[PPT] - One Eilenberg Theorem to Rule Them All Stefan Milius joint work PowerPoint Presentation

SLIDE 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

SLIDE 2

Overview

Algebraic language theory: Automata/languages vs. algebraic structures

One Eilenberg Theorem to Rule Them All December 6, 2016 2 / 28

SLIDE 3

Overview

Algebraic language theory: Automata/languages vs. algebraic structures Categorical perspective: Id

η

− → T

µ

← − T 2 Automata via algebras and coalgebras. Languages via initial algebras and final coalgebras. Algebra via Lawvere theories and monads.

One Eilenberg Theorem to Rule Them All December 6, 2016 2 / 28

SLIDE 4

Overview

Algebraic language theory: Automata/languages vs. algebraic structures Categorical perspective: Id

η

− → T

µ

← − T 2 Automata via algebras and coalgebras. Languages via initial algebras and final coalgebras. Algebra via Lawvere theories and monads. Our goal: Categorical Algebraic Language Theory!

One Eilenberg Theorem to Rule Them All December 6, 2016 2 / 28

SLIDE 5

Eilenberg’s Variety Theorem (1976)

varieties of languages

∼

= pseudovarieties of monoids

One Eilenberg Theorem to Rule Them All

December 6, 2016 3 / 28

SLIDE 6

Eilenberg’s Variety Theorem (1976)

varieties of languages

∼

= pseudovarieties of 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

SLIDE 7

Eilenberg’s Variety Theorem (1976)

varieties of languages

∼

= pseudovarieties of monoids

Variety of languages

For each alphabet Σ a set VΣ ⊆ Reg(Σ) closed under ∪, ∩, (−)∁ derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free monoid morphisms f : ∆∗ → Σ∗, i.e. L ∈ VΣ ⇒ f −1[L] ∈ V∆ 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

SLIDE 8

Other Eilenberg-Type Theorems

One Eilenberg Theorem to Rule Them All

December 6, 2016 4 / 28

SLIDE 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

SLIDE 10

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

Other types of languages:

Weighted languages Reutenauer 1980 Infinite words Wilke 1991, Pin 1998 Ordered words Bedon et. al. 1998, 2005 Ranked trees Almeida 1990, Steinby 1992 Binary trees Salehi, Steinby 2008 Cost functions Daviaud, Kuperberg, Pin 2016

One Eilenberg Theorem to Rule Them All December 6, 2016 4 / 28

SLIDE 11

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

Other types of languages:

Weighted languages Reutenauer 1980 Infinite words Wilke 1991, Pin 1998 Ordered words Bedon et. al. 1998, 2005 Ranked trees Almeida 1990, Steinby 1992 Binary trees 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

SLIDE 12

Big Picture

General Variety Theorem = Monads + Duality

One Eilenberg Theorem to Rule Them All December 6, 2016 5 / 28

SLIDE 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

SLIDE 14

Big Picture

General Variety Theorem = Monads + Duality Use monads to model the type of languages and the algebras recognizing them.

Boja´ nczyk, DLT 2015

Use duality to relate varieties of languages to pseudovarieties of finite algebras.

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

SLIDE 15

Big Picture

General Variety Theorem = Monads + Duality Use monads to model the type of languages and the algebras recognizing them. Use duality to relate varieties of languages to pseudovarieties of finite algebras.

One Eilenberg Theorem to Rule Them All December 6, 2016 6 / 28

SLIDE 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

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Languages

Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : TΣ → 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

SLIDE 18

Languages

Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : TΣ → 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 TΣ = Σ∗ on Set and O = {0, 1}.

One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

SLIDE 19

Languages

Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : TΣ → 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 TΣ = Σ∗ on Set and O = {0, 1}. Languages of finite and infinite words: free ω-semigroup monad T(Σ, ∅) = (Σ+, Σω) on Set2 and O = ({0, 1}, {0, 1}).

One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

SLIDE 20

Languages

Fix a monad T on a locally finite variety D (with finitely many sorts). Definition Language = morphism L : TΣ → 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 TΣ = Σ∗ on Set and O = {0, 1}. Languages of finite and infinite words: free ω-semigroup monad T(Σ, ∅) = (Σ+, Σω) on Set2 and O = ({0, 1}, {0, 1}). Weighted languages (D = vector spaces), tree languages (D = Set3), cost functions (D = posets), . . .

One Eilenberg Theorem to Rule Them All December 6, 2016 7 / 28

SLIDE 21

Algebraic recognition

Definition A language L : TΣ → O is recognizable if it factors through some finite quotient algebra of the free T-algebra TΣ = (TΣ, µΣ). TΣ

L

∃e

O A

∃p

Languages of finite words: free monoid monad

TΣ = 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

SLIDE 22

Algebraic recognition

Definition A language L : TΣ → O is recognizable if it factors through some finite quotient algebra of the free T-algebra TΣ = (TΣ, µΣ). TΣ

L

∃e

O A

∃p

Languages of finite and infinite words: free ω-semigroup monad

T(Σ, ∅) = (Σ+, Σω) on Set2 and O = ({0, 1}, {0, 1}). Recognizable languages = regular ∞-languages

One Eilenberg Theorem to Rule Them All December 6, 2016 9 / 28

SLIDE 23

Big Picture

General Variety Theorem = Monads + Duality Use monads to model the type of languages and the algebras recognizing them. Use duality to relate varieties of languages to pseudovarieties of finite algebras.

One Eilenberg Theorem to Rule Them All December 6, 2016 10 / 28

SLIDE 24

Profinite Words

Consider Stone duality between boolean algebras and Stone spaces: BAop

≃

Stone

Pro(Setf )

One Eilenberg Theorem to Rule Them All December 6, 2016 11 / 28

SLIDE 25

Profinite Words

Consider Stone duality between boolean algebras and Stone spaces: BAop

≃

Stone

Pro(Setf ) 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

SLIDE 26

Profinite Words

Consider Stone duality between boolean algebras and Stone spaces: BAop

≃

Stone

Pro(Setf ) 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

SLIDE 27

Profinite Words

Consider Stone duality between boolean algebras and Stone spaces: BAop

≃

Stone

Pro(Setf ) 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 TΣ = Σ∗ to arbitrary monads T!

One Eilenberg Theorem to Rule Them All December 6, 2016 11 / 28

SLIDE 28

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df )

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 29

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 30

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces ˆ TΣ ∈ D: inverse limit of all finite quotient T-algebras TΣ ։ A.

ˆ T : D → D is the profinite monad of T

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 31

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces ˆ TΣ ∈ D: inverse limit of all finite quotient T-algebras TΣ ։ A.

ˆ T : D → D is the profinite monad of T

Now O := (dual of 1), with 1 the free one-generated object in C.

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 32

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces ˆ TΣ ∈ D: inverse limit of all finite quotient T-algebras TΣ ։ A.

ˆ T : D → D is the profinite monad of T

Now O := (dual of 1), with 1 the free one-generated object in C. Rec(Σ) ∼ = D( ˆ TΣ, O)

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 33

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces ˆ TΣ ∈ D: inverse limit of all finite quotient T-algebras TΣ ։ A.

ˆ T : D → D is the profinite monad of T

Now O := (dual of 1), with 1 the free one-generated object in C. Rec(Σ) ∼ = D( ˆ TΣ, O) ∼ = C(1, (dual of ˆ TΣ)) ∼ =

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 34

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces ˆ TΣ ∈ D: inverse limit of all finite quotient T-algebras TΣ ։ A.

ˆ T : D → D is the profinite monad of T

Now O := (dual of 1), with 1 the free one-generated object in C. Rec(Σ) ∼ = D( ˆ TΣ, O) ∼ = C(1, (dual of ˆ TΣ)) ∼ =

dual of ˆ

TΣ

One Eilenberg Theorem to Rule Them All

December 6, 2016 12 / 28

SLIDE 35

General Duality

Monad T on D as before. Additionally, let C be a locally finite variety with: Cop

≃

D Pro(Df ) C D

D

boolean algebras sets Stone spaces distributive lattices posets Priestley spaces vector spaces vector spaces Stone vector spaces ˆ TΣ ∈ D: inverse limit of all finite quotient T-algebras TΣ ։ A.

ˆ T : D → D is the profinite monad of T

Now O := (dual of 1), with 1 the free one-generated object in C. Rec(Σ) ∼ = D( ˆ TΣ, O) ∼ = C(1, (dual of ˆ TΣ)) ∼ =

dual of ˆ

TΣ

Thus Rec(Σ) can be viewed as an object of C!

One Eilenberg Theorem to Rule Them All December 6, 2016 12 / 28

SLIDE 36

Eilenberg’s Variety Theorem (1976)

varieties of languages

∼

= pseudovarieties of monoids

Variety of languages

For each alphabet Σ a set VΣ ⊆ Reg(Σ) closed under ∪, ∩, (−)∁ derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free monoid morphisms f : ∆∗ → Σ∗, i.e. L ∈ VΣ ⇒ f −1[L] ∈ V∆ 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 13 / 28

SLIDE 37

General Variety Theorem (2016)

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a set VΣ ⊆ Reg(Σ) closed under ∪, ∩, (−)∁ derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free monoid morphisms f : ∆∗ → Σ∗, i.e. L ∈ VΣ ⇒ f −1[L] ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 14 / 28

SLIDE 38

General Variety Theorem (2016)

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a set VΣ ⊆ Reg(Σ) closed under ∪, ∩, (−)∁ derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free monoid morphisms f : ∆∗ → Σ∗, i.e. L ∈ VΣ ⇒ f −1[L] ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 15 / 28

SLIDE 39

General Variety Theorem (2016)

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free monoid morphisms f : ∆∗ → Σ∗, i.e. L ∈ VΣ ⇒ f −1[L] ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 16 / 28

SLIDE 40

General Variety Theorem (2016)

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free monoid morphisms f : ∆∗ → Σ∗, i.e. L ∈ VΣ ⇒ f −1[L] ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 17 / 28

SLIDE 41

General Variety Theorem (2016)

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free T-algebra morphisms f : T∆ → TΣ, i.e. (TΣ L − → O) ∈ VΣ ⇒ (T∆ f − → TΣ L − → O) ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 18 / 28

SLIDE 42

General Variety Theorem (2016)

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives (?) x−1Ly−1 = {w : xwy ∈ L} preimages of free T-algebra morphisms f : T∆ → TΣ, i.e. (TΣ L − → O) ∈ VΣ ⇒ (T∆ f − → TΣ L − → O) ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 19 / 28

SLIDE 43

Derivatives: Monoid Case

Consider the unary operations Σ∗ x(−)y − − − − → Σ∗ (x, y ∈ Σ∗).

One Eilenberg Theorem to Rule Them All December 6, 2016 20 / 28

SLIDE 44

Derivatives: Monoid Case

Consider the unary operations Σ∗ x(−)y − − − − → Σ∗ (x, y ∈ Σ∗). For a language Σ∗ L − → {0, 1}, x−1Ly−1 = ( Σ∗

x(−)y

Σ∗

L {0, 1} ).

One Eilenberg Theorem to Rule Them All December 6, 2016 20 / 28

SLIDE 45

Derivatives: Monoid Case

Consider the unary operations Σ∗ x(−)y − − − − → Σ∗ (x, y ∈ Σ∗). For a language Σ∗ L − → {0, 1}, x−1Ly−1 = ( Σ∗

x(−)y

Σ∗

L {0, 1} ).

For any surjective map e : Σ∗ ։ A, e carries a quotient monoid of Σ∗ ⇐ ⇒ all Σ∗ x(−)y − − − − → Σ∗ lift along e. Σ∗

e x(−)y Σ∗ e

A

∃

A

One Eilenberg Theorem to Rule Them All December 6, 2016 20 / 28

SLIDE 46

Derivatives: General Case

Definition Unary presentation U = { TΣ u − → TΣ }: for any quotient e : TΣ ։ A, e carries a quotient T-algebra of TΣ ⇐ ⇒ all u ∈ U lift along e. TΣ

e u

TΣ

e

A

∃

A

One Eilenberg Theorem to Rule Them All December 6, 2016 21 / 28

SLIDE 47

Derivatives: General Case

Definition Unary presentation U = { TΣ u − → TΣ }: for any quotient e : TΣ ։ A, e carries a quotient T-algebra of TΣ ⇐ ⇒ all u ∈ U lift along e. TΣ

e u

TΣ

e

A

∃

A

Definition For a language TΣ L − → O and TΣ u − → TΣ in U, we have the derivative u−1L := ( TΣ

u TΣ L O ).

One Eilenberg Theorem to Rule Them All December 6, 2016 21 / 28

SLIDE 48

General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives x−1Ly−1 = {w : xwy ∈ L} preimages of free T-algebra morphisms f : T∆ → TΣ, i.e. (TΣ L − → O) ∈ VΣ ⇒(T∆ f − → TΣ L − → O) ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 22 / 28

SLIDE 49

General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives: for all u ∈ U, L ∈ VΣ ⇒ u−1L ∈ VΣ. preimages of free T-algebra morphisms f : T∆ → TΣ, i.e. (TΣ L − → O) ∈ VΣ ⇒(T∆ f − → TΣ L − → O) ∈ V∆ Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 23 / 28

SLIDE 50

General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives and T-preimages. Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products.

One Eilenberg Theorem to Rule Them All December 6, 2016 24 / 28

SLIDE 51

General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives and T-preimages. Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products. How to prove the theorem?

One Eilenberg Theorem to Rule Them All December 6, 2016 24 / 28

SLIDE 52

General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

Variety of languages

For each alphabet Σ a subobject VΣ ⊆ Rec(Σ) in C closed under derivatives and T-preimages. Pseudovariety of T-algebras A class of finite T-algebras closed under quotients, subalgebras and finite products. How to prove the theorem?

Dualize!

One Eilenberg Theorem to Rule Them All December 6, 2016 24 / 28

SLIDE 53

Applications

Cop ∼ = ˆ D

T
U
· · ·
General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

One Eilenberg Theorem to Rule Them All

December 6, 2016 25 / 28

SLIDE 54

Applications

Cop ∼ = ˆ D

T
U
· · ·
General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

More than a dozen variety

theorems known in the literature.

One Eilenberg Theorem to Rule Them All December 6, 2016 25 / 28

SLIDE 55

Some results covered by the General Variety Theorem

Languages of finite words:

∪, ∩, (−)∁ Eilenberg 1976 Only ∪, ∩ Pin 1995 Only ∪ Pol´ ak 2001 Only ⊕ Reutenauer 1980 Fewer monoid morphisms Straubing 2002 Fixed alphabet, no preimages Gehrke, Grigorieff, Pin 2008

Other types of languages:

Weighted languages Reutenauer 1980 Infinite words Wilke 1991, Pin 1998 Ordered words Bedon et. al. 1998, 2005 Ranked trees Almeida 1990, Steinby 1992 Binary trees Salehi, Steinby 2008 Cost functions Daviaud, Kuperberg, Pin 2016

One Eilenberg Theorem to Rule Them All December 6, 2016 26 / 28

SLIDE 56

Applications

C ∼ = ˆ D

T
U
· · ·
General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

More than a dozen variety

theorems known in the literature.

One Eilenberg Theorem to Rule Them All December 6, 2016 27 / 28

SLIDE 57

Applications

C ∼ = ˆ D

T
U
· · ·
General Variety Theorem

varieties of languages

∼

= pseudovarieties of T-algebras

More than a dozen variety

theorems known in the literature. New results, e.g. extending work

f Gehrke, Grigorieff, Pin (2008)

from finite words to infinite words, trees, cost functions, . . . .

One Eilenberg Theorem to Rule Them All December 6, 2016 27 / 28

SLIDE 58

Conclusions and Further Work

Eilenberg = Monads + Duality

One Eilenberg Theorem to Rule Them All December 6, 2016 28 / 28

SLIDE 59

Conclusions and Further Work

Eilenberg = Monads + Duality

Categorical approach to algebraic language theory using monads

joining Boja´ nczyk, DLT 2015 and Ad´ amek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015

One Eilenberg Theorem to Rule Them All December 6, 2016 28 / 28

SLIDE 60

Conclusions and Further Work

Eilenberg = Monads + Duality

Categorical approach to algebraic language theory using monads

joining Boja´ nczyk, DLT 2015 and Ad´ amek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015

A General Eilenberg Theorem with many applications Isolates the algebraic part of the proof of Eilenberg-type correspondences

Nontrivial work lies in finding the right monad and unary presentation

One Eilenberg Theorem to Rule Them All December 6, 2016 28 / 28

SLIDE 61

Conclusions and Further Work

Eilenberg = Monads + Duality

Categorical approach to algebraic language theory using monads

joining Boja´ nczyk, DLT 2015 and Ad´ amek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015

A General Eilenberg Theorem with many applications Isolates the algebraic part of the proof of Eilenberg-type correspondences

Nontrivial work lies in finding the right monad and unary presentation

Further work:

General Reiterman Theorem: pseudovarieties vs. profinite equations

Chen, Ad´ amek, Milius, Urbat, FoSSaCS 2016

One Eilenberg Theorem to Rule Them All December 6, 2016 28 / 28

SLIDE 62

Conclusions and Further Work

Eilenberg = Monads + Duality

Categorical approach to algebraic language theory using monads

joining Boja´ nczyk, DLT 2015 and Ad´ amek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015

A General Eilenberg Theorem with many applications Isolates the algebraic part of the proof of Eilenberg-type correspondences

Nontrivial work lies in finding the right monad and unary presentation

Further work:

General Reiterman Theorem: pseudovarieties vs. profinite equations

Chen, Ad´ amek, Milius, Urbat, FoSSaCS 2016

Non-regular languages ?

Ballester-Bolinches, Cosme-Llopez, Rutten 2015, Behle, Krebs, Reifferscheid 2011

One Eilenberg Theorem to Rule Them All December 6, 2016 28 / 28

SLIDE 63

Conclusions and Further Work

Eilenberg = Monads + Duality

Categorical approach to algebraic language theory using monads

joining Boja´ nczyk, DLT 2015 and Ad´ amek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015

A General Eilenberg Theorem with many applications Isolates the algebraic part of the proof of Eilenberg-type correspondences

Nontrivial work lies in finding the right monad and unary presentation

Further work:

General Reiterman Theorem: pseudovarieties vs. profinite equations

Chen, Ad´ amek, Milius, Urbat, FoSSaCS 2016

Non-regular languages ?

Ballester-Bolinches, Cosme-Llopez, Rutten 2015, Behle, Krebs, Reifferscheid 2011

Nominal Stone duality and data languages??

Gabbay, Litak, Petri¸ san 2009

One Eilenberg Theorem to Rule Them All December 6, 2016 28 / 28

SLIDE 64

Conclusions and Further Work

Eilenberg = Monads + Duality

Categorical approach to algebraic language theory using monads

joining Boja´ nczyk, DLT 2015 and Ad´ amek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015

A General Eilenberg Theorem with many applications Isolates the algebraic part of the proof of Eilenberg-type correspondences

Nontrivial work lies in finding the right monad and unary presentation

Further work:

General Reiterman Theorem: pseudovarieties vs. profinite equations

Chen, Ad´ amek, Milius, Urbat, FoSSaCS 2016

Non-regular languages ?

Ballester-Bolinches, Cosme-Llopez, Rutten 2015, Behle, Krebs, Reifferscheid 2011

Nominal Stone duality and data languages??

Gabbay, Litak, Petri¸ san 2009

Monadic second order logic for a monad?

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