Commutative closures of regular Antoine Delignat-Lavaud languages - PowerPoint PPT Presentation

Commutative closures of regular languages Commutative closures of regular Antoine Delignat-Lavaud languages Jul, 28. 2009 Outline Trace monoids and recognizability The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References Antoine Delignat-Lavaud Computer Science Department, École Normale Supérieure de Cachan 1

Commutative Outline closures of regular languages Antoine Delignat-Lavaud Trace monoids and recognizability 1 The free partially commutative monoid Recognizability in non-free monoids Outline Trace monoids and recognizability The free partially Varieties 2 commutative monoid Recognizability in non-free Relationship with recognizable languages monoids Varieties Relationship with recognizable languages Commutative closures Commutative closures 3 Closure under P 4 Closure under P 4 References 4 References 2

Commutative closures of regular languages Antoine Delignat-Lavaud Commutation relation • An independence or commutation relation is a symmetric and irreflexive relation I . Outline Trace monoids and recognizability The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 3

Commutative closures of regular languages Antoine Delignat-Lavaud Commutation relation • An independence or commutation relation is a symmetric and irreflexive relation I . Outline • (Σ , I ) is the independence alphabet Trace monoids and recognizability The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 3

Commutative closures of regular languages Antoine Delignat-Lavaud Commutation relation • An independence or commutation relation is a symmetric and irreflexive relation I . Outline • (Σ , I ) is the independence alphabet Trace monoids and • Can be represented as undirected commutation graph . recognizability The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 3

Commutative closures of regular languages Antoine Delignat-Lavaud Commutation relation • An independence or commutation relation is a symmetric and irreflexive relation I . Outline • (Σ , I ) is the independence alphabet Trace monoids and • Can be represented as undirected commutation graph . recognizability The free partially commutative monoid • Complement of I is the dependence relation D . Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 3

Commutative closures of regular languages Antoine Delignat-Lavaud Commutation relation • An independence or commutation relation is a symmetric and irreflexive relation I . Outline • (Σ , I ) is the independence alphabet Trace monoids and • Can be represented as undirected commutation graph . recognizability The free partially commutative monoid • Complement of I is the dependence relation D . Recognizability in non-free monoids Varieties Relationship with recognizable languages a c Commutative closures �������� a c b d Closure under P 4 References b d I D 3

Commutative closures of regular languages Antoine Delignat-Lavaud Trace monoid • The set of finite sequences of letters from an alphabet Σ is the free monoid Σ ∗ . Outline Trace monoids and recognizability The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 4

Commutative closures of regular languages Antoine Delignat-Lavaud 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 Outline equivalence ∼ I over Σ ∗ is the least congruence such that Trace monoids and recognizability ab ∼ I ba for all ( a , b ) ∈ I . The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 4

Commutative closures of regular languages Antoine Delignat-Lavaud 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 Outline equivalence ∼ I over Σ ∗ is the least congruence such that Trace monoids and recognizability ab ∼ I ba for all ( a , b ) ∈ I . The free partially commutative monoid Recognizability in non-free • The quotient M (Σ , I ) = Σ ∗ / ∼ I is the free partially monoids I-commutative monoid or trace monoid . Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 4

Commutative closures of regular languages Antoine Delignat-Lavaud 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 Outline equivalence ∼ I over Σ ∗ is the least congruence such that Trace monoids and recognizability ab ∼ I ba for all ( a , b ) ∈ I . The free partially commutative monoid Recognizability in non-free • The quotient M (Σ , I ) = Σ ∗ / ∼ I is the free partially monoids I-commutative monoid or trace monoid . Varieties Relationship with • Elements from M are called traces. recognizable languages Commutative closures Closure under P 4 References 4

Commutative closures of regular languages Antoine Delignat-Lavaud 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 Outline equivalence ∼ I over Σ ∗ is the least congruence such that Trace monoids and recognizability ab ∼ I ba for all ( a , b ) ∈ I . The free partially commutative monoid Recognizability in non-free • The quotient M (Σ , I ) = Σ ∗ / ∼ I is the free partially monoids I-commutative monoid or trace monoid . Varieties Relationship with • Elements from M are called traces. recognizable languages Commutative closures • φ denotes the canonical quotient homomorphism. Closure under P 4 References 4

Commutative closures of regular languages Antoine Delignat-Lavaud 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 Outline equivalence ∼ I over Σ ∗ is the least congruence such that Trace monoids and recognizability ab ∼ I ba for all ( a , b ) ∈ I . The free partially commutative monoid Recognizability in non-free • The quotient M (Σ , I ) = Σ ∗ / ∼ I is the free partially monoids I-commutative monoid or trace monoid . Varieties Relationship with • Elements from M are called traces. recognizable languages Commutative closures • φ denotes the canonical quotient homomorphism. Closure under P 4 References • [ · ] I denotes the closure operator φ − 1 ◦ φ . 4

Commutative closures of regular languages Antoine Delignat-Lavaud 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 Outline equivalence ∼ I over Σ ∗ is the least congruence such that Trace monoids and recognizability ab ∼ I ba for all ( a , b ) ∈ I . The free partially commutative monoid Recognizability in non-free • The quotient M (Σ , I ) = Σ ∗ / ∼ I is the free partially monoids I-commutative monoid or trace monoid . Varieties Relationship with • Elements from M are called traces. recognizable languages Commutative closures • φ denotes the canonical quotient homomorphism. Closure under P 4 References • [ · ] I denotes the closure operator φ − 1 ◦ φ . • If t ∈ M , a word w such that t = [ w ] I is a linearization of t . 4

Commutative closures of regular The free monoid case languages Antoine • A recognizable language L ⊆ Σ ∗ is a language accepted Delignat-Lavaud by a (deterministic or nondeterministic) finite state machine A = (Σ , Q , I , ∆ , F ) . Outline Trace monoids and recognizability The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 5

Commutative closures of regular The free monoid case languages Antoine • A recognizable language L ⊆ Σ ∗ is a language accepted Delignat-Lavaud 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 Outline Trace monoids and under product, union and iteration. (the rational recognizability languages). The free partially commutative monoid Recognizability in non-free monoids Varieties Relationship with recognizable languages Commutative closures Closure under P 4 References 5