Duality and Automata Theory Mai Gehrke Universit e Paris VII and - - PowerPoint PPT Presentation

Duality and Automata Theory

Duality and Automata Theory

Mai Gehrke

Universit´ e Paris VII and CNRS Joint work with Serge Grigorieff and Jean-´ Eric Pin

Duality and Automata Theory Elements of automata theory

A finite automaton

1 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 automata

1 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 u such 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 finite automaton. A language is rational provided it belongs to the smallest class of languages containing the finite languages which is closed under union, 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-th letter 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 that the 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 language A∗aA∗bA∗. The formula ∃x ∀y [(x < y) ∨ (x = y)] ∧ ax defines the language aA∗.

Duality and Automata Theory Connection to logic on words

Defining the set of words of even length

Macros: (x < y) ∨ (x = y) means x y ∀y x y means x = 1 ∀y y x means x = |u| x < y ∧ ∀z (x < z → y 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: (B¨ uchi ’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 using the Boolean operations on languages and lifted concatenation product 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 of associating with each finite automaton, A, a finite monoid, (MA, ·, 1). Theorem: [Sch¨ utzenberger ’65] A recognizable language is star-free if and only if the associated monoid is aperiodic, i.e., M is such that there exists n > 0 with xn = xn+1 for each x ∈ M. Submonoid generated by x:

1 x x2 x3 . . . xi+p = xi xi+1 xi+2 xi+p−1

This makes starfreeness decidable!

Duality and Automata Theory The algebraic theory of automata

Eilenberg-Reiterman theory

Varieties

• f languages

Profinite identities Varieties of finite monoids

Decidability Eilenberg Reiterman In good cases

A variety of monoids here means a class of finite monoids closed under homomorphic images, submonoids, and finite products Various generalisations: [Pin 1995], [Pin-Weil 1996], [Pippenger 1997], [Pol´ ak 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-regular languages

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) and Ly−1= {u ∈ A∗ | uy ∈ L}(= L/{y}) Given a language L ⊆ A∗, let B(L) be the Boolean algebra of languages generated by

• x−1Ly−1 | x, y ∈ A∗

NB! B(L) is closed under quotients since the quotient operations commute 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 since all the languages are recognized by the same machine with varying sets of initial and final states.

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 since all the languages are recognized by the same machine with varying sets of initial and final states. Since B(L) is finite it is also closed under residuation with respect to arbitrary denominators and is thus a bi-module for P(A∗). For any K ∈ B(L) and any S ∈ P(A∗) S\K =

• u∈S

u−1K ∈ B(L) K/S =

• u∈S

Ku−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.

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 the algebra (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 = 0 is join-irreducible iff x = y ∨ z ⇒ (x = y or x = z) D J(D) Join-irreducibles

Duality and Automata Theory Duality – the finite case

Finite lattices and join-irreducibles

x = 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 power set (P)

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 power set (P) a b c

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 power set (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 enough completely join irreducibles, with complete homomorphisms POS — partially ordered sets with order preserving maps

• +
• 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 subalgebra

• f P(A∗) generated by these four languages.

Duality and Automata Theory Duality – the finite case

Boolean subalgebras correspond to partitions

Let B = <(ab)∗, a(ba)∗, b(ab)∗, (ba)∗> be the Boolean subalgebra

• f 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

Duality and Automata Theory Duality – the finite case

Boolean subalgebras correspond to partitions

Let B = <(ab)∗, a(ba)∗, b(ab)∗, (ba)∗> be the Boolean subalgebra

• f 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
• perator

♦ : Dn → D

• R ⊆ X × Xn

relation ♦ → R♦ = {(x, x) | x ♦(x)}

• ♦R : S → 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 ∧) in each coordinate? dual operator : Dn → D

• S ⊆ M × Mn

relation

• →

S = {(m, m) | m (m)}

• S : u →
• S−1[↑u ∩ Mn]
• ←

S where M = M(D) and we use the duality with respect to meet-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 → in the first coordinate sends → in the second coordinate R(X, Y, Z) ⇐ ⇒ X\(Zc) ⊆ Y c ⇐ ⇒ Y ⊆ X\Zc ⇐ ⇒ XY ⊆ Zc ⇐ ⇒ XY ∩ Z = ∅

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 1 1 a ba b ab a a a ab ba ba ba b b b ba b b ab ab a ab

Syntactic monoid This monoid is aperiodic since 1 = 12, a2 = 0 = a3, ba = ba2, b2 = 0 = b3, ab = ab2, and 0 = 02

Duality and Automata Theory Duality – the finite case

The syntactic monoid

L = (ab)∗ − → M(L) =

· 1 a ba b ab 1 1 a ba b ab a a a ab ba ba ba b b b ba b b ab ab a ab

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 sub residuation bi-module) B(L) ֒ → P(A∗) is a quotient monoid A∗ ։ Atoms(B(L)) This dual correspondence goes through to the setting of topological duality!

Duality and Automata Theory Stone representation and Priestley duality

Not enough join-irreducibles

1

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

◮ 1 ∈ F and 0 ∈ F ◮ a ∈ F and a b ∈ D implies b ∈ F ◮ 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

• r 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 → η(a) = {F ∈ XD | a ∈ F} is an injective lattice homomorphism

◮ It is easy to verify that η preserves 0, 1, ∧, and ∨ ◮ Injectivity uses Stone’s Prime Filter Theorem:

Let I be an ideal of D and F a filter of D. If I ∩ F = ∅, then there is a prime filter F ′ with F ⊆ F ′ and I ∩ F ′ = ∅

Duality and Automata Theory Stone representation and Priestley duality

Priestley duality

Let D be a DL, the Priestley dual of D is (XD, πD, σ) where πD = <η(a), (η(a))c | a ∈ D> Then the image of η is characterized as the clopen downsets

Duality and Automata Theory Stone representation and Priestley duality

Priestley duality

Let D be a DL, the Priestley dual of D is (XD, πD, σ) where πD = <η(a), (η(a))c | a ∈ D> Then the image of η is characterized as the clopen downsets

Duality and Automata Theory Stone representation and Priestley duality

Priestley duality

Let D be a DL, the Priestley dual of D is (XD, πD, σ) 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 and

TOD = totally order disconnected: ∀x, y (x y = ⇒ [∃U ∈ ClopDown(X)y ∈ U but x ∈ 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 dual

• f f is

R ⊆ X × Xn satisfying

◮ ◦R ◦ ()[n] ◮ For each x ∈ X the set R[x] is closed in the product topology ◮ For U1, . . . , Un ⊆ X clopen downsets, the set

R−1[U1 × . . . × Un] is clopen NB! If f is unary, then R is a function if and only if f is a homomorphism

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 a Boolean (Priestley) space so that all the operation of the algebra are continuous (in the product topology)

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 a Boolean (Priestley) space so that all the operation of the algebra are continuous (in the product topology) Theorem: Every Priestley topological algebra is the dual space of some bounded distributive lattice with additional operations (up to rearranging the order of the coordinates).

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 a Boolean (Priestley) space so that all the operation of the algebra are continuous (in the product topology) Theorem: Every Priestley topological algebra is the dual space of some bounded distributive lattice with additional operations (up to rearranging the order of the coordinates). Theorem: The Boolean topological algebras are precisely the extended Boolean dual spaces with functional relations.

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 a Boolean (Priestley) space so that all the operation of the algebra are continuous (in the product topology) Theorem: Every Priestley topological algebra is the dual space of some bounded distributive lattice with additional operations (up to rearranging the order of the coordinates). Theorem: The Boolean topological algebras are precisely the extended Boolean dual spaces with functional relations. Theorem: The Boolean topological algebras are, up to isomorphism, precisely the duals of Boolean residuation algebras for 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 bounded sublattice of B. Furthermore let X be the extended dual space of B 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.

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 bounded sublattice of B. Furthermore let X be the extended dual space of B 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 the Priestley topological algebra quotients of X are in one-to-one correspondence with the residuation ideals of the dual to X.

Duality and Automata Theory Duality and automata II

Recognition by monoids

A language L ⊆ A∗ is recognized by a finite monoid M provided there 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 a quotient 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∗ G F 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 and continuous.

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 locally finite with respect to residuation ideals provided each finite subset

• f B generates a finite Boolean residuation ideal.

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 locally finite with respect to residuation ideals provided each finite subset

• f B generates a finite Boolean residuation ideal.

Theorem: A Boolean topological algebra is profinite if and only if the dual residuation algebra is locally finite with respect to residuation ideals.

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 locally finite with respect to residuation ideals provided each finite subset

• f B generates a finite Boolean residuation ideal.

Theorem: A Boolean topological algebra is profinite if and only if the dual residuation algebra is locally finite with respect to residuation ideals. Corollary: Boolean topological quotients of a profinite algebra are profinite.

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 և −

• A∗

և − 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 the Eilenberg-Reiterman theory for recognizable languages

Closed under Equations Definition ∪, ∩ u → v ˆ ϕ(v) ∈ ϕ(L) ⇒ ˆ ϕ(u) ∈ ϕ(L) quotienting u 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 variables all 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

• f 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 between subalgebras (possibly with additional operations) and dual quotient spaces

Duality and Automata Theory

Conclusions

◮ The dual space of (B(L), ∩, ∪, ( )c, 0, 1, \, /) is the syntactic

monoid of L

◮ Every Priestley topological algebra is a dual space, and

Boolean topological algebras are precisely the Boolean dual spaces whose relations are operations

◮ The Priestley topological algebra quotients of a Priestley

topological algebra are in one-to-one correspondence with the residuation ideals of its dual

◮ A Boolean topological algebra is profinite if and only if its

dual is locally finite wrt residuation ideals

◮ In formal language theory, duality for sublattices generalizes

Reiterman-Eilenberg theory