The profinite theory of rational languages Laure Daviaud LIP, ENS - - PowerPoint PPT Presentation

The profinite theory of rational languages

Laure Daviaud

LIP, ENS Lyon

Toulouse, 22th June 2016

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

The 3 reasons I am here...

→ 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∗: set of words ⊆ L: rational language (A∗, d): metric space ⊆ L: subset satisfying some topological properties

3/20

Profinite theory

→ a topological approach for the study of rational languages. A: finite alphabet A∗: set of words ⊆ L: rational language (A∗, d): metric space ⊆ 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∗: set of words ⊆ L: rational language (A∗, d): metric space ⊆ L: subset satisfying some topological properties

automata logic rational expressions monoids

3/20

Profinite theory

Monoid: a set with an associative operation and a neutral element. Idempotent: e2 = e bla In a finite monoid, every element has a unique idempotent power bla x ∈ M − → x|M|! is idempotent

A few things to know about monoids...

4/20

Finite monoids and rational languages

Monoid: a set with an associative operation and a neutral element. Idempotent: e2 = e bla In a finite monoid, every element has a unique idempotent power bla x ∈ M − → x|M|! is idempotent

A few things to know about monoids...

A monoid M recognises a language L if there is a morphism ϕ : A∗ → M and P ⊆ M s.t. L = ϕ−1(P). A∗ M ϕ ⊆ ⊆ ϕ−1(P) = L P

4/20

Finite monoids and rational languages

Monoid: a set with an associative operation and a neutral element. Idempotent: e2 = e bla In a finite monoid, every element has a unique idempotent power bla x ∈ M − → x|M|! is idempotent

A few things to know about monoids...

A monoid M recognises a language L if there is a morphism ϕ : A∗ → M and P ⊆ M s.t. L = ϕ−1(P). A∗ M ϕ ⊆ ⊆ ϕ−1(P) = L P A language is rational iff it is recognised by a finite monoid.

4/20

Finite monoids and rational languages

A∗ M ϕ ⊆ ⊆ ϕ−1(P) = L P Example: L = {w ∈ A∗ | |w| is even}

5/20

Examples and syntactic monoid

A∗ M ϕ ⊆ ⊆ ϕ−1(P) = L P Example: L = {w ∈ A∗ | |w| is even} (Z/2Z, +) - ϕ(A) = {1} A∗ Z/2Z ϕ ⊆ ⊆ ϕ−1({0}) = L {0}

5/20

Examples and syntactic monoid

A∗ M ϕ ⊆ ⊆ ϕ−1(P) = L P Example: L = {w ∈ A∗ | |w| is even} (Z/2Z, +) - ϕ(A) = {1} A∗ Z/2Z ϕ ⊆ ⊆ ϕ−1({0}) = L {0} Syntactic monoid: the smallest monoid recognising L. = Monoid of transitions of a minimal deterministic automaton.

5/20

Examples and syntactic monoid

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 a99 and a100?

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 a99 and a100? Z/2Z

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 a99 and a100? Z/2Z Example 3: u ∈ A∗, n ∈ N - separate un! 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 a99 and a100? Z/2Z Example 3: u ∈ A∗, n ∈ N - separate un! 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)! un! and u(n+1)! cannot be separated by a monoid of size less than n

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).

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 un! and u(n+1)! are closer and closer...

7/20

Distance over words

Profinite monoid A∗ : completion of A∗ with respect to the distance d.

Definition

Monoid if u and v sequences of words, (u.v)n = unvn Metric space A∗ dense subset Compact

8/20

Profinite monoid

Idempotent power

uω = lim

n→∞ un!

9/20

V.I.P. words (very important profinite words)

Idempotent power

uω = lim

n→∞ un!

Zero (Reilly-Zhang 2000, Almeida-Volkov 2003) |A| 2 u0, u1, . . . an enumeration of the words of A∗ v0 = u0, vn+1 = (vnun+1vn)(n+1)!

ρA = lim

n→∞ vn

9/20

V.I.P. words (very important profinite words)

Universal property M a finite monoid. Every morphism ϕ : A∗ → M can be uniquely extended to a continuous morphism ϕ : A∗ → M.

• A∗

M

• ϕ

⊆ ⊆

• 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 continuous morphism ϕ : A∗ → M.

• A∗

M

• ϕ

⊆ ⊆

• 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 continuous morphism ϕ : A∗ → M.

• A∗

M

• ϕ

⊆ ⊆

• 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 continuous morphism ϕ : A∗ → M.

• A∗

M

• ϕ

⊆ ⊆

• ϕ−1(P) = L

P A language L is rational iff L is open and closed in A∗.

10/20

Profinite monoid and rational languages

Idempotent power

uω = lim

n→∞ un!

11/20

V.I.P. words (very important profinite words)

Idempotent power

uω = lim

n→∞ un!

− → For all morphisms ϕ : A∗ → M (finite monoid): − → ϕ(uω) is the idempotent power of ϕ(u) in M.

11/20

V.I.P. words (very important profinite words)

Idempotent power

uω = lim

n→∞ un!

− → For all morphisms ϕ : A∗ → M (finite monoid): − → ϕ(uω) is the idempotent power of ϕ(u) in M. Zero (Reilly-Zhang 2000, Almeida-Volkov 2003) |A| 2 u0, u1, . . . an enumeration of the words of A∗ v0 = u0, vn+1 = (vnun+1vn)(n+1)!

ρA = lim

n→∞ vn

11/20

V.I.P. words (very important profinite words)

Idempotent power

uω = lim

n→∞ un!

− → For all morphisms ϕ : A∗ → M (finite monoid): − → ϕ(uω) is the idempotent power of ϕ(u) in M. Zero (Reilly-Zhang 2000, Almeida-Volkov 2003) |A| 2 u0, u1, . . . an enumeration of the words of A∗ v0 = u0, vn+1 = (vnun+1vn)(n+1)!

ρA = lim

n→∞ vn

− → For all morphisms ϕ : A∗ → M (finite monoid): − → if M has a zero then ϕ(ρA) = 0.

11/20

V.I.P. words (very important profinite words)

Birkhoff variety of monoids: class of monoids closed under: direct product submonoid quotient N quotient of M: M

ϕ

− → N with ϕ a surjective morphism.

12/20

Study of classes of rational languages

Birkhoff variety of monoids: class of monoids closed under: direct product submonoid quotient N quotient of M: M

ϕ

− → N with ϕ a surjective morphism. A monoid T satisfies a word-identity u = v with u, v ∈ A∗, if for all morphisms ϕ : A∗ → T, ϕ(u) = ϕ(v).

12/20

Study of classes of rational languages

Birkhoff variety of monoids: class of monoids closed under: direct product submonoid quotient N quotient of M: M

ϕ

− → N with ϕ a surjective morphism. A monoid T satisfies a word-identity u = v with u, v ∈ A∗, if for all morphisms ϕ : A∗ → T, ϕ(u) = ϕ(v). Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff]

12/20

Study of classes of rational languages

Birkhoff variety of monoids: class of monoids closed under: direct product submonoid quotient N quotient of M: M

ϕ

− → N with ϕ a surjective morphism. A monoid T satisfies a word-identity u = v with u, v ∈ A∗, if for all morphisms ϕ : A∗ → T, ϕ(u) = ϕ(v). Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff] Pseudovariety of finite monoids: class of finite monoids closed under: finite direct product submonoid quotient

12/20

Study of classes of rational languages

Birkhoff variety of monoids: class of monoids closed under: direct product submonoid quotient N quotient of M: M

ϕ

− → N with ϕ a surjective morphism. A monoid T satisfies a word-identity u = v with u, v ∈ A∗, if for all morphisms ϕ : A∗ → T, ϕ(u) = ϕ(v). Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff] Pseudovariety of finite monoids: class of finite monoids closed under: finite direct product submonoid quotient Varieties of finite monoids ← → varieties of rational languages [Eilenberg]

12/20

Study of classes of rational languages

Birkhoff variety of monoids: class of monoids closed under: direct product submonoid quotient N quotient of M: M

ϕ

− → N with ϕ a surjective morphism. A monoid T satisfies a word-identity u = v with u, v ∈ A∗, if for all morphisms ϕ : A∗ → T, ϕ(u) = ϕ(v). Birkhoff varieties of monoids are defined by a set of identities. [Birkhoff] Pseudovariety of finite monoids: class of finite monoids closed under: finite direct product submonoid quotient Varieties of finite monoids ← → varieties of rational languages [Eilenberg] → Equations for pseudovarieties? Profinite equations! [Reiterman]

12/20

Study of classes of rational languages

Lattice (union, intersection) Boolean algebra (lattice, complement) Lattice closed under quotient Boolean algebra closed under quotient

quotient : u−1Lv −1 = {w | uwv ∈ L}

13/20

Classes of rational languages

Given two profinite words u, v, a rational language L satisfies u → v if u ∈ ¯ L implies v ∈ ¯ L

Definition

a, b ∈ A Equation ab → aba {L ⊆ A∗ | ab / ∈ L} ∪ {L ⊆ A∗ | ab, aba ∈ L}

14/20

Equations

Given two profinite words u, v, a rational language L satisfies u ↔ v if u ∈ ¯ L if and only if v ∈ ¯ L

Definition

a, b ∈ A Equation ab ↔ aba {L ⊆ A∗ | ab, aba / ∈ L} ∪ {L ⊆ A∗ | ab, aba ∈ L}

14/20

Equations

Given two profinite words u, v, a rational language L satisfies u v if for all w, w′ ∈ A∗, wuw′ ∈ ¯ L implies wvw′ ∈ ¯ L

Definition

a, b ∈ A Equation ab aba {L ⊆ A∗ | for all w, w′, if wabw′ ∈ L then wabaw′ ∈ L}

14/20

Equations

Given two profinite words u, v, a rational language L satisfies u = v if for all w, w′ ∈ A∗, wuw′ ∈ ¯ L if and only if wvw′ ∈ ¯ L

Definition

a, b ∈ A Equation ab = aba {L ⊆ A∗ | for all w, w′, wabw′ ∈ L iff wabaw′ ∈ L}

14/20

Equations

Classes of rational languages Lattice (union, intersection): → Boolean algebra (lattice, complement): ↔ Lattice closed under quotient: Boolean algebra closed under quotient: = Theorem [Gehrke, Grigorieff, Pin 2008]

quotient : u−1Lv −1 = {w | uwv ∈ L}

15/20

Characterisation by equations on profinite words

A alphabet Example 1: Commutative languages A language L is commutative if for all u ∈ L, com(u) ⊆ L.

16/20

Examples

A alphabet Example 1: Commutative languages A language L is commutative if for all u ∈ L, com(u) ⊆ L. uv = vu → Boolean algebra closed under quotient Decidability?

16/20

Examples

A alphabet Example 1: Commutative languages A language L is commutative if for all u ∈ L, com(u) ⊆ L. uv = vu → Boolean algebra closed under quotient Decidability? A language L satisfies u = v if and only if ϕ(u) = ϕ(v) in M where M is the syntactic monoid of L and ϕ is its syntactic morphism

16/20

Examples

A alphabet Example 1: Commutative languages A language L is commutative if for all u ∈ L, com(u) ⊆ L. uv = vu → Boolean algebra closed under quotient Decidability? A language L satisfies u = v if and only if ϕ(u) = ϕ(v) in M where M is the syntactic monoid of L and ϕ is its syntactic morphism Example 2: Existence of a zero

16/20

Examples

A alphabet Example 1: Commutative languages A language L is commutative if for all u ∈ L, com(u) ⊆ L. uv = vu → Boolean algebra closed under quotient Decidability? A language L satisfies u = v if and only if ϕ(u) = ϕ(v) in M where M is the syntactic monoid of L and ϕ is its syntactic morphism Example 2: Existence of a zero {ρAu = uρA = ρA | u ∈ A∗}

16/20

Examples

Rational expressions: 1, a ∈ A, E ∪ F, E ∩ F, E.F, cE, E ∗

17/20

Generalised star-height problem

Rational expressions: 1, a ∈ A, E ∪ F, E ∩ F, E.F, cE, E ∗ Given a rational language L, what is the minimal number of nested stars needed to describe L by such an expression?

17/20

Generalised star-height problem

Rational expressions: 1, a ∈ A, E ∪ F, E ∩ F, E.F, cE, E ∗ Given a rational language L, what is the minimal number of nested stars needed to describe L by such an expression? → Star-height 0 [Schützenberger, McNaughton-Papert] Star-free languages, aperiodic monoid xω+1 = xω, FO[<]

17/20

Generalised star-height problem

Rational expressions: 1, a ∈ A, E ∪ F, E ∩ F, E.F, cE, E ∗ Given a rational language L, what is the minimal number of nested stars needed to describe L by such an expression? → Star-height 0 [Schützenberger, McNaughton-Papert] Star-free languages, aperiodic monoid xω+1 = xω, FO[<] → Star-height 1 Example: (aa)∗ - Is there a nontrivial identity for this class ?

17/20

Generalised star-height problem

Pu =

• p prefix of u

u∗p and Su =

• s suffix of u

su∗

xωyω = 0 for x, y ∈ A∗ such that xy = yx (E1) xωy = 0 for x, y ∈ A∗ such that y / ∈ Px (E2) yxω = 0 for x, y ∈ A∗ such that y / ∈ Sx (E3) xω 1 for x ∈ A∗ (E4) xℓ ↔ xω+ℓ for x ∈ A∗, ℓ > 0 (E5) x → xℓ for x ∈ A∗, ℓ > 0 (E6) xα ↔ xβ for all (α, β) ∈ Γ (E7) DECIDABLE

18/20

Equations for u∗ (joint work with Charles Paperman)

Pu =

• p prefix of u

u∗p and Su =

• s suffix of u

su∗

xωyω = 0 for x, y ∈ A∗ such that xy = yx (E1) xωy = 0 for x, y ∈ A∗ such that y / ∈ Px (E2) yxω = 0 for x, y ∈ A∗ such that y / ∈ Sx (E3) xω 1 for x ∈ A∗ (E4) xℓ ↔ xω+ℓ for x ∈ A∗, ℓ > 0 (E5) x → xℓ for x ∈ A∗, ℓ > 0 (E6) xα ↔ xβ for all (α, β) ∈ Γ (E7) DECIDABLE Lattice

18/20

Equations for u∗ (joint work with Charles Paperman)

Pu =

• p prefix of u

u∗p and Su =

• s suffix of u

su∗

xωyω = 0 for x, y ∈ A∗ such that xy = yx (E1) xωy = 0 for x, y ∈ A∗ such that y / ∈ Px (E2) yxω = 0 for x, y ∈ A∗ such that y / ∈ Sx (E3) xω 1 for x ∈ A∗ (E4) xℓ ↔ xω+ℓ for x ∈ A∗, ℓ > 0 (E5) x → xℓ for x ∈ A∗, ℓ > 0 (E6) xα ↔ xβ for all (α, β) ∈ Γ (E7) DECIDABLE Lattice closed under quotients

18/20

Equations for u∗ (joint work with Charles Paperman)

Pu =

• p prefix of u

u∗p and Su =

• s suffix of u

su∗

xωyω = 0 for x, y ∈ A∗ such that xy = yx (E1) xωy = 0 for x, y ∈ A∗ such that y / ∈ Px (E2) yxω = 0 for x, y ∈ A∗ such that y / ∈ Sx (E3) xω 1 for x ∈ A∗ (E4) xℓ ↔ xω+ℓ for x ∈ A∗, ℓ > 0 (E5) x → xℓ for x ∈ A∗, ℓ > 0 (E6) xα ↔ xβ for all (α, β) ∈ Γ (E7) DECIDABLE Boolean algebra closed under quotients

18/20

Equations for u∗ (joint work with Charles Paperman)

Pu =

• p prefix of u

u∗p and Su =

• s suffix of u

su∗

xωyω = 0 for x, y ∈ A∗ such that xy = yx (E1) xωy = 0 for x, y ∈ A∗ such that y / ∈ Px (E2) yxω = 0 for x, y ∈ A∗ such that y / ∈ Sx (E3) xω 1 for x ∈ A∗ (E4) xℓ ↔ xω+ℓ for x ∈ A∗, ℓ > 0 (E5) x → xℓ for x ∈ A∗, ℓ > 0 (E6) xα ↔ xβ for all (α, β) ∈ Γ (E7) DECIDABLE Boolean algebra

18/20

Equations for u∗ (joint work with Charles Paperman)

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6) (u6)∗u2 ⊆ L if and only if (u6)∗u4 ⊆ L

19/20

The Boolean algebra

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . . 2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6) (u6)∗u2 ⊆ L if and only if (u6)∗u4 ⊆ L

19/20

The Boolean algebra

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . . Equivalence relation over the integers r ≡m s if and only if gcd(r, m) = gcd(s, m) (um)∗ur ⊆ L if and only if (um)∗us ⊆ L 2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6) (u6)∗u2 ⊆ L if and only if (u6)∗u4 ⊆ L

19/20

The Boolean algebra

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . . Equivalence relation over the integers r ≡m s if and only if gcd(r, m) = gcd(s, m) (um)∗ur ⊆ L if and only if (um)∗us ⊆ L 2 ≡6 4 since gcd(2, 6) = 2 = gcd(4, 6) (u6)∗u2 ⊆ L if and only if (u6)∗u4 ⊆ L

19/20

The Boolean algebra

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . . Equivalence relation over the integers r ≡m s if and only if gcd(r, m) = gcd(s, m) (um)∗ur ⊆ L if and only if (um)∗us ⊆ L xα ↔ xβ for α and β representing sequences of integers (km + r)k and (km + s)k with r ≡m s...

19/20

The Boolean algebra

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . . Equivalence relation over the integers r ≡m s if and only if gcd(r, m) = gcd(s, m) (um)∗ur ⊆ L if and only if (um)∗us ⊆ L xα ↔ xβ for α and β profinite numbers in N = {a}∗ satisfying some specific conditions...

19/20

The Boolean algebra

xα ↔ xβ for all (α, β) ∈ Γ (E7) An example: (a2)∗ − (a6)∗ = (a6)∗a2 ∪ (a6)∗a4 1 a a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 . . . Γ is the set of all the pairs of profinite numbers (dzP, dpzP) s.t.: P is a cofinite sequence of prime numbers {p1, p2, . . .} zP = limn(p1p2 . . . pn)n! p ∈ P if q divides d then q / ∈ P xα ↔ xβ for all (α, β) ∈ Γ (E7)

19/20

The Boolean algebra

Topology Languages Thank you for your attention

20/20

Conclusion