = ( v n u n + 1 v n ) ( n + 1 )! v 0 = u 0 , ρ A = lim n →∞ v n 11/20
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