Duality and Automata Theory

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1 Duality and Automata Theory Mai Gehrke Université Paris VII and CNRS Joint work with Serge Grigorieff and Jean-Éric Pin

2 Elements of automata theory A finite automaton a 1 2 b b a 3 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

3 Elements of automata theory Recognition by automata a 1 2 b b a 3 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.

4 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.

5 Connection to logic on words Logic on words To each non-empty word u is associated a structure M u = ({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 M u = ({1, 2, 3, 4, 5, 6}, <, (a, b)) where a = {1, 4, 5} and b = {2, 3, 6}.

6 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.

7 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} = (A 2 )

8 Connection to logic on words Monadic second order Only second order quantifiers over unary predicates are allowed. Theorem: (Büchi 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???

9 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, (M A,, 1). Theorem: [Schützenberger 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 x n = x n+1 for each x M. Submonoid generated by x: 1 x x 2 x 3 x i+p = x i... x i+1 x i+2 This makes starfreeness decidable! x i+p 1

10 The algebraic theory of automata Eilenberg-Reiterman theory Varieties of finite monoids Eilenberg Reiterman Varieties of languages Profinite identities In good cases Decidability 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], [Polák 2001], [Esik 2002], [Straubing 2002], [Kunc 2003]

11 Duality and automata I Eilenberg, Reiterman, and Stone Classes of monoids (1) (2) algebras of languages (3) equational theories (1) Eilenberg theorems (2) Reiterman theorems (3) extended Stone/Priestley duality (3) allows generalisation to non-varieties and even to non-regular languages

12 Duality and automata I Assigning a Boolean algebra to each language For x A and L A, define the quotient x 1 L= {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 1 Ly 1 x, y A } NB! B(L) is closed under quotients since the quotient operations commute with the Boolean operations.

13 Duality and automata I Quotients of a recognizable language a 1 2 b L(A) = (ab) (ab) a b a 1 L= {u A au L} = (ba) b (ba) La 1 = {u A ua L} = (ab) b 1 L= {u A bu L} = 3 a, b a NB! These are recognized by the same underlying machine.

14 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.

15 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 1 K B(L) K/S = u S Ku 1 B(L)

16 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.

17 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!

18 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

19 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

20 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)

21 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

22 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) X {a,b} {a,c} {b,c} a b c {a} {b} {c}

23 Duality the finite case Categorical Duality DL + complete and completely distributive lattices with enough completely join irreducibles, with complete homomorphisms P OS 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

24 Duality the finite case Boolean subalgebras correspond to partitions Let B = <(ab), a(ba), b(ab), (ba) > be the Boolean subalgebra of P(A ) generated by these four languages.

25 Duality the finite case Boolean subalgebras correspond to partitions Let B = <(ab), a(ba), b(ab), (ba) > be the Boolean subalgebra of 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

26 Duality the finite case Boolean subalgebras correspond to partitions Let B = <(ab), a(ba), b(ab), (ba) > be the Boolean subalgebra of 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}

27 Duality the finite case Duality for operators + D operator : D n D R X X n relation ( R : S R 1 [S 1... S n ] ) R R = {(x, x) x (x)}

28 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 : D n D S M M n relation ( S : u ) S 1 [ u M n ] S = {(m, m) m (m)} where M = M(D) and we use the duality with respect to meet-irreducibles instead of duality with respect to join-irreducibles S

29 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 X c R(X, Y, Z) X\(Z c ) Y c X B(L) Y X\Z c XY Z c XY Z

30 Duality the finite case The syntactic monoid 1 a ba b ab a ba b ab 0 L = (ab) M(L) = 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 Syntactic monoid This monoid is aperiodic since 1 = 1 2, a 2 = 0 = a 3, ba = ba 2, b 2 = 0 = b 3, ab = ab 2, and 0 = 0 2

31 Duality the finite case The syntactic monoid 1 a ba b ab a ba b ab 0 L = (ab) M(L) = 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 Syntactic monoid This monoid is aperiodic since 1 = 1 2, a 2 = 0 = a 3, ba = ba 2, b 2 = 0 = b 3, ab = ab 2, and 0 = 0 2 Indeed, L is star-free since L c = ba A a A aaa A bba and A = c

32 Duality the finite case Duality for Boolean residuation ideals The dual of a Boolean residuation ideal (= Boolean sub residuation bi-module) is a quotient monoid B(L) P(A ) A Atoms(B(L)) This dual correspondence goes through to the setting of topological duality!

33 Stone representation and Priestley duality Not enough join-irreducibles 1 0 D = 0 (N op N op ) has no join irreducible elements whatsoever!

34 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 or b F ) (Order dual notion is that of a prime ideal)

35 Stone representation and Priestley duality The Stone representation theorem Let D be a distributive lattice, X D the set of prime filters of D, then η : D P(X D ) a η(a) = {F X D 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 =

36 Stone representation and Priestley duality Priestley duality Let D be a DL, the Priestley dual of D is (X D, π D, σ ) where π D = <η(a), (η(a)) c a D> Then the image of η is characterized as the clopen downsets

37 Stone representation and Priestley duality Priestley duality Let D be a DL, the Priestley dual of D is (X D, π D, σ ) where π D = <η(a), (η(a)) c a D> Then the image of η is characterized as the clopen downsets

38 Stone representation and Priestley duality Priestley duality Let D be a DL, the Priestley dual of D is (X D, π D, σ ) where π D = <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 T OD = totally order disconnected: x, y (x y = [ U ClopDown(X)y U but x U]) with continuous and order preserving maps

39 Stone representation and Priestley duality Priestley duality for additional operations If f : D n D preserves and 0 in each coordinate, then the dual of f is R X X n satisfying R ( ) [n] For each x X the set R[x] is closed in the product topology For U 1,..., U n X clopen downsets, the set R 1 [U 1... U n ] is clopen NB! If f is unary, then R is a function if and only if f is a homomorphism

40 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)

41 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).

42 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.

43 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.

44 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.

45 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.

46 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)

47 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} A lim F A = Â H G The inverse limit system F A F P(A ) lim G A =Rec(A ) P(H) P(G) The direct limit system G A P(F )

48 Profinite completions and recognizable subsets The dual of (Rec(A ), /, \) Corollary: The dual space of (Rec(A )\, /) is the profinite completion  with its monoid operations. In particular, the dual of the residual operations is functional and continuous.

49 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 of B generates a finite Boolean residuation ideal.

50 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 of 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.

51 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 of 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.

52 Duality and automata III Classes of languages C a class of recognizable languages closed under and DUALLY C Rec(A ) P(A ) X C Â β(a ) That is, C is described dually by EQUATING elements of Â. This is Reiterman s theorem in a very general form.

53 Duality and automata III A fully modular Eilenberg-Reiterman theorem Using the fact that sublattices of Rec(A ) correspond to Stone quotients of  we get a vast generalization of the Eilenberg-Reiterman theory for recognizable languages Closed under Equations Definition, u v ˆϕ(v) ϕ(l) ˆϕ(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

54 Duality and automata III Equational theory of lattices of languages Varieties of languages Lattices of regular languages Lattices of languages Extended duality Profinite identities Profinite equations Procompact equations The (two outer) theorems are proved using the duality between subalgebras (possibly with additional operations) and dual quotient spaces

55 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

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