Duality and Automata Theory - Universiteit...

Duality and Automata Theory


Mai Gehrke

Universite Paris VII and CNRS

Joint work with Serge Grigorieff and Jean-Eric Pin


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



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.



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.


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}.



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∗.



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)∗



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???


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!


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 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



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.



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.



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)



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 – 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



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



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



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



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 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 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



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



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 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!


Stone representation and Priestley duality

Not enough join-irreducibles

1

0

D = 0⊕ (Nop × Nop) has no join irreducible elements whatsoever!



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)



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 ′ = ∅



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



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


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 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 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)


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!)


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 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 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.



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



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


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

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