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One Eilenberg Theorem to Rule Them All Stefan Milius joint work with Ji r Ad amek, Liang-Ting Chen, Henning Urbat December 6, 2016 Overview Algebraic language theory: Automata/languages vs. algebraic structures One Eilenberg

  1. One Eilenberg Theorem to Rule Them All Stefan Milius joint work with Jiˇ r´ ı Ad´ amek, Liang-Ting Chen, Henning Urbat December 6, 2016

  2. Overview Algebraic language theory: Automata/languages vs. algebraic structures One Eilenberg Theorem to Rule Them All December 6, 2016 2 / 28

  3. Overview Algebraic language theory: Automata/languages vs. algebraic structures Categorical perspective: Automata via algebras and coalgebras. Languages via initial algebras and final coalgebras. Algebra via Lawvere theories and monads. η − T 2 µ Id → T − ← One Eilenberg Theorem to Rule Them All December 6, 2016 2 / 28

  4. Overview Algebraic language theory: Automata/languages vs. algebraic structures Categorical perspective: Automata via algebras and coalgebras. Languages via initial algebras and final coalgebras. Algebra via Lawvere theories and monads. η − T 2 µ Id → T − ← Our goal: Categorical Algebraic Language Theory! One Eilenberg Theorem to Rule Them All December 6, 2016 2 / 28

  5. Eilenberg’s Variety Theorem (1976) � varieties of � � pseudovarieties of � ∼ = languages monoids One Eilenberg Theorem to Rule Them All December 6, 2016 3 / 28

  6. Eilenberg’s Variety Theorem (1976) � varieties of � � pseudovarieties of � ∼ = languages monoids Pseudovariety of monoids A class of finite monoids closed under quotients, submonoids and finite products. One Eilenberg Theorem to Rule Them All December 6, 2016 3 / 28

  7. Eilenberg’s Variety Theorem (1976) � varieties of � � pseudovarieties of � ∼ = languages monoids Variety of languages Pseudovariety of monoids For each alphabet Σ a set A class of finite monoids closed V Σ ⊆ Reg (Σ) closed under under quotients, submonoids and ∪ , ∩ , ( − ) ∁ finite products. derivatives x − 1 Ly − 1 = { w : xwy ∈ L } preimages of free monoid morphisms f : ∆ ∗ → Σ ∗ , i.e. L ∈ V Σ ⇒ f − 1 [ L ] ∈ V ∆ One Eilenberg Theorem to Rule Them All December 6, 2016 3 / 28

  8. � � Other Eilenberg-Type Theorems One Eilenberg Theorem to Rule Them All December 6, 2016 4 / 28

  9. � � Other Eilenberg-Type Theorems Weaker closure properties: Only ∪ , ∩ Pin 1995 Only ∪ Pol´ ak 2001 Only ⊕ Reutenauer 1980 Fewer monoid morphisms Straubing 2002 Fixed alphabet, no preimages Gehrke, Grigorieff, Pin 2008 One Eilenberg Theorem to Rule Them All December 6, 2016 4 / 28

  10. � � Other Eilenberg-Type Theorems Weaker closure properties: Other types of languages: Only ∪ , ∩ Weighted languages Pin 1995 Reutenauer 1980 Only ∪ Infinite words Pol´ ak 2001 Wilke 1991, Pin 1998 Only ⊕ Ordered words Reutenauer 1980 Bedon et. al. 1998, 2005 Fewer monoid morphisms Ranked trees Straubing 2002 Almeida 1990, Steinby 1992 Fixed alphabet, no preimages Binary trees Gehrke, Grigorieff, Pin 2008 Salehi, Steinby 2008 Cost functions Daviaud, Kuperberg, Pin 2016 One Eilenberg Theorem to Rule Them All December 6, 2016 4 / 28

  11. � � Other Eilenberg-Type Theorems Weaker closure properties: Other types of languages: Only ∪ , ∩ Weighted languages Pin 1995 Reutenauer 1980 Only ∪ Infinite words Pol´ ak 2001 Wilke 1991, Pin 1998 Only ⊕ Ordered words Reutenauer 1980 Bedon et. al. 1998, 2005 Fewer monoid morphisms Ranked trees Straubing 2002 Almeida 1990, Steinby 1992 Fixed alphabet, no preimages Binary trees Gehrke, Grigorieff, Pin 2008 Salehi, Steinby 2008 Cost functions Daviaud, Kuperberg, Pin 2016 � � This talk A General Variety Theorem that covers them all! One Eilenberg Theorem to Rule Them All December 6, 2016 4 / 28

  12. Big Picture General Variety Theorem = Monads + Duality One Eilenberg Theorem to Rule Them All December 6, 2016 5 / 28

  13. Big Picture General Variety Theorem = Monads + Duality Use monads to model the type of languages and the algebras recognizing them. Boja´ nczyk, DLT 2015 One Eilenberg Theorem to Rule Them All December 6, 2016 5 / 28

  14. Big Picture General Variety Theorem = Monads + Duality Use monads to model the type of Use duality to relate varieties of languages and the algebras languages to pseudovarieties of recognizing them. finite algebras. Boja´ nczyk, DLT 2015 Gehrke, Grigorieff, Pin, ICALP 2008 Ad´ amek, Milius, Myers, Urbat, FoSSaCS 2014, LICS 2015 One Eilenberg Theorem to Rule Them All December 6, 2016 5 / 28

  15. Big Picture General Variety Theorem = Monads + Duality Use monads to model the type of Use duality to relate varieties of languages and the algebras languages to pseudovarieties of recognizing them. finite algebras. One Eilenberg Theorem to Rule Them All December 6, 2016 6 / 28

  16. Languages Fix a monad T on a locally finite variety D (with finitely many sorts). One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

  17. Languages Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : T Σ → O in D Σ: free finite object of D (“alphabet”) O : finite object of D (“object of outputs”) One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

  18. Languages Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : T Σ → O in D Σ: free finite object of D (“alphabet”) O : finite object of D (“object of outputs”) Languages of finite words: free monoid monad T Σ = Σ ∗ on Set and O = { 0 , 1 } . One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

  19. Languages Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : T Σ → O in D Σ: free finite object of D (“alphabet”) O : finite object of D (“object of outputs”) Languages of finite words: free monoid monad T Σ = Σ ∗ on Set and O = { 0 , 1 } . Languages of finite and infinite words: free ω -semigroup monad T (Σ , ∅ ) = (Σ + , Σ ω ) on Set 2 and O = ( { 0 , 1 } , { 0 , 1 } ) . One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

  20. Languages Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : T Σ → O in D Σ: free finite object of D (“alphabet”) O : finite object of D (“object of outputs”) Languages of finite words: free monoid monad T Σ = Σ ∗ on Set and O = { 0 , 1 } . Languages of finite and infinite words: free ω -semigroup monad T (Σ , ∅ ) = (Σ + , Σ ω ) on Set 2 and O = ( { 0 , 1 } , { 0 , 1 } ) . Weighted languages ( D = vector spaces), tree languages ( D = Set 3 ), cost functions ( D = posets), . . . One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

  21. � � Algebraic recognition Definition A language L : T Σ → O is recognizable if it factors through some finite quotient algebra of the free T -algebra T Σ = ( T Σ , µ Σ ). L T Σ O ∃ e �� ∃ p A Languages of finite words: free monoid monad T Σ = Σ ∗ on Set and O = { 0 , 1 } . Recognizable languages = regular languages of finite words One Eilenberg Theorem to Rule Them All December 6, 2016 8 / 28

  22. � � Algebraic recognition Definition A language L : T Σ → O is recognizable if it factors through some finite quotient algebra of the free T -algebra T Σ = ( T Σ , µ Σ ). L T Σ O ∃ e �� ∃ p A Languages of finite and infinite words: free ω -semigroup monad T (Σ , ∅ ) = (Σ + , Σ ω ) on Set 2 and O = ( { 0 , 1 } , { 0 , 1 } ) . Recognizable languages = regular ∞ -languages One Eilenberg Theorem to Rule Them All December 6, 2016 9 / 28

  23. Big Picture General Variety Theorem = Monads + Duality Use monads to model the type of Use duality to relate varieties of languages and the algebras languages to pseudovarieties of recognizing them. finite algebras. One Eilenberg Theorem to Rule Them All December 6, 2016 10 / 28

  24. Profinite Words Consider Stone duality between boolean algebras and Stone spaces: ≃ � Stone BA op Pro( Set f ) One Eilenberg Theorem to Rule Them All December 6, 2016 11 / 28

  25. Profinite Words Consider Stone duality between boolean algebras and Stone spaces: ≃ � Stone BA op Pro( Set f ) Stone space of profinite words: Σ ∗ = inverse limit of all finite quotient monoids e : Σ ∗ ։ M . � One Eilenberg Theorem to Rule Them All December 6, 2016 11 / 28

  26. Profinite Words Consider Stone duality between boolean algebras and Stone spaces: ≃ � Stone BA op Pro( Set f ) Stone space of profinite words: Σ ∗ = inverse limit of all finite quotient monoids e : Σ ∗ ։ M . � Dual boolean algebra (Pippenger 1997): Reg (Σ) = regular languages over Σ . One Eilenberg Theorem to Rule Them All December 6, 2016 11 / 28

  27. Profinite Words Consider Stone duality between boolean algebras and Stone spaces: ≃ � Stone BA op Pro( Set f ) Stone space of profinite words: Σ ∗ = inverse limit of all finite quotient monoids e : Σ ∗ ։ M . � Dual boolean algebra (Pippenger 1997): Reg (Σ) = regular languages over Σ . This generalizes from T Σ = Σ ∗ to arbitrary monads T ! One Eilenberg Theorem to Rule Them All December 6, 2016 11 / 28

  28. General Duality Monad T on D as before. Additionally, let C be a locally finite variety with: ≃ � � C op D Pro( D f ) One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

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