The Theory of Languages Highlights in London September 12-15, 2017 Paul Brunet University College London
Introduction Outline I. Introduction II. Free Representation III. Main results IV. Outlook Paul Brunet 2/18 Language Algebra
Introduction Universal laws a ∪ b = b ∪ a (commutativity of union) a · ( b · c ) = ( a · b ) · c (associativity of concatenation) Paul Brunet 3/18 Language Algebra
Introduction Universal laws ∀ Σ , ∀ a , b , c ⊆ Σ ⋆ a ∪ b = b ∪ a (commutativity of union) a · ( b · c ) = ( a · b ) · c (associativity of concatenation) Paul Brunet 3/18 Language Algebra
Introduction Universal laws ∀ Σ , ∀ a , b , c ⊆ Σ ⋆ a ∪ b = b ∪ a (commutativity of union) a · ( b · c ) = ( a · b ) · c (associativity of concatenation) | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f Paul Brunet 3/18 Language Algebra
Introduction Universal laws ∀ Σ , ∀ a , b , c ⊆ Σ ⋆ a ∪ b = b ∪ a (commutativity of union) a · ( b · c ) = ( a · b ) · c (associativity of concatenation) | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f Language equivalence Lang | = e ≃ f iff ∀ Σ , ∀ σ : X → P (Σ ⋆ ) , � σ ( e ) = � σ ( f ) . Paul Brunet 3/18 Language Algebra
Introduction representation r : E X → R Paul Brunet 4/18 Language Algebra
Introduction Effective representation r : E X → R Computable Decidable Paul Brunet 4/18 Language Algebra
Introduction Effective free representation r : E X → R Computable Decidable r ( e ) = r ( f ) Lang | = e ≃ f Paul Brunet 4/18 Language Algebra
Introduction Effective free representation r : E X → R Computable Decidable r ( e ) = r ( f ) Lang | = e ≃ f Ax ⊢ e = f Paul Brunet 4/18 Language Algebra
Introduction Effective free representation r : E X → R Computable Decidable r ( e ) = r ( f ) Lang | = e ≃ f Ax ⊢ e = f Kleene Algebra, KA with Tests, Kleene lattices, Allegories, Monoids,... Paul Brunet 4/18 Language Algebra
Introduction Outline I. Introduction II. Free Representation III. Main results IV. Outlook Paul Brunet 5/18 Language Algebra
Free Representation Outline I. Introduction II. Free Representation III. Main results IV. Outlook Paul Brunet 6/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : � σ (( 1 ∩ a ) · b ) = σ ( b · ( 1 ∩ a )) = � ◮ If ε / ∈ σ ( a ) : � σ (( 1 ∩ a ) · b ) = � σ ( b · ( 1 ∩ a )) = Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : then � σ ( 1 ∩ a ) = { ε } , thus: � σ (( 1 ∩ a ) · b ) = σ ( b · ( 1 ∩ a )) = � ◮ If ε / ∈ σ ( a ) : � σ (( 1 ∩ a ) · b ) = � σ ( b · ( 1 ∩ a )) = Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : then � σ ( 1 ∩ a ) = { ε } , thus: � σ (( 1 ∩ a ) · b ) = { ε } · σ ( b ) = σ ( b ) . σ ( b · ( 1 ∩ a )) = σ ( b ) · { ε } = σ ( b ) . � ◮ If ε / ∈ σ ( a ) : � σ (( 1 ∩ a ) · b ) = � σ ( b · ( 1 ∩ a )) = Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : then � σ ( 1 ∩ a ) = { ε } , thus: � σ (( 1 ∩ a ) · b ) = { ε } · σ ( b ) = σ ( b ) . σ ( b · ( 1 ∩ a )) = σ ( b ) · { ε } = σ ( b ) . � ◮ If ε / ∈ σ ( a ) : then � σ ( 1 ∩ a ) = ∅ , thus: � σ (( 1 ∩ a ) · b ) = � σ ( b · ( 1 ∩ a )) = Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : then � σ ( 1 ∩ a ) = { ε } , thus: � σ (( 1 ∩ a ) · b ) = { ε } · σ ( b ) = σ ( b ) . σ ( b · ( 1 ∩ a )) = σ ( b ) · { ε } = σ ( b ) . � ◮ If ε / ∈ σ ( a ) : then � σ ( 1 ∩ a ) = ∅ , thus: � σ (( 1 ∩ a ) · b ) = ∅ · σ ( b ) = ∅ . � σ ( b · ( 1 ∩ a )) = σ ( b ) · ∅ = ∅ . Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : then � σ ( 1 ∩ a ) = { ε } , thus: � σ (( 1 ∩ a ) · b ) = { ε } · σ ( b ) = σ ( b ) . � σ ( b · ( 1 ∩ a )) = σ ( b ) · { ε } = σ ( b ) . ◮ If ε / ∈ σ ( a ) : then � σ ( 1 ∩ a ) = ∅ , thus: � σ (( 1 ∩ a ) · b ) = ∅ · σ ( b ) = ∅ . � σ ( b · ( 1 ∩ a )) = σ ( b ) · ∅ = ∅ . � Paul Brunet 7/18 Language Algebra
Free Representation Example Lang | = ( 1 ∩ a ) · b ≃ b · ( 1 ∩ a ) Proof. Let σ : { a , b } → P (Σ ⋆ ) . ◮ If ε ∈ σ ( a ) : then � σ ( 1 ∩ a ) = { ε } , thus: � σ (( 1 ∩ a ) · b ) = { ε } · σ ( b ) = σ ( b ) . � σ ( b · ( 1 ∩ a )) = σ ( b ) · { ε } = σ ( b ) . ◮ If ε / ∈ σ ( a ) : then � σ ( 1 ∩ a ) = ∅ , thus: � σ (( 1 ∩ a ) · b ) = ∅ · σ ( b ) = ∅ . � σ ( b · ( 1 ∩ a )) = σ ( b ) · ∅ = ∅ . � Idea Compare 1-free terms under the assumption that certain variables contain ε . Paul Brunet 7/18 Language Algebra
Free Representation Weak graphs Definition A weak graph is a pair of a graph and a set of tests. Paul Brunet 8/18 Language Algebra
Free Representation Weak graphs Definition A weak graph is a pair of a graph and a set of tests. Weak graph preorder � G , A � ◭ � H , B � if B ⊆ A and there is an A -weak morphism from H to G . Paul Brunet 8/18 Language Algebra
Free Representation Weak graphs Definition A weak graph is a pair of a graph and a set of tests. Weak graph preorder � G , A � ◭ � H , B � if B ⊆ A and there is an A -weak morphism from H to G . A = { a } a b H : a c b G : a c Paul Brunet 8/18 Language Algebra
Free Representation Characterisation Theorem u , v ∈ T X ::= 1 | a | u · v | u ∩ v Paul Brunet 9/18 Language Algebra
Free Representation Characterisation Theorem u , v ∈ T X ::= 1 | a | u · v | u ∩ v For every term u ∈ T X we can build a weak graph G ( u ) . Paul Brunet 9/18 Language Algebra
Free Representation Characterisation Theorem u , v ∈ T X ::= 1 | a | u · v | u ∩ v For every term u ∈ T X we can build a weak graph G ( u ) . Corollary Lang | = u ⊆ v ⇔ G ( u ) ◭ G ( v ) . Paul Brunet 9/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X T P ( T X ∪ X ′ ) Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X T P ( T X ∪ X ′ ) P ( G ) P ( WeakGraph X ∪ X ′ ) Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X T P ( T X ∪ X ′ ) P ( G ) P ( WeakGraph X ∪ X ′ ) ◭ _ P ( WeakGraph X ∪ X ′ ) Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X T P ( T X ∪ X ′ ) P ( G ) R P ( WeakGraph X ∪ X ′ ) ◭ _ P ( WeakGraph X ∪ X ′ ) Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X T P ( T X ∪ X ′ ) P ( G ) R P ( WeakGraph X ∪ X ′ ) ◭ _ P ( WeakGraph X ∪ X ′ ) Theorem Lang | = e ≃ f ⇔ R ( e ) = R ( f ) Paul Brunet 10/18 Language Algebra
Free Representation Free representation of expressions | e � | e ⋆ . e , f ∈ E X ::= 0 | 1 | a | e ∪ f | e ∩ f | e · f E X T P ( T X ∪ X ′ ) Lemma P ( G ) R If e doesn’t use the Kleene star, then T ( e ) is finite. P ( WeakGraph X ∪ X ′ ) ◭ _ P ( WeakGraph X ∪ X ′ ) Theorem Lang | = e ≃ f ⇔ R ( e ) = R ( f ) Paul Brunet 10/18 Language Algebra
Main results Outline I. Introduction II. Free Representation III. Main results IV. Outlook Paul Brunet 11/18 Language Algebra
Recommend
More recommend
