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Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

1

Commutative closures of regular languages

Jul, 28. 2009 Antoine Delignat-Lavaud Computer Science Department, École Normale Supérieure de Cachan

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

2

Outline

1

Trace monoids and recognizability The free partially commutative monoid Recognizability in non-free monoids

2

Varieties Relationship with recognizable languages

3

Commutative closures Closure under P4

4

References

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

3

Commutation relation

• An independence or commutation relation is a symmetric

and irreflexive relation I.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

3

Commutation relation

• An independence or commutation relation is a symmetric

and irreflexive relation I.

• (Σ, I) is the independence alphabet

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

3

Commutation relation

• An independence or commutation relation is a symmetric

and irreflexive relation I.

• (Σ, I) is the independence alphabet
• Can be represented as undirected commutation graph.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

3

Commutation relation

• An independence or commutation relation is a symmetric

and irreflexive relation I.

• (Σ, I) is the independence alphabet
• Can be represented as undirected commutation graph.
• Complement of I is the dependence relation D.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

3

Commutation relation

• An independence or commutation relation is a symmetric

and irreflexive relation I.

• (Σ, I) is the independence alphabet
• Can be represented as undirected commutation graph.
• Complement of I is the dependence relation D.

a b c d I b a d c D

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

• Given a commutation relation I, the commutation

equivalence ∼I over Σ∗ is the least congruence such that ab ∼I ba for all (a, b) ∈ I.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

• Given a commutation relation I, the commutation

equivalence ∼I over Σ∗ is the least congruence such that ab ∼I ba for all (a, b) ∈ I.

• The quotient M(Σ, I) = Σ∗/∼I is the free partially

I-commutative monoid or trace monoid.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

• Given a commutation relation I, the commutation

equivalence ∼I over Σ∗ is the least congruence such that ab ∼I ba for all (a, b) ∈ I.

• The quotient M(Σ, I) = Σ∗/∼I is the free partially

I-commutative monoid or trace monoid.

• Elements from M are called traces.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

• Given a commutation relation I, the commutation

equivalence ∼I over Σ∗ is the least congruence such that ab ∼I ba for all (a, b) ∈ I.

• The quotient M(Σ, I) = Σ∗/∼I is the free partially

I-commutative monoid or trace monoid.

• Elements from M are called traces.
• φ denotes the canonical quotient homomorphism.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

• Given a commutation relation I, the commutation

equivalence ∼I over Σ∗ is the least congruence such that ab ∼I ba for all (a, b) ∈ I.

• The quotient M(Σ, I) = Σ∗/∼I is the free partially

I-commutative monoid or trace monoid.

• Elements from M are called traces.
• φ denotes the canonical quotient homomorphism.
• [·]I denotes the closure operator φ−1 ◦ φ.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

4

Trace monoid

• The set of finite sequences of letters from an alphabet Σ is

the free monoid Σ∗.

• Given a commutation relation I, the commutation

equivalence ∼I over Σ∗ is the least congruence such that ab ∼I ba for all (a, b) ∈ I.

• The quotient M(Σ, I) = Σ∗/∼I is the free partially

I-commutative monoid or trace monoid.

• Elements from M are called traces.
• φ denotes the canonical quotient homomorphism.
• [·]I denotes the closure operator φ−1 ◦ φ.
• If t ∈ M, a word w such that t = [w]I is a linearization of t.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

5

The free monoid case

• A recognizable language L ⊆ Σ∗ is a language accepted

by a (deterministic or nondeterministic) finite state machine A = (Σ, Q, I, ∆, F).

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

5

The free monoid case

• A recognizable language L ⊆ Σ∗ is a language accepted

by a (deterministic or nondeterministic) finite state machine A = (Σ, Q, I, ∆, F).

• Kleene’s theorem: the class of recognizable word

languages is the closure of the class of finite languages under product, union and iteration. (the rational languages).

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

5

The free monoid case

• A recognizable language L ⊆ Σ∗ is a language accepted

by a (deterministic or nondeterministic) finite state machine A = (Σ, Q, I, ∆, F).

• Kleene’s theorem: the class of recognizable word

languages is the closure of the class of finite languages under product, union and iteration. (the rational languages). (a∗(ba)∗(b(a + c)b∗b)∗)∗

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

6

Recognizability in non-free monoids

• An M-automaton is a tuple A = (Q, δ, F).

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

6

Recognizability in non-free monoids

• An M-automaton is a tuple A = (Q, δ, F).
• Q is a finite monoid, F ⊆ Q.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

6

Recognizability in non-free monoids

• An M-automaton is a tuple A = (Q, δ, F).
• Q is a finite monoid, F ⊆ Q.
• δ : M → Q is an homomorphism of monoids.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

6

Recognizability in non-free monoids

• An M-automaton is a tuple A = (Q, δ, F).
• Q is a finite monoid, F ⊆ Q.
• δ : M → Q is an homomorphism of monoids.
• The subset of M recognized by A is δ−1(F).

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

6

Recognizability in non-free monoids

• An M-automaton is a tuple A = (Q, δ, F).
• Q is a finite monoid, F ⊆ Q.
• δ : M → Q is an homomorphism of monoids.
• The subset of M recognized by A is δ−1(F).

Recognizable = rational

(Σ, I) = a—b, L = (ab)∗. φ−1(L) = {w ∈ {a, b}∗ | |w|a = |w|b}

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

7

Syntactic monoid

Let M be a monoid and T ⊆ M. The syntactic preorder over T, ≤T is defined by: x ≤T y if ∀u, v ∈ M, uyv ∈ T ⇒ uxv ∈ T The syntactic equivalence ≡T is defined by x ≡T y ⇔ x ≤T y ∧ y ≤T x The syntactic monoid of T is the quotient MT = M/≡T.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

7

Syntactic monoid

Let M be a monoid and T ⊆ M. The syntactic preorder over T, ≤T is defined by: x ≤T y if ∀u, v ∈ M, uyv ∈ T ⇒ uxv ∈ T The syntactic equivalence ≡T is defined by x ≡T y ⇔ x ≤T y ∧ y ≤T x The syntactic monoid of T is the quotient MT = M/≡T.

Characterization of recognizable languages

A subset T ⊆ M is recognizable if and only if, its syntactic monoid MT is finite.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

8

Varieties of monoids and recognizable languages

Variety of monoids

A class of monoids closed under taking submonoids, left and right quotients and direct products. A pseudovariety of monoids is a variety of finite monoids.

Example

The class of all finite monoids is a pseudovariety. The class of finite groups is a pseudovariety denoted G.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

9

Variety of languages

A variety of languages V(Σ) over some alphabet Σ is a family

• f recognizable languages such that for all Σ,
• V(Σ) is a boolean algebra

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

9

Variety of languages

A variety of languages V(Σ) over some alphabet Σ is a family

• f recognizable languages such that for all Σ,
• V(Σ) is a boolean algebra
• V(Σ) is closed under inverse of morphisms.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

9

Variety of languages

A variety of languages V(Σ) over some alphabet Σ is a family

• f recognizable languages such that for all Σ,
• V(Σ) is a boolean algebra
• V(Σ) is closed under inverse of morphisms.
• V(Σ) is closed under residuals

a−1L, La−1, a ∈ Σ, L ∈ V(Σ).

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

9

Variety of languages

A variety of languages V(Σ) over some alphabet Σ is a family

• f recognizable languages such that for all Σ,
• V(Σ) is a boolean algebra
• V(Σ) is closed under inverse of morphisms.
• V(Σ) is closed under residuals

a−1L, La−1, a ∈ Σ, L ∈ V(Σ).

Variety theorem

The correspondance V → V that maps to a variety of monoids V the variety of languages whose syntactic monoid is in V is

• ne to one.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

10

Polynomial closure

The polynomial closure Pol(L) of a class of language L over Σ is the set finite unions of L0a1L1a2 · · · anLn whith ai ∈ Σ and Li ∈ L.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

10

Polynomial closure

The polynomial closure Pol(L) of a class of language L over Σ is the set finite unions of L0a1L1a2 · · · anLn whith ai ∈ Σ and Li ∈ L.

Polynomials of group languages

• Pol(G) (the polynomial closure of the variety associated to

G) is closed unter total commutation.

• It is closed under I-commutation if (Σ, I) does not have
• ne of the two following induced subgraphs:

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

10

Polynomial closure

The polynomial closure Pol(L) of a class of language L over Σ is the set finite unions of L0a1L1a2 · · · anLn whith ai ∈ Σ and Li ∈ L.

Polynomials of group languages

• Pol(G) (the polynomial closure of the variety associated to

G) is closed unter total commutation.

• It is closed under I-commutation if (Σ, I) does not have
• ne of the two following induced subgraphs:
• P4
• Paw

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

11

P4-closure of a group language

During the internship, we proved the following result: The P4-closure of a group language is recognizable.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

11

P4-closure of a group language

During the internship, we proved the following result: The P4-closure of a group language is recognizable. b∗ (a + c)∗a d(b + d)∗d c∗ b∗ a(a + c)∗a

• · · ·

· · ·

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

11

P4-closure of a group language

During the internship, we proved the following result: The P4-closure of a group language is recognizable. b∗ (a + c)∗a d(b + d)∗d c∗ b∗ a(a + c)∗a

• · · ·

· · ·

Proof

• Proof relies on structure of traces.
• Recognizability is shown using MSO logic
• See the internship report for more details.

Commutative closures of regular languages Antoine Delignat-Lavaud Outline Trace monoids and recognizability

The free partially commutative monoid Recognizability in non-free monoids

Varieties

Relationship with recognizable languages

Commutative closures

Closure under P4

References

12

References Gómez, Antonio Cano and Guaiana, Giovanna and Pin, Jean-Éric When Does Partial Commutative Closure Preserve Regularity? ICALP’2008