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Duality and Automata Theory
Duality and Automata Theory
Mai Gehrke
Universite Paris VII and CNRS
Joint work with Serge Grigorieff and Jean-Eric Pin
Duality and Automata Theory
Elements of automata theory
A finite automaton1 2
3
a
b
b a
a, b
The states are {1, 2, 3}.The initial state is 1, the final states are 1 and 2.The alphabet is A = {a, b} The transitions are
1 · a = 2 2 · a = 3 3 · a = 3
1 · b = 3 2 · b = 1 3 · b = 3
Duality and Automata Theory
Elements of automata theory
Recognition by automata1 2
3
a
b
b a
a, b
Transitions extend to words: 1 · aba = 2, 1 · abb = 3.The language recognized by the automaton is the set of words usuch that 1 · u is a final state. Here:
L(A) = (ab)∗ ∪ (ab)∗a
where ∗ means arbitrary iteration of the product.
Duality and Automata Theory
Elements of automata theory
Rational and recognizable languages
A language is recognizable provided it is recognized by some finiteautomaton.
A language is rational provided it belongs to the smallest class oflanguages containing the finite languages which is closed underunion, product and star.
Theorem: [Kleene ’54] A language is rational iff it is recognizable.
Example: L(A) = (ab)∗ ∪ (ab)∗a.
Duality and Automata Theory
Connection to logic on words
Logic on words
To each non-empty word u is associated a structure
Mu = ({1, 2, . . . , |u|}, <, (a)a∈A)
where a is interpreted as the set of integers i such that the i-thletter of u is an a, and < as the usual order on integers.
Example:
Let u = abbaab then
Mu = ({1, 2, 3, 4, 5, 6}, <, (a,b))
where a = {1, 4, 5} and b = {2, 3, 6}.
Duality and Automata Theory
Connection to logic on words
Some examples
The formula φ = ∃x ax interprets as:
There exists a position x in u such thatthe letter in position x is an a.
This defines the language L(φ) = A∗aA∗.
The formula ∃x ∃y (x < y) ∧ ax ∧ by defines the languageA∗aA∗bA∗.
The formula ∃x ∀y [(x < y) ∨ (x = y)] ∧ ax defines the languageaA∗.
Duality and Automata Theory
Connection to logic on words
Defining the set of words of even length
Macros:
(x < y) ∨ (x = y) means x 6 y∀y x 6 y means x = 1
∀y y 6 x means x = |u|x < y ∧ ∀z (x < z → y 6 z) means y = x+ 1
Let φ = ∃X (1 /∈ X ∧ |u| ∈ X ∧ ∀x (x ∈ X ↔ x+ 1 /∈ X))
Then 1 /∈ X, 2 ∈ X, 3 /∈ X, 4 ∈ X, . . . , |u| ∈ X. Thus
L(φ) = {u | |u| is even} = (A2)∗
Duality and Automata Theory
Connection to logic on words
Monadic second order
Only second order quantifiers over unary predicates are allowed.
Theorem: (Buchi ’60, Elgot ’61)
Monadic second order captures exactly the recognizable languages.
Theorem: (McNaughton-Papert ’71)
First order captures star-free languages
(star-free = the ones that can be obtained from the alphabet usingthe Boolean operations on languages and lifted concatenationproduct only).
How does one decide the complexity of a given language???
Duality and Automata Theory
The algebraic theory of automata
Algebraic theory of automata
Theorem: [Myhill ’53, Rabin-Scott ’59] There is an effective way ofassociating with each finite automaton, A, a finite monoid,(MA, ·, 1).
Theorem: [Schutzenberger ’65] A recognizable language is star-freeif and only if the associated monoid is aperiodic, i.e., M is suchthat there exists n > 0 with xn = xn+1 for each x ∈M .
Submonoid generated by x:
1 x x2 x3
. . .x i+p = x i
x i+1 x i+2
x i+p!1
This makes starfreeness decidable!
Duality and Automata Theory
The algebraic theory of automata
Eilenberg-Reiterman theory
Varieties
of languagesProfinite
identities
Varieties of
finite monoids
Decidability
Eilenberg Reiterman
In good
cases
A variety of monoids here means a class of finite monoids closedunder homomorphic images, submonoids, and finite products
Various generalisations: [Pin 1995], [Pin-Weil 1996], [Pippenger1997], [Polak 2001], [Esik 2002], [Straubing 2002], [Kunc 2003]
Duality and Automata Theory
Duality and automata I
Eilenberg, Reiterman, and Stone
Classes of monoids
algebras of languages equational theories
(1) (2)
(3)
(1) Eilenberg theorems(2) Reiterman theorems(3) extended Stone/Priestley duality
(3) allows generalisation to non-varieties and even to non-regularlanguages
Duality and Automata Theory
Duality and automata I
Assigning a Boolean algebra to each language
For x ∈ A∗ and L ⊆ A∗, define the quotient
x−1L= {u ∈ A∗ | xu ∈ L}(= {x}\L)
andLy−1= {u ∈ A∗ | uy ∈ L}(= L/{y})
Given a language L ⊆ A∗, let B(L) be the Boolean algebra oflanguages generated by
{x−1Ly−1 | x, y ∈ A∗
}
NB! B(L) is closed under quotients since the quotient operationscommute with the Boolean operations.
Duality and Automata Theory
Duality and automata I
Quotients of a recognizable language
1 2
3
a
b
b a
a, b
L(A) = (ab)∗ ∪ (ab)∗a
a−1L= {u ∈ A∗ | au ∈ L} = (ba)∗b ∪ (ba)∗
La−1= {u ∈ A∗ | ua ∈ L} = (ab)∗
b−1L= {u ∈ A∗ | bu ∈ L} = ∅
NB! These are recognized by the same underlying machine.
Duality and Automata Theory
Duality and automata I
B(L) for a recognizable language
If L is recognizable then the generating set of B(L) is finite sinceall the languages are recognized by the same machine with varyingsets of initial and final states.
Since B(L) is finite it is also closed under residuation with respectto arbitrary denominators and is thus a bi-module for P(A∗).
For any K ∈ B(L) and any S ∈ P(A∗)
S\K =⋂
u∈Su−1K ∈ B(L)
K/S =⋂
u∈SKu−1 ∈ B(L)
Duality and Automata Theory
Duality and automata I
B(L) for a recognizable language
If L is recognizable then the generating set of B(L) is finite sinceall the languages are recognized by the same machine with varyingsets of initial and final states.
Since B(L) is finite it is also closed under residuation with respectto arbitrary denominators and is thus a bi-module for P(A∗).
For any K ∈ B(L) and any S ∈ P(A∗)
S\K =⋂
u∈Su−1K ∈ B(L)
K/S =⋂
u∈SKu−1 ∈ B(L)
Duality and Automata Theory
Duality and automata I
The syntactic monoid via duality
For a recognizable language L, the algebra
(B(L),∩,∪, ( )c, 0, 1, \, /) ⊆ P(A∗)is the Boolean residuation bi-module generated by L.
Theorem: For a recognizable language L, the dual space of thealgebra (B(L),∩,∪, ( )c, 0, 1, \, /) is the syntactic monoid of L.
– including the product operation!
Duality and Automata Theory
Duality and automata I
The syntactic monoid via duality
For a recognizable language L, the algebra
(B(L),∩,∪, ( )c, 0, 1, \, /) ⊆ P(A∗)is the Boolean residuation bi-module generated by L.
Theorem: For a recognizable language L, the dual space of thealgebra (B(L),∩,∪, ( )c, 0, 1, \, /) is the syntactic monoid of L.
– including the product operation!
Duality and Automata Theory
Duality – the finite case
Finite lattices and join-irreducibles
x 6= 0 is join-irreducible iff x = y ∨ z ⇒ (x = y or x = z)
D
J(D)Join-irreducibles
D(J(D))Downset lattice
Duality and Automata Theory
Duality – the finite case
Finite lattices and join-irreducibles
x 6= 0 is join-irreducible iff x = y ∨ z ⇒ (x = y or x = z)
D
J(D)Join-irreducibles
D(J(D))Downset lattice
Duality and Automata Theory
Duality – the finite case
The finite Boolean case
In the Boolean case, the join-irreducibles (J) are exactly the atoms(At) and the downset lattice (D) of the atoms is just the powerset (P)
∅
{a} {b} {c}
{a, b} {a, c} {b, c}
X
Duality and Automata Theory
Duality – the finite case
The finite Boolean case
In the Boolean case, the join-irreducibles (J) are exactly the atoms(At) and the downset lattice (D) of the atoms is just the powerset (P)
a b c
∅
{a} {b} {c}
{a, b} {a, c} {b, c}
X
Duality and Automata Theory
Duality – the finite case
The finite Boolean case
In the Boolean case, the join-irreducibles (J) are exactly the atoms(At) and the downset lattice (D) of the atoms is just the powerset (P)
a b c
∅
{a} {b} {c}
{a, b} {a, c} {b, c}
X
Duality and Automata Theory
Duality – the finite case
Categorical Duality
DL+ — complete and completely distributive lattices with enoughcompletely join irreducibles, with complete homomorphisms
POS — partially ordered sets with order preserving maps
+
+
D
D( ( )) != (D( )) !=
: " ! ( ) # ( ) : ( )
D( ) # D( ) : D( ) ! : "
D(J(D)) ∼= D J(D(P )) ∼= P
h : D → E ! J(D)← J(E) : J(h)
D(P )← D(Q) : D(f) ! f : P → Q
Dual of a morphism is given by lower/upper adjoint
Duality and Automata Theory
Duality – the finite case
Boolean subalgebras correspond to partitions
Let B = <(ab)∗, a(ba)∗, b(ab)∗, (ba)∗> be the Boolean subalgebraof P(A∗) generated by these four languages.
The corresponding equivalence relation on Atoms(P(A∗)) = A∗
gives the partition
{(ab)+, {ε}, (ba)+, a(ba)∗, b(ab)∗, Z}
where Z is the complement of the union of all the others
The dual of the embedding B ↪→ P(A∗) is the quotient map
q : A∗ � {(ab)+, {ε}, (ba)+, a(ba)∗, b(ab)∗, Z}
Duality and Automata Theory
Duality – the finite case
Boolean subalgebras correspond to partitions
Let B = <(ab)∗, a(ba)∗, b(ab)∗, (ba)∗> be the Boolean subalgebraof P(A∗) generated by these four languages.
The corresponding equivalence relation on Atoms(P(A∗)) = A∗
gives the partition
{(ab)+, {ε}, (ba)+, a(ba)∗, b(ab)∗, Z}
where Z is the complement of the union of all the others
The dual of the embedding B ↪→ P(A∗) is the quotient map
q : A∗ � {(ab)+, {ε}, (ba)+, a(ba)∗, b(ab)∗, Z}
Duality and Automata Theory
Duality – the finite case
Boolean subalgebras correspond to partitions
Let B = <(ab)∗, a(ba)∗, b(ab)∗, (ba)∗> be the Boolean subalgebraof P(A∗) generated by these four languages.
The corresponding equivalence relation on Atoms(P(A∗)) = A∗
gives the partition
{(ab)+, {ε}, (ba)+, a(ba)∗, b(ab)∗, Z}
where Z is the complement of the union of all the others
The dual of the embedding B ↪→ P(A∗) is the quotient map
q : A∗ � {(ab)+, {ε}, (ba)+, a(ba)∗, b(ab)∗, Z}
Duality and Automata Theory
Duality – the finite case
Duality for operators
+
+
D
D( ( )) != (D( )) !=
: " ! ( ) # ( ) : ( )
D( ) # D( ) : D( ) ! : "operator ♦ : Dn → D ! R ⊆ X ×Xn relation
♦ 7→ R♦ = {(x, x) | x 6 ♦(x)}(♦R : S 7→ R−1[S1 × . . . Sn]
)←[ R
Duality and Automata Theory
Duality – the finite case
Duality for dual operators
What do we do for a � that preserves finite meets (1 and ∧) ineach coordinate?
dual operator � : Dn → D ! S ⊆M ×Mn relation
� 7→ S� = {(m,m) | m > �(m)}(�S : u 7→
∧S−1[↑u ∩Mn]
)←[ S
where M =M(D) and we use the duality with respect tomeet-irreducibles instead of duality with respect to join-irreducibles
Duality and Automata Theory
Duality – the finite case
The dual of residuation operations
The residuation operation \ : B × B → B
sends∨ 7→ ∧
in the first coordinatesends
∧ 7→ ∧in the second coordinate
R(X,Y, Z) ⇐⇒ X\(Zc) ⊆ Y c
⇐⇒ Y 6⊆ X\Zc
⇐⇒ XY 6⊆ Zc
⇐⇒ XY ∩ Z 6= ∅
Part I: Duality and recognition
The syntactic monoid of a recognizable language and duality
The dual of residuation operations
The residuation operation \ : B ! B " B
sends! #" "
in the first coordinatesends
" #" "in the second coordinate
R(X ,Y ,Z ) $% X\(Z c) & Y c
$% Y '& X\Z c
$% XY '& Z c
X
B(L)
X c
Duality and Automata Theory
Duality – the finite case
The syntactic monoid
L = (ab)∗ −→ M(L) =
· 1 a ba b ab 0
1 1 a ba b ab 0
a a 0 a ab 0 0
ba ba 0 ba b 0 0
b b ba b 0 b 0
ab ab a 0 0 ab 0
0 0 0 0 0 0 0
Syntactic monoid
This monoid is aperiodic since 1 = 12, a2 = 0 = a3, ba = ba2,b2 = 0 = b3, ab = ab2, and 0 = 02
Indeed, L is star-free since Lc = bA∗ ∪A∗a ∪A∗aaA∗ ∪A∗bbA∗and A∗ = ∅c
Duality and Automata Theory
Duality – the finite case
The syntactic monoid
L = (ab)∗ −→ M(L) =
· 1 a ba b ab 0
1 1 a ba b ab 0
a a 0 a ab 0 0
ba ba 0 ba b 0 0
b b ba b 0 b 0
ab ab a 0 0 ab 0
0 0 0 0 0 0 0
Syntactic monoid
This monoid is aperiodic since 1 = 12, a2 = 0 = a3, ba = ba2,b2 = 0 = b3, ab = ab2, and 0 = 02
Indeed, L is star-free since Lc = bA∗ ∪A∗a ∪A∗aaA∗ ∪A∗bbA∗and A∗ = ∅c
Duality and Automata Theory
Duality – the finite case
Duality for Boolean residuation ideals
The dual of a Boolean residuation ideal (= Boolean subresiduation bi-module)
B(L) ↪→ P(A∗)
is a quotient monoid
A∗ � Atoms(B(L))
This dual correspondence goes through to the setting oftopological duality!
Duality and Automata Theory
Stone representation and Priestley duality
Not enough join-irreducibles
1
0
D = 0⊕ (Nop × Nop) has no join irreducible elements whatsoever!
Duality and Automata Theory
Stone representation and Priestley duality
Adding limits
Prime filters are subsets witnessing missing join-irreducibles (i.e.maximal searches for join-irreducibles)
F ⊆ D is a filter provided
I 1 ∈ F and 0 6∈ FI a ∈ F and a 6 b ∈ D implies b ∈ FI a, b ∈ F implies a ∧ b ∈ F
(Order dual notion is that of an ideal)
A filter F ⊆ D is prime provided, for a, b ∈ D
a ∨ b ∈ F =⇒ (a ∈ F or b ∈ F )
(Order dual notion is that of a prime ideal)
Duality and Automata Theory
Stone representation and Priestley duality
The Stone representation theorem
Let D be a distributive lattice, XD the set of prime filters of D,then
η : D → P(XD)
a 7→ η(a) = {F ∈ XD | a ∈ F}
is an injective lattice homomorphism
I It is easy to verify that η preserves 0, 1,∧, and ∨I Injectivity uses Stone’s Prime Filter Theorem:
Let I be an ideal of D and F a filter of D. If I ∩ F = ∅, thenthere is a prime filter F ′ with F ⊆ F ′ and I ∩ F ′ = ∅
Duality and Automata Theory
Stone representation and Priestley duality
Priestley dualityLet D be a DL, the Priestley dual of D is
(XD, πD,6σ)
where πD = <η(a), (η(a))c | a ∈ D>Then the image of η is characterized as the clopen downsets
( , ! , !!)
! = <"( ), ("( )) | ! >
"
==
" , ( ! # [$ ! ( ) ! %! ])
The dual category are the spaces that are C = Compact andTOD = totally order disconnected:
∀x, y (x � y =⇒ [∃U ∈ ClopDown(X)y ∈ U but x 6∈ U ])
with continuous and order preserving maps
Duality and Automata Theory
Stone representation and Priestley duality
Priestley dualityLet D be a DL, the Priestley dual of D is
(XD, πD,6σ)
where πD = <η(a), (η(a))c | a ∈ D>Then the image of η is characterized as the clopen downsets
( , ! , !!)
! = <"( ), ("( )) | ! >
"
==
" , ( ! # [$ ! ( ) ! %! ])
The dual category are the spaces that are C = Compact andTOD = totally order disconnected:
∀x, y (x � y =⇒ [∃U ∈ ClopDown(X)y ∈ U but x 6∈ U ])
with continuous and order preserving maps
Duality and Automata Theory
Stone representation and Priestley duality
Priestley dualityLet D be a DL, the Priestley dual of D is
(XD, πD,6σ)
where πD = <η(a), (η(a))c | a ∈ D>Then the image of η is characterized as the clopen downsets
( , ! , !!)
! = <"( ), ("( )) | ! >
"
==
" , ( ! # [$ ! ( ) ! %! ])
The dual category are the spaces that are C = Compact andTOD = totally order disconnected:
∀x, y (x � y =⇒ [∃U ∈ ClopDown(X)y ∈ U but x 6∈ U ])
with continuous and order preserving maps
Duality and Automata Theory
Stone representation and Priestley duality
Priestley duality for additional operations
If f : Dn → D preserves ∨ and 0 in each coordinate, then the dualof f is
R ⊆ X ×Xn
satisfying
I 6 ◦R ◦ (6)[n]I For each x ∈ X the set R[x] is closed in the product topology
I For U1, . . . , Un ⊆ X clopen downsets, the setR−1[U1 × . . .× Un] is clopen
NB! If f is unary, then R is a function if and only if f is ahomomorphism
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results I
Let τ be an abstract algebra type. A Boolean (Priestley)topological algebra of type τ is an algebra of type τ based on aBoolean (Priestley) space so that all the operation of the algebraare continuous (in the product topology)
Theorem: Every Priestley topological algebra is the dual space ofsome bounded distributive lattice with additional operations (up torearranging the order of the coordinates).
Theorem: The Boolean topological algebras are precisely theextended Boolean dual spaces with functional relations.
Theorem: The Boolean topological algebras are, up toisomorphism, precisely the duals of Boolean residuation algebrasfor which residuation is “join preserving at primes”.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results I
Let τ be an abstract algebra type. A Boolean (Priestley)topological algebra of type τ is an algebra of type τ based on aBoolean (Priestley) space so that all the operation of the algebraare continuous (in the product topology)
Theorem: Every Priestley topological algebra is the dual space ofsome bounded distributive lattice with additional operations (up torearranging the order of the coordinates).
Theorem: The Boolean topological algebras are precisely theextended Boolean dual spaces with functional relations.
Theorem: The Boolean topological algebras are, up toisomorphism, precisely the duals of Boolean residuation algebrasfor which residuation is “join preserving at primes”.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results I
Let τ be an abstract algebra type. A Boolean (Priestley)topological algebra of type τ is an algebra of type τ based on aBoolean (Priestley) space so that all the operation of the algebraare continuous (in the product topology)
Theorem: Every Priestley topological algebra is the dual space ofsome bounded distributive lattice with additional operations (up torearranging the order of the coordinates).
Theorem: The Boolean topological algebras are precisely theextended Boolean dual spaces with functional relations.
Theorem: The Boolean topological algebras are, up toisomorphism, precisely the duals of Boolean residuation algebrasfor which residuation is “join preserving at primes”.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results I
Let τ be an abstract algebra type. A Boolean (Priestley)topological algebra of type τ is an algebra of type τ based on aBoolean (Priestley) space so that all the operation of the algebraare continuous (in the product topology)
Theorem: Every Priestley topological algebra is the dual space ofsome bounded distributive lattice with additional operations (up torearranging the order of the coordinates).
Theorem: The Boolean topological algebras are precisely theextended Boolean dual spaces with functional relations.
Theorem: The Boolean topological algebras are, up toisomorphism, precisely the duals of Boolean residuation algebrasfor which residuation is “join preserving at primes”.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results II
Let X be a Priestley space and � a compatible quasiorder on X(dual of a sublattice). We say � is a congruence for R provided
[y � x and R(x, z)] =⇒ ∃z′ [R(y, z′) and z′ � z].
Theorem: Let B be a residuation algebra and C a boundedsublattice of B. Furthermore let X be the extended dual space ofB and � the compatible quasiorder on X corresponding to C.Then C is a residuation ideal of B if and only if the quasiorder �is a congruence on X.
Theorem: Let X be a Priestley topological algebra. Then thePriestley topological algebra quotients of X are in one-to-onecorrespondence with the residuation ideals of the dual to X.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results II
Let X be a Priestley space and � a compatible quasiorder on X(dual of a sublattice). We say � is a congruence for R provided
[y � x and R(x, z)] =⇒ ∃z′ [R(y, z′) and z′ � z].
Theorem: Let B be a residuation algebra and C a boundedsublattice of B. Furthermore let X be the extended dual space ofB and � the compatible quasiorder on X corresponding to C.Then C is a residuation ideal of B if and only if the quasiorder �is a congruence on X.
Theorem: Let X be a Priestley topological algebra. Then thePriestley topological algebra quotients of X are in one-to-onecorrespondence with the residuation ideals of the dual to X.
Duality and Automata Theory
Duality and automata II
Recognition by monoidsA language L ⊆ A∗ is recognized by a finite monoid M providedthere is a monoid morphism ϕ : A∗ →M with ϕ−1(ϕ(L)) = L.
L recognized by a finite automaton
=⇒ B(L) ↪→ P(A∗) finite residuation ideal
=⇒ A∗ �M(L) finite monoid quotient
=⇒ L is recognized by a finite monoid
=⇒ L recognized by a finite automaton
And M(L) is the minimum recognizer among monoids (being aquotient of the others)
Duality and Automata Theory
Profinite completions and recognizable subsets
The dual of the recognizable subsets of A∗Rec(A∗) = {ϕ−1(S) | ϕ : A∗ → F hom, F finite, S ⊆ F}
The inverse limit system FA!
lim!" FA! = !A!
GF
H
A!
The direct limit system GA!
lim!"GA! = Rec(A!)P(G )
P(F )P(H)
P(A!)
Duality and Automata Theory
Profinite completions and recognizable subsets
The dual of (Rec(A∗), /, \)
Corollary: The dual space of
(Rec(A∗)\, /)
is the profinite completion A∗ with its monoid operations.
In particular, the dual of the residual operations is functional andcontinuous.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results III
Let B be a Boolean residuation algebra. We say that B is locallyfinite with respect to residuation ideals provided each finite subsetof B generates a finite Boolean residuation ideal.
Theorem: A Boolean topological algebra is profinite if and only ifthe dual residuation algebra is locally finite with respect toresiduation ideals.
Corollary: Boolean topological quotients of a profinite algebra areprofinite.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results III
Let B be a Boolean residuation algebra. We say that B is locallyfinite with respect to residuation ideals provided each finite subsetof B generates a finite Boolean residuation ideal.
Theorem: A Boolean topological algebra is profinite if and only ifthe dual residuation algebra is locally finite with respect toresiduation ideals.
Corollary: Boolean topological quotients of a profinite algebra areprofinite.
Duality and Automata Theory
Boolean topological algebras as dual spaces
Duality results III
Let B be a Boolean residuation algebra. We say that B is locallyfinite with respect to residuation ideals provided each finite subsetof B generates a finite Boolean residuation ideal.
Theorem: A Boolean topological algebra is profinite if and only ifthe dual residuation algebra is locally finite with respect toresiduation ideals.
Corollary: Boolean topological quotients of a profinite algebra areprofinite.
Duality and Automata Theory
Duality and automata III
Classes of languages
C a class of recognizable languages closed under ∩ and ∪
C ↪−→ Rec(A∗) ↪−→ P(A∗)DUALLY
XC �− A∗ �− β(A∗)
That is, C is described dually by EQUATING elements of A∗.
This is Reiterman’s theorem in a very general form.
Duality and Automata Theory
Duality and automata III
A fully modular Eilenberg-Reiterman theorem
Using the fact that sublattices of Rec(A∗) correspond to Stone
quotients of A∗ we get a vast generalization of theEilenberg-Reiterman theory for recognizable languages
Closed under Equations Definition∪,∩ u→ v ϕ(v) ∈ ϕ(L)⇒ ϕ(u) ∈ ϕ(L)
quotienting u 6 v for all x, y, xuy → xvy
complement u↔ v u→ v and v → u
quotienting and complement u = v for all x, y, xuy ↔ xvy
Closed under inverses of morphisms Interpretation of variablesall morphisms words
non-erasing morphisms nonempty words
length multiplying morphisms words of equal length
length preserving morphisms letters
Duality and Automata Theory
Duality and automata III
Equational theory of lattices of languages
Varieties of finite monoids
Reiterman’s theorem
Eilenberg’s theorem
Varieties of languages
Profinite identities
Profinite equations Extended duality Lattices of regular
languages
Varieties of languages
Profinite identities
Profinite equations
Extended duality
Lattices of regular languages
Lattices of languages Procompact equations
The (two outer) theorems are proved using the duality betweensubalgebras (possibly with additional operations) and dual quotientspaces
Duality and Automata Theory
Conclusions
I The dual space of (B(L),∩,∪, ( )c, 0, 1, \, /) is the syntacticmonoid of L
I Every Priestley topological algebra is a dual space, andBoolean topological algebras are precisely the Boolean dualspaces whose relations are operations
I The Priestley topological algebra quotients of a Priestleytopological algebra are in one-to-one correspondence with theresiduation ideals of its dual
I A Boolean topological algebra is profinite if and only if itsdual is locally finite wrt residuation ideals
I In formal language theory, duality for sublattices generalizesReiterman-Eilenberg theory