Regu l ar Tree A lg ebras Achim Bl u men s a t h joint work wit h T. - PowerPoint PPT Presentation

Regu l ar Tree A lg ebras Achim Bl u men s a t h joint work wit h T. Colcombe t , M . B oj a ń czy k , B . Klin

Algeb r aic Lang u age T eo ry Recognisabilit y φ ∶ f ree a lg ebra → fini te a lg ebra L = φ −  [ P ]

Algeb r aic Lang u age T eo ry R ecog nis ab ilit y φ ∶ f ree a lg ebra → fini te a lg ebra L = φ −  [ P ] W hi c h a lg eb r a s? fini te w o rds monoi ds, se mig r o ups infini te w o rds ω - se mig r o ups fini te trees c lon es, prec lon es, ter m a lg ebras, fo rest a lg ebras,... infini te trees ?

G e n era l Fo r m a li s m ⟨ A , π ⟩ w h e r e π ∶ T A → A a n d A lg eb r a s fini t e w o r d s T A = A ∗ infini t e w o r d s T A = A ∞ fini t e tr ee s T A fini t e A -l abe ll ed tr ee s infini t e tr ee s T A fini t e a n d infini t e A -l abe ll ed tr ee s (p l us a x iom s fo r a ss o c i a t i v i ty : T mon ad, π ∶ T A → A E il e n be r g–Moo r e a lg eb r a )

G e n era l Fo r m a li s m ⟨ A , π ⟩ w h e r e π ∶ T A → A a n d A lg eb r a s fini t e w o r d s T A = A ∗ infini t e w o r d s T A = A ∞ fini t e tr ee s T A fini t e A -l abe ll ed tr ee s infini t e tr ee s T A fini t e a n d infini t e A -l abe ll ed tr ee s Tr ee A lg eb r a s

R eg u l a r Tr ee A lg eb r a s Pr oblem Fini tary tree a lg ebras ca n rec ogni se non- re g u l ar l a ng ua g es. R eg u la r algeb r a s ⟨ A , π ⟩ i s r eg u la r if π ca n be eva l uated by a tree aut om at on .

Regu l ar Tree A lg ebras Prob l e m Fini tary tree a lg ebras ca n rec ogni se non- re g u l ar l a ng ua g es. Re g u l ar a lg ebras ⟨ A , π ⟩ i s re g u l ar if π ca n be eva l uated by a tree aut om at on . Te o re m A fini tary a lg ebra ⟨ A , π ⟩ i s re g u l ar if , a n d onl y if , every l a ng ua g e rec ogni sed by ⟨ A , π ⟩ i s re g u l ar. Te o re m A l a ng ua g e L i s re g u l ar if , a n d onl y if , i t i s rec ogni sed by a re g u l ar tree a lg ebra.

Syntact i c a lg ebras Teorem T e c l a ss of r e g u l a r tr ee a lg eb r a s i s a ps e u d o- va r i e t y . I t i s t h e unique ps e u d o- va r i e t y a ss o c i a t ed w i t h t h e c l a ss of r e g u l a r tr ee l a ng u a g e s.

Syntact i c a lg ebras Teorem T e c l a ss of r e g u l a r tr ee a lg eb r a s i s a ps e u d o- va r i e t y . I t i s t h e unique ps e u d o- va r i e t y a ss o c i a t ed w i t h t h e c l a ss of r e g u l a r tr ee l a ng u a g e s. Teorem Eve r y r e g u l a r tr ee l a ng u a g e L h a s a s y n t ac t i c a lg eb r a S y n ( L ) w hi c h i s a r e g u l a r tr ee a lg eb r a .