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Regular Tree Algebras
Achim Blumensath
joint work with
- T. Colcombet, M. Bojańczyk, B. Klin
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. - - 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 czy k , B . Klin Algeb r aic Lang u age T eo ry Recognisabilit y f ree a lg ebra fini te a lg ebra L = [ P ] Algeb r aic Lang u age
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Algebraic Language Teory
Recognisability φ ∶ free algebra → finite algebra L = φ−[P] Which algebras? finite words monoids, semigroups infinite words ω-semigroups finite trees clones, preclones, term algebras, forest algebras,... infinite trees ?
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General Formalism
Algebras ⟨A, π⟩ where π ∶ TA → A and finite words TA = A∗ infinite words TA = A∞ finite trees TA finite A-labelled trees infinite trees TA finite and infinite A-labelled trees (plus axioms for associativity: T monad, π ∶ TA → A Eilenberg–Moore algebra)
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General Formalism
Algebras ⟨A, π⟩ where π ∶ TA → A and finite words TA = A∗ infinite words TA = A∞ finite trees TA finite A-labelled trees infinite trees TA finite and infinite A-labelled trees Tree Algebras
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Regular Tree Algebras
Problem Finitary tree algebras can recognise non-regular languages. Regular algebras ⟨A, π⟩ is regular if π can be evaluated by a tree automaton.
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Regular Tree Algebras
Problem Finitary tree algebras can recognise non-regular languages. Regular algebras ⟨A, π⟩ is regular if π can be evaluated by a tree automaton. Teorem A finitary algebra ⟨A, π⟩ is regular if, and only if, every language recognised by ⟨A, π⟩ is regular. Teorem A language L is regular if, and only if, it is recognised by a regular tree algebra.
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Syntactic algebras
Teorem Te class of regular tree algebras is a pseudo-variety. It is the unique pseudo-variety associated with the class of regular tree languages.
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