Equivalence of regular expressions with converse on relations An - PowerPoint PPT Presentation

Equivalence of regular expressions with converse on relations An alternative presentation of the proof by Bloom , Ésik and Stefanescu Paul Brunet and Damien Pous ENS de Lyon 17 juin 2013 Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 1 / 22

Converse Converse on languages : Mirror 1 ∨ ≔ 1 ( x · w ) ∨ ≔ w ∨ · x L ∨ ≔ { w ∨ | w ∈ L } Converse on relations R ∨ ≔ { ( y , x ) | ( x , y ) ∈ R } a b c d Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 2 / 22

Converse Converse on languages : Mirror 1 ∨ ≔ 1 ( x · w ) ∨ ≔ w ∨ · x L ∨ ≔ { w ∨ | w ∈ L } Converse on relations R ∨ ≔ { ( y , x ) | ( x , y ) ∈ R } a b c d Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 2 / 22

Plan Kleene Algebrae with converse 1 Construction of the closure of an automaton 2 On examples 3 Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 3 / 22

CKA Table of Contents Kleene Algebrae with converse 1 Construction of the closure of an automaton 2 On examples 3 Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 4 / 22

CKA Equivalence We’ll use different notions of equivalence on expressions e , f on an alphabet X : We write � e � for the language denoted by a regular expression e . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 5 / 22

CKA Equivalence We’ll use different notions of equivalence on expressions e , f on an alphabet X : Language equality : � e � = � f � ; We write � e � for the language denoted by a regular expression e . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 5 / 22

CKA Equivalence We’ll use different notions of equivalence on expressions e , f on an alphabet X : Language equality : � e � = � f � ; Equivalence for language models : ∀ Σ , ∀ σ ∈ Σ ∗ X , ˆ σ ( f ) : ; σ ( e ) = ˆ e ≡ Lang f We write � e � for the language denoted by a regular expression e . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 5 / 22

CKA Equivalence We’ll use different notions of equivalence on expressions e , f on an alphabet X : Language equality : � e � = � f � ; Equivalence for language models : ∀ Σ , ∀ σ ∈ Σ ∗ X , ˆ σ ( f ) : ; σ ( e ) = ˆ e ≡ Lang f S 2 � X , ˆ Equivalence for relation models : ∀ S , ∀ σ ∈ P � σ ( f ) : e ≡ Rel f . σ ( e ) = ˆ We write � e � for the language denoted by a regular expression e . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 5 / 22

CKA Kleene Algebrae I A Kleene Algebra (i) is an algebraic structure � K , + , · , ∗ , 0 , 1 � satifying : � K , + , · , 0 , 1 � is an idempotent semiring : 1 � K , + , 0 � is a  a + ( b + c ) = ( a + b ) + c  commutative  a + b = b + a  idempotent a + 0 = a  monoid  a + a = a   a ( bc ) = ( ab ) c � K , · , 1 � is a  1 a = a monoid a 1 = a   a ( b + c ) = ab + ac  Distributivity  ( a + b ) c = ac + bc  laws 0 a = 0   a 0 = 0  Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 6 / 22

CKA Kleene Algebrae II The ∗ operation satisfy : 2 1 + aa ∗ � a ∗ 1 + a ∗ a � a ∗ b + ax � x ⇒ a ∗ b � x b + xa � x ⇒ ba ∗ � x ∆ Where a � b ⇒ a + b = b . ⇐ The last axioms can be replaced by a number of things. (i). As presented in [Koz94]. Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 7 / 22

CKA All is well � e � = � f � Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 8 / 22

CKA All is well � e � = � f � ⇔ KA ⊢ e = f Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 8 / 22

CKA All is well � e � = � f � ⇔ KA ⊢ e = f ⇔ e ≡ Lang f Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 8 / 22

CKA All is well � e � = � f � ⇔ KA ⊢ e = f ⇔ e ≡ Lang f ⇔ e ≡ Rel f Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 8 / 22

CKA Kleene Algebra with Converse Theorem [BES95] A complete axiomatization of the variety L ∨ generated by regular languages with converse consists of the axioms for KA and the following : ( a + b ) ∨ = a ∨ + b ∨ ( a · b ) ∨ = b ∨ · a ∨ ( a ∗ ) ∨ = ( a ∨ ) ∗ a ∨∨ = a . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 9 / 22

CKA Equivalence in L ∨ Let e , f ∈ Reg ∨ ( X ) . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 10 / 22

CKA Equivalence in L ∨ Let e , f ∈ Reg ∨ ( X ) . We compute τ ( e ) , τ ( f ) ∈ Reg ( X ∪ X ′ ) . Where ∀ x ∈ X , ν ( x ) = x ′ and ∀ x ∈ X , τ ( x ) = x τ ( 1 ) = 1 ν ( 1 ) = 1 τ ( e 1 · e 2 ) = τ ( e 1 ) · τ ( e 2 ) ν ( e 1 · e 2 ) = ν ( e 2 ) · ν ( e 1 ) τ ( e 1 + e 2 ) = τ ( e 1 ) + τ ( e 2 ) ν ( e 1 + e 2 ) = ν ( e 1 ) + ν ( e 2 ) τ ( e ∗ ) = τ ( e ) ∗ ν ( e ∗ ) = ν ( e ) ∗ τ ( e ∨ ) = ν ( e ) ν ( e ∨ ) = τ ( e ) X ′ ≔ { x ′ | x ∈ X } is a disjoint copy of X . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 10 / 22

CKA Equivalence in L ∨ Let e , f ∈ Reg ∨ ( X ) . We compute τ ( e ) , τ ( f ) ∈ Reg ( X ∪ X ′ ) . Where ∀ x ∈ X , ν ( x ) = x ′ and ∀ x ∈ X , τ ( x ) = x τ ( 1 ) = 1 ν ( 1 ) = 1 τ ( e 1 · e 2 ) = τ ( e 1 ) · τ ( e 2 ) ν ( e 1 · e 2 ) = ν ( e 2 ) · ν ( e 1 ) τ ( e 1 + e 2 ) = τ ( e 1 ) + τ ( e 2 ) ν ( e 1 + e 2 ) = ν ( e 1 ) + ν ( e 2 ) τ ( e ∗ ) = τ ( e ) ∗ ν ( e ∗ ) = ν ( e ) ∗ τ ( e ∨ ) = ν ( e ) ν ( e ∨ ) = τ ( e ) X ′ ≔ { x ′ | x ∈ X } is a disjoint copy of X . Furthermore � τ ( e ) � = � τ ( f ) � ⇔ e ≡ Lang f Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 10 / 22

CKA Languages vs. Relations a � aa ∨ a ??? in L ∨ in REL ∨ a � aa ∨ a = aaa if xRy then xRyR ∨ xRy which means xRR ∨ Ry so R ⊆ RR ∨ R Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 11 / 22

CKA Languages vs. Relations a � aa ∨ a ??? in L ∨ in REL ∨ a � aa ∨ a = aaa if xRy then xRyR ∨ xRy which means xRR ∨ Ry so R ⊆ RR ∨ R Theorem [EB95] A complete set of axioms for the variety REL ∨ generated by regular relations with converse consists on the axioms for L ∨ and the axiom a � aa ∨ a . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 11 / 22

CKA Equivalence in REL ∨ Theorem [BES95] cl ( � τ ( e ) � ) = cl ( � τ ( f ) � ) ⇔ e ≡ Rel f Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 12 / 22

CKA Equivalence in REL ∨ Theorem [BES95] cl ( � τ ( e ) � ) = cl ( � τ ( f ) � ) ⇔ e ≡ Rel f  ( X ∪ X ′ ) ∗ ( X ∪ X ′ ) ∗ →   ǫ �→ ǫ  Let ¯ . be the function wx ′ �→ xw   x ′ w �→ wx  Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 12 / 22

CKA Equivalence in REL ∨ Theorem [BES95] cl ( � τ ( e ) � ) = cl ( � τ ( f ) � ) ⇔ e ≡ Rel f  ( X ∪ X ′ ) ∗ ( X ∪ X ′ ) ∗ →   ǫ �→ ǫ  Let ¯ . be the function wx ′ �→ xw   x ′ w �→ wx  We define the relation � on ( X ∪ X ′ ) ∗ : ∆ ⇐ ⇒ ∃ u 1 , w , u 2 : u = u 1 wwwu 2 ∧ v = u 1 wu 2 . u � v Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 12 / 22

CKA Equivalence in REL ∨ Theorem [BES95] cl ( � τ ( e ) � ) = cl ( � τ ( f ) � ) ⇔ e ≡ Rel f  ( X ∪ X ′ ) ∗ ( X ∪ X ′ ) ∗ →   ǫ �→ ǫ  Let ¯ . be the function wx ′ �→ xw   x ′ w �→ wx  We define the relation � on ( X ∪ X ′ ) ∗ : ∆ ⇐ ⇒ ∃ u 1 , w , u 2 : u = u 1 wwwu 2 ∧ v = u 1 wu 2 . u � v cl ( A ) is the closure of A for � : cl ( A ) = { v | ∃ u ∈ A : u � ∗ v } . Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 12 / 22

Construction Table of Contents Kleene Algebrae with converse 1 Construction of the closure of an automaton 2 On examples 3 Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 13 / 22

Construction Closure of an automaton a ′ a a 0 1 2 3 Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 14 / 22

Construction Closure of an automaton a a ′ a a 0 1 2 3 Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 14 / 22

Construction Closure of an automaton a a ′ a a 0 1 2 3 a a ′ a ′ a a a 0 1 2 3 4 5 6 b Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 14 / 22

Construction Closure of an automaton a a ′ a a 0 1 2 3 a a a ′ a ′ a a a 0 1 2 3 4 5 6 b Decidability in REL ∨ Paul Brunet and Damien Pous (ENS de Lyon) 17 juin 2013 14 / 22