One Eilenberg Theorem to Rule Them All
Stefan Milius
joint work with Jiřı́ Adámek, Liang-Ting Chen, Henning Urbat
December 6, 2016
Overview
Algebraic language theory:
Automata/languages vs. algebraic structures
One Eilenberg Theorem to Rule Them All
December 6, 2016
2 / 28
Overview
Algebraic language theory:
Automata/languages vs. algebraic structures
Categorical perspective:
Automata via algebras and coalgebras.
Languages via initial algebras and final
coalgebras.
η
µ
Algebra via Lawvere theories and monads.
Id −
→T ←
− T2
One Eilenberg Theorem to Rule Them All
December 6, 2016
2 / 28
Overview
Algebraic language theory:
Automata/languages vs. algebraic structures
Categorical perspective:
Automata via algebras and coalgebras.
Languages via initial algebras and final
coalgebras.
η
µ
Algebra via Lawvere theories and monads.
Id −
→T ←
− T2
Our goal: Categorical Algebraic Language Theory!
One Eilenberg Theorem to Rule Them All
December 6, 2016
2 / 28
Eilenberg’s Variety Theorem (1976)
varieties of
languages
∼
=
pseudovarieties of
monoids
One Eilenberg Theorem to Rule Them All
December 6, 2016
3 / 28
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
Eilenberg’s Variety Theorem (1976)
varieties of
languages
Variety of languages
For each alphabet Σ a set
VΣ ⊆ Reg(Σ) closed under
∪, ∩, (−){
∼
=
pseudovarieties of
monoids
Pseudovariety of monoids
A class of finite monoids closed
under quotients, submonoids and
finite products.
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
preimages of free monoid
morphisms f : ∆∗ → Σ∗ , i.e.
L ∈ VΣ ⇒ f −1 [L] ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
3 / 28
Other Eilenberg-Type Theorems
q
-
One Eilenberg Theorem to Rule Them All
December 6, 2016
4 / 28
Other Eilenberg-Type Theorems
q
-
Weaker closure properties:
Only ∪, ∩
Pin 1995
Only ∪
Polák 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
Other Eilenberg-Type Theorems
q
-
Weaker closure properties:
Other types of languages:
Only ∪, ∩
Pin 1995
Weighted languages
Reutenauer 1980
Only ∪
Polák 2001
Infinite words
Wilke 1991, Pin 1998
Only ⊕
Reutenauer 1980
Ordered words
Bedon et. al. 1998, 2005
Fewer monoid morphisms
Straubing 2002
Ranked trees
Almeida 1990, Steinby 1992
Fixed alphabet, no preimages
Gehrke, Grigorieff, Pin 2008
Binary trees
Salehi, Steinby 2008
Cost functions
Daviaud, Kuperberg, Pin 2016
One Eilenberg Theorem to Rule Them All
December 6, 2016
4 / 28
Other Eilenberg-Type Theorems
q
-
Weaker closure properties:
Other types of languages:
Only ∪, ∩
Pin 1995
Weighted languages
Reutenauer 1980
Only ∪
Polák 2001
Infinite words
Wilke 1991, Pin 1998
Only ⊕
Reutenauer 1980
Ordered words
Bedon et. al. 1998, 2005
Fewer monoid morphisms
Straubing 2002
Ranked trees
Almeida 1990, Steinby 1992
Fixed alphabet, no preimages
Gehrke, Grigorieff, Pin 2008
Binary trees
Salehi, Steinby 2008
Cost functions
Daviaud, Kuperberg, Pin 2016
- q
This talk
A General Variety Theorem that covers them all!
One Eilenberg Theorem to Rule Them All
December 6, 2016
4 / 28
Big Picture
General Variety Theorem
=
Monads
+
Duality
One Eilenberg Theorem to Rule Them All
December 6, 2016
5 / 28
Big Picture
General Variety Theorem
=
Monads
+
Duality
Use monads to model the type of
languages and the algebras
recognizing them.
Bojańczyk, DLT 2015
One Eilenberg Theorem to Rule Them All
December 6, 2016
5 / 28
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.
Bojańczyk, DLT 2015
Gehrke, Grigorieff, Pin, ICALP 2008
Adámek, Milius, Myers, Urbat,
FoSSaCS 2014, LICS 2015
One Eilenberg Theorem to Rule Them All
December 6, 2016
5 / 28
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
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
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
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
TΣ = Σ∗ on Set
and O = {0, 1}.
One Eilenberg Theorem to Rule Them All
December 6, 2016
7 / 28
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
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
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
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
Algebraic recognition
Definition
A language L : T Σ → O is recognizable if it factors through some finite
quotient algebra of the free T-algebra TΣ = (T Σ, µΣ ).
TΣ
∃e
L
/O
=
∃p
A
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
Algebraic recognition
Definition
A language L : T Σ → O is recognizable if it factors through some finite
quotient algebra of the free T-algebra TΣ = (T Σ, µΣ ).
TΣ
∃e
L
/O
=
∃p
A
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
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
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
Profinite Words
Consider Stone duality between boolean algebras and Stone spaces:
BAop
'
/ Stone
Pro(Setf )
Stone space of profinite words:
c∗ = inverse limit of all finite quotient monoids e : Σ∗ M.
Σ
One Eilenberg Theorem to Rule Them All
December 6, 2016
11 / 28
Profinite Words
Consider Stone duality between boolean algebras and Stone spaces:
BAop
'
/ Stone
Pro(Setf )
Stone space of profinite words:
c∗ = inverse limit of all finite quotient monoids e : Σ∗ M.
Σ
Dual boolean algebra (Pippenger 1997):
Reg(Σ) = regular languages over Σ.
One Eilenberg Theorem to Rule Them All
December 6, 2016
11 / 28
Profinite Words
Consider Stone duality between boolean algebras and Stone spaces:
BAop
'
/ Stone
Pro(Setf )
Stone space of profinite words:
c∗ = inverse limit of all finite quotient monoids e : Σ∗ M.
Σ
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
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
'
b
/D
Pro(Df )
One Eilenberg Theorem to Rule Them All
December 6, 2016
12 / 28
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
One Eilenberg Theorem to Rule Them All
December 6, 2016
12 / 28
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
b inverse limit of all finite quotient T-algebras TΣ A.
T̂ Σ ∈ D:
b →D
b is the profinite monad of T
T̂ : D
One Eilenberg Theorem to Rule Them All
December 6, 2016
12 / 28
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
b inverse limit of all finite quotient T-algebras TΣ A.
T̂ Σ ∈ D:
b →D
b is the profinite monad of T
T̂ : D
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
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
b inverse limit of all finite quotient T-algebras TΣ A.
T̂ Σ ∈ D:
b →D
b is the profinite monad of T
T̂ : D
Now O := (dual of 1), with 1 the free one-generated object in C.
b T̂ Σ, O)
Rec(Σ) ∼
= D(
One Eilenberg Theorem to Rule Them All
December 6, 2016
12 / 28
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
b inverse limit of all finite quotient T-algebras TΣ A.
T̂ Σ ∈ D:
b →D
b is the profinite monad of T
T̂ : D
Now O := (dual of 1), with 1 the free one-generated object in C.
b T̂ Σ, O) ∼
Rec(Σ) ∼
= D(
= C(1, (dual of T̂ Σ)) ∼
=
One Eilenberg Theorem to Rule Them All
December 6, 2016
12 / 28
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
b inverse limit of all finite quotient T-algebras TΣ A.
T̂ Σ ∈ D:
b →D
b is the profinite monad of T
T̂ : D
Now O := (dual of 1), with 1 the free one-generated
b T̂ Σ, O) ∼
Rec(Σ) ∼
= D(
= C(1, (dual of T̂ Σ)) ∼
=
One Eilenberg Theorem to Rule Them All
object in C.
dual of T̂ Σ
December 6, 2016
12 / 28
General Duality
Monad T on D as before. Additionally, let C be a locally finite variety with:
C op
C
boolean algebras
distributive lattices
vector spaces
'
b
/D
Pro(Df )
D
sets
posets
vector spaces
b
D
Stone spaces
Priestley spaces
Stone vector spaces
b inverse limit of all finite quotient T-algebras TΣ A.
T̂ Σ ∈ D:
b →D
b is the profinite monad of T
T̂ : D
Now O := (dual of 1), with 1 the free one-generated
b T̂ Σ, O) ∼
Rec(Σ) ∼
= D(
= C(1, (dual of T̂ Σ)) ∼
=
object in C.
dual of T̂ Σ
Thus Rec(Σ) can be viewed as an object of C!
One Eilenberg Theorem to Rule Them All
December 6, 2016
12 / 28
Eilenberg’s Variety Theorem (1976)
varieties of
languages
Variety of languages
For each alphabet Σ a set
VΣ ⊆ Reg(Σ) closed under
∪, ∩, (−){
∼
=
pseudovarieties of
monoids
Pseudovariety of monoids
A class of finite monoids closed
under quotients, submonoids and
finite products.
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
preimages of free monoid
morphisms f : ∆∗ → Σ∗ , i.e.
L ∈ VΣ ⇒ f −1 [L] ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
13 / 28
General Variety Theorem (2016)
varieties of
languages
Variety of languages
For each alphabet Σ a set
VΣ ⊆ Reg(Σ) closed under
∪, ∩, (−){
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
preimages of free monoid
morphisms f : ∆∗ → Σ∗ , i.e.
L ∈ VΣ ⇒ f −1 [L] ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
14 / 28
General Variety Theorem (2016)
varieties of
languages
Variety of languages
For each alphabet Σ a set
VΣ ⊆ Reg(Σ) closed under
∪, ∩, (−){
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
preimages of free monoid
morphisms f : ∆∗ → Σ∗ , i.e.
L ∈ VΣ ⇒ f −1 [L] ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
15 / 28
General Variety Theorem (2016)
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
preimages of free monoid
morphisms f : ∆∗ → Σ∗ , i.e.
L ∈ VΣ ⇒ f −1 [L] ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
16 / 28
General Variety Theorem (2016)
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
preimages of free monoid
morphisms f : ∆∗ → Σ∗ , i.e.
L ∈ VΣ ⇒ f −1 [L] ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
17 / 28
General Variety Theorem (2016)
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
preimages of free T-algebra
morphisms f : T∆ → TΣ, i.e.
L
(T Σ −
→ O) ∈ VΣ
f
L
⇒ (T ∆ −
→ TΣ −
→ O) ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
18 / 28
General Variety Theorem (2016)
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives (?)
x −1 Ly −1 = {w : xwy ∈ L}
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
preimages of free T-algebra
morphisms f : T∆ → TΣ, i.e.
L
(T Σ −
→ O) ∈ VΣ
f
L
⇒ (T ∆ −
→ TΣ −
→ O) ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
19 / 28
Derivatives: Monoid Case
Consider the unary operations
x(−)y
Σ∗ −−−−→ Σ∗
One Eilenberg Theorem to Rule Them All
(x, y ∈ Σ∗ ).
December 6, 2016
20 / 28
Derivatives: Monoid Case
Consider the unary operations
x(−)y
Σ∗ −−−−→ Σ∗
(x, y ∈ Σ∗ ).
L
For a language Σ∗ −
→ {0, 1},
x −1 Ly −1 = ( Σ∗
x(−)y
/ Σ∗ L / {0, 1} ).
One Eilenberg Theorem to Rule Them All
December 6, 2016
20 / 28
Derivatives: Monoid Case
Consider the unary operations
x(−)y
Σ∗ −−−−→ Σ∗
(x, y ∈ Σ∗ ).
L
For a language Σ∗ −
→ {0, 1},
x −1 Ly −1 = ( Σ∗
x(−)y
/ Σ∗ L / {0, 1} ).
For any surjective map e : Σ∗ A,
x(−)y
e carries a quotient monoid of Σ∗ ⇐⇒ all Σ∗ −−−−→ Σ∗ lift along e.
Σ∗
e
x(−)y
A
/ Σ∗
∃
e
/A
One Eilenberg Theorem to Rule Them All
December 6, 2016
20 / 28
Derivatives: General Case
Definition
u
Unary presentation U = { T Σ −
→ T Σ }: for any quotient e : T Σ A,
e carries a quotient T-algebra of TΣ ⇐⇒ all u ∈ U lift along e.
TΣ
e
u
A
/ TΣ
∃
e
/A
One Eilenberg Theorem to Rule Them All
December 6, 2016
21 / 28
Derivatives: General Case
Definition
u
Unary presentation U = { T Σ −
→ T Σ }: for any quotient e : T Σ A,
e carries a quotient T-algebra of TΣ ⇐⇒ all u ∈ U lift along e.
e
/ TΣ
u
TΣ
A
∃
e
/A
Definition
L
u
For a language T Σ −
→ O and T Σ −
→ T Σ in U, we have the derivative
u −1 L := ( T Σ
u
/ T Σ L / O ).
One Eilenberg Theorem to Rule Them All
December 6, 2016
21 / 28
General Variety Theorem
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives
x −1 Ly −1 = {w : xwy ∈ L}
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
preimages of free T-algebra
morphisms f : T∆ → TΣ, i.e.
L
(T Σ −
→ O) ∈ VΣ
f
L
⇒(T ∆ −
→ TΣ −
→ O) ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
22 / 28
General Variety Theorem
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives: for all u ∈ U,
L ∈ VΣ ⇒ u −1 L ∈ VΣ .
∼
=
pseudovarieties of
T-algebras
Pseudovariety of T-algebras
A class of finite T-algebras closed
under quotients, subalgebras and
finite products.
preimages of free T-algebra
morphisms f : T∆ → TΣ, i.e.
L
(T Σ −
→ O) ∈ VΣ
f
L
⇒(T ∆ −
→ TΣ −
→ O) ∈ V∆
One Eilenberg Theorem to Rule Them All
December 6, 2016
23 / 28
General Variety Theorem
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives and T-preimages.
∼
=
pseudovarieties of
T-algebras
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
General Variety Theorem
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives and T-preimages.
∼
=
pseudovarieties of
T-algebras
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
General Variety Theorem
varieties of
languages
Variety of languages
For each alphabet Σ a subobject
VΣ ⊆ Rec(Σ) in C closed under
derivatives and T-preimages.
∼
=
pseudovarieties of
T-algebras
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
Applications
C op ∼
= D̂
T
U
···
General Variety Theorem
varieties of
pseudovarieties of
∼
=
languages
T-algebras
q
-
One Eilenberg Theorem to Rule Them All
December 6, 2016
25 / 28
Applications
C op ∼
= D̂
T
U
···
General Variety Theorem
varieties of
pseudovarieties of
∼
=
languages
T-algebras
q
-
More than a dozen variety
theorems known in the literature.
One Eilenberg Theorem to Rule Them All
December 6, 2016
25 / 28
Some results covered by the General Variety Theorem
Languages of finite words:
Other types of languages:
∪, ∩,
Eilenberg 1976
Weighted languages
Reutenauer 1980
Only ∪, ∩
Pin 1995
Infinite words
Wilke 1991, Pin 1998
Only ∪
Polák 2001
Ordered words
Bedon et. al. 1998, 2005
Only ⊕
Reutenauer 1980
Ranked trees
Almeida 1990, Steinby 1992
Fewer monoid morphisms
Straubing 2002
Binary trees
Salehi, Steinby 2008
Fixed alphabet, no preimages
Gehrke, Grigorieff, Pin 2008
Cost functions
Daviaud, Kuperberg, Pin 2016
(−){
One Eilenberg Theorem to Rule Them All
December 6, 2016
26 / 28
Applications
C∼
= D̂
T
U
···
General Variety Theorem
varieties of
pseudovarieties of
∼
=
languages
T-algebras
q
-
More than a dozen variety
theorems known in the literature.
One Eilenberg Theorem to Rule Them All
December 6, 2016
27 / 28
Applications
C∼
= D̂
T
U
···
General Variety Theorem
varieties of
pseudovarieties of
∼
=
languages
T-algebras
q
-
More than a dozen variety
theorems known in the literature.
New results, e.g. extending work
of 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
Conclusions and Further Work
Eilenberg = Monads + Duality
One Eilenberg Theorem to Rule Them All
December 6, 2016
28 / 28
Conclusions and Further Work
Eilenberg = Monads + Duality
Categorical approach to algebraic language theory using monads
joining Bojańczyk, DLT 2015 and Adámek, M, Myers, Urbat, FoSSaCS 2014/LICS 2015
One Eilenberg Theorem to Rule Them All
December 6, 2016
28 / 28
Conclusions and Further Work
Eilenberg = Monads + Duality
Categorical approach to algebraic language theory using monads
joining Bojańczyk, DLT 2015 and Adámek, 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
Conclusions and Further Work
Eilenberg = Monads + Duality
Categorical approach to algebraic language theory using monads
joining Bojańczyk, DLT 2015 and Adámek, 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, Adámek, Milius, Urbat, FoSSaCS 2016
One Eilenberg Theorem to Rule Them All
December 6, 2016
28 / 28
Conclusions and Further Work
Eilenberg = Monads + Duality
Categorical approach to algebraic language theory using monads
joining Bojańczyk, DLT 2015 and Adámek, 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, Adámek, 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
Conclusions and Further Work
Eilenberg = Monads + Duality
Categorical approach to algebraic language theory using monads
joining Bojańczyk, DLT 2015 and Adámek, 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, Adámek, 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, Petrişan 2009
One Eilenberg Theorem to Rule Them All
December 6, 2016
28 / 28
Conclusions and Further Work
Eilenberg = Monads + Duality
Categorical approach to algebraic language theory using monads
joining Bojańczyk, DLT 2015 and Adámek, 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, Adámek, 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, Petrişan 2009
Monadic second order logic for a monad?
One Eilenberg Theorem to Rule Them All
December 6, 2016
28 / 28