Duality and Automata Theory

Mai Gehrke

Universit´eParis VII and CNRS

Joint work with Serge Grigorieﬀ and Jean-Eric´ Pin Duality and Automata Theory Elements of automata theory

a A ﬁnite automaton 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 =22 a =33 a =3 · · · 1 b =32 b =13 b =3 · · · Duality and Automata Theory 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( ) = (ab)∗ (ab)∗a 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( ) = (ab) (ab) a. A ∗ ∪ ∗ Duality and Automata Theory Connection to logic on words

Logic on words To each non-empty word u is associated a structure

u = ( 1, 2,..., u , <, (a)a A) M { | |} ∈

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

= ( 1, 2, 3, 4, 5, 6 , <, (a, b)) Mu { }

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 ∨ 6 y x y means x = 1 ∀ 6 y y x means x = u ∀ 6 | | x < y z (x < z y z) means y = x + 1 ∧ ∀ → 6

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 ∈ ∈ ∈ ∈ | | ∈ 2 L(φ) = u u is even = (A )∗ { | | | } 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 finite monoid, A (M , , 1). A · 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 xn = xn+1 for each x M. ∈ Submonoid generated by x: xi+1 xi+2

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

xi+p 1 This makes starfreeness decidable! − Duality and Automata Theory The algebraic theory of automata

Eilenberg-Reiterman theory

Varieties of ﬁnite monoids

Eilenberg Reiterman

Varieties In good Proﬁnite Decidability of languages identities 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

(1) (2)

algebrasoflanguages equationaltheories (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 ∈ ∗ ⊆ ∗ 1 x− L= u A∗ xu L (= x L) { ∈ | ∈ } { }\ and 1 Ly− = u A∗ uy L (= L/ y ) { ∈ | ∈ } { }

Given a language L A , let (L) be the Boolean algebra of ⊆ ∗ B languages generated by

1 1 x− Ly− x, y A∗ | ∈ NB! (L) is closed under quotients since the quotient operations B commute with the Boolean operations. Duality and Automata Theory Duality and automata I

Quotients of a recognizable language a

1 2

L( ) = (ab) (ab) a b a A ∗ ∪ ∗ 1 3 a− L= u A∗ au L = (ba)∗b (ba)∗ { ∈ | ∈ } ∪ 1 La− = u A∗ ua L = (ab)∗ { ∈ | ∈ } 1 a, b b− L= u A∗ bu L = { ∈ | ∈ } ∅ NB! These are recognized by the same underlying machine. Since (L) is ﬁnite it is also closed under residuation with respect B to arbitrary denominators and is thus a bi-module for (A ). P ∗ For any K (L) and any S (A ) ∈ B ∈ P ∗ 1 S K = u− K (L) \ ∈ B u S \∈ 1 K/S = Ku− (L) ∈ B u S \∈

Duality and Automata Theory Duality and automata I

(L) for a recognizable language B

If L is recognizable then the generating set of (L) is ﬁnite since B 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

(L) for a recognizable language B

If L is recognizable then the generating set of (L) is ﬁnite since B all the languages are recognized by the same machine with varying sets of initial and ﬁnal states.

Since (L) is ﬁnite it is also closed under residuation with respect B to arbitrary denominators and is thus a bi-module for (A ). P ∗ For any K (L) and any S (A ) ∈ B ∈ P ∗ 1 S K = u− K (L) \ ∈ B u S \∈ 1 K/S = Ku− (L) ∈ B u S \∈ Theorem: For a recognizable language L, the dual space of the algebra ( (L), , , ()c, 0, 1, ,/) is the syntactic monoid of L. B ∩ ∪ \ – including the product operation!

Duality and Automata Theory Duality and automata I

The syntactic monoid via duality

For a recognizable language L, the algebra

c ( (L), , , () , 0, 1, ,/) (A∗) B ∩ ∪ \ ⊆ P 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

c ( (L), , , () , 0, 1, ,/) (A∗) B ∩ ∪ \ ⊆ P is the Boolean residuation bi-module generated by L.

Theorem: For a recognizable language L, the dual space of the algebra ( (L), , , ()c, 0, 1, ,/) is the syntactic monoid of L. B ∩ ∪ \ – including the product operation! D(J(D)) Downset lattice

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) 6 ∨ ⇒

J(D) 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) 6 ∨ ⇒

J(D) D(J(D)) D Join-irreducibles Downset lattice X

{a, b} {a,c} {b, c}

{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( ) of the atoms is just the power D set( ) P X

{a, b} {a,c} {b, c}

{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( ) of the atoms is just the power D 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( ) of the atoms is just the power D set( ) P X

{a, b} {a,c} {b, c}

{a} {b} {c} a b c

∅ Duality and Automata Theory Duality – the ﬁnite case

.mHBiv BM *i2;Q`B+H .mHBiv GQ;B+ Categorical Duality G2+im`2 R + DL — complete.G+ ě U+QKTH2i2 and completely M/ +QKTH2i2HvV distributive /Bbi`B#miBp2 lattices HiiB+2b rBi? with enoughGQ;B+ M/ HiiB+2b completely2MQm;? join DQBM irreducibles, B``2/m+B#H2b- with rBi? U+QKTH2i2V complete ?QKQKQ`T?BbKb homomorphisms .mHBiv POS —SPa partiallyě T`iBHHv ordered Q`/2`2/ sets b2ib with rBi? order Q`/2` preserving T`2b2`pBM; KTb maps BMi`Q/m+iBQM aiQM2 `2T`2@ D b2MiiBQM

.GX+ SPa 1H2K2Mib Q7 C iQTQHQ;v aiQM2 M/ S`B2biH2v (J(D))(C=(.))DJ= . C(( (S(P))))= S= P D D ∼ ∼ DD ∼ ∼ /mHBiv ? : . 1 C(.) C(1):C(?) h : D E → !! J(D) ← J(E): J(h) *QKTH2i2M2bb → ← i?2Q`2Kb (S) (Z): (7) ! 7 : S Z (P ) D (Q←):D (f)D f : P→ Q D ← D D ! → Dual of a morphism is given by lower/upper adjoint The corresponding equivalence relation on Atoms( (A )) = A P ∗ ∗ 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 , (A ) is the quotient map B → P ∗ + + q : A∗ (ab) , ε , (ba) , a(ba)∗, b(ab)∗,Z { { } }

Duality and Automata Theory Duality – the ﬁnite case

Boolean subalgebras correspond to partitions

Let = <(ab) , a(ba) , b(ab) , (ba) > be the Boolean subalgebra B ∗ ∗ ∗ ∗ of (A ) generated by these four languages. P ∗ The dual of the embedding , (A ) is the quotient map B → P ∗ + + q : A∗ (ab) , ε , (ba) , a(ba)∗, b(ab)∗,Z { { } }

Duality and Automata Theory Duality – the ﬁnite case

Boolean subalgebras correspond to partitions

Let = <(ab) , a(ba) , b(ab) , (ba) > be the Boolean subalgebra B ∗ ∗ ∗ ∗ of (A ) generated by these four languages. P ∗ The corresponding equivalence relation on Atoms( (A )) = A P ∗ ∗ 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

+ + (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 , (A ) is the quotient map B → P ∗ + + q : A∗ (ab) , ε , (ba) , a(ba)∗, b(ab)∗,Z { { } } .mHBiv BM Duality and Automata Theory *i2;Q`B+H .mHBiv GQ;B+ Duality – the ﬁnite case G2+im`2 R

.G+ ě U+QKTH2i2 M/ +QKTH2i2HvV /Bbi`B#miBp2 HiiB+2b rBi? GQ;B+ M/ HiiB+2b Duality for2MQm;? operators DQBM B``2/m+B#H2b- rBi? U+QKTH2i2V ?QKQKQ`T?BbKb .mHBiv SPa ě T`iBHHv Q`/2`2/ b2ib rBi? Q`/2` T`2b2`pBM; KTb BMi`Q/m+iBQM

aiQM2 `2T`2@ D b2MiiBQM

.GX+ SPa 1H2K2Mib Q7 C iQTQHQ;v aiQM2 M/ S`B2biH2v (C(.)) = . C( (S)) = S D ∼ D ∼ /mHBiv ? : . n 1 C(.) C(1):C(?) n operator : D→ D! ←R X X relation*QKTH2i2M2bb ♦ → ! ⊆ × i?2Q`2Kb (S) (Z): (7) 7 : S Z D ←D D ! → R = (x, x) x (x) ♦ 7→ ♦ { | 6 ♦ } 1 ♦R : S R− [S1 ...Sn] R 7→ × ←[ 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 M n relation → ! ⊆ × S = (m, m) m (m) 7→ { | > }

1 n S : u S− [ u M ] S 7→ ↑ ∩ ←[ ^ where M = M(D) and we use the duality with respect to meet-irreducibles instead of duality with respect to join-irreducibles DualityPart I: andDuality Automata and recognition Theory DualityThe syntactic – the ﬁnite monoid case of a recognizable language and duality

TheThe dual dual of residuation residuation operations operations

The residuation operation : The residuation operation\ B: ×B→B \ B × B → B sends in the ﬁrst coordinate X c sends #→ in the ﬁrst coordinate sends 7→ in the second coordinate sends ! #→ " in the second coordinate W" 7→ "V V V R(X , Y , Z) X (Z c ) Y c X ⇐⇒ \ c ⊆ c R(X,Y,Z) X (Z ) Y (L) ⇐⇒ \ ⊆ B Y X Z c ⇐⇒ Y '⊆ X\ Zc ⇐⇒ 6⊆ \ XY Z c ⇐⇒ XY '⊆ Zc ⇐⇒ 6⊆ XY Z = ⇐⇒ ∩ 6 ∅ 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 1 a ba b ab 0 · 1 1 a ba b ab 0 a a 0 a ab 0 0 L = (ab)∗ (L)= −→ ba ba 0 ba b 0 0 M 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 Duality and Automata Theory Duality – the ﬁnite case

The syntactic monoid 1 a ba b ab 0 · 1 1 a ba b ab 0 a a 0 a ab 0 0 L = (ab)∗ (L)= −→ ba ba 0 ba b 0 0 M 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 sub residuation bi-module)

(L) , (A∗) B → P is a quotient monoid

A∗ Atoms( (L)) B This dual correspondence goes through to the setting of topological duality! Duality and Automata Theory Stone representation and Priestley duality

Not enough join-irreducibles

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 F ∈ 6∈ I a F and a b D implies b F ∈ 6 ∈ ∈ I 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 (X ) → P D a η(a) = F X a F 7→ { ∈ D | ∈ } 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 = , then ∩ ∅ there is a prime ﬁlter F with F F and I F = 0 ⊆ 0 ∩ 0 ∅ .mHBiv BM S`B2biH2v /mHBiv GQ;B+ G2+im`2 R G2i . #2 .G- i?2 S`B2biH2v /mH Q7 . Bb GQ;B+ M/ HiiB+2b (s.,π., !σ) + .mHBiv r?2`2 π. = <η(), (η()) .> | ∈ BMi`Q/m+iBQM h?2M i?2 BK;2 Q7 η Bb +?`+i2`Bx2/ b i?2 +HQT2M /QrMb2ib aiQM2 `2T`2@ *HQT2M.QrM b2MiiBQM

X 1H2K2Mib Q7 .G *hP. iQTQHQ;v S` The dual category are the spaces that are C = Compact and aiQM2 M/ S`B2biH2v TOD = totally order disconnected: h?2 /mH +i2;Q`v `2 i?2 bT+2b i?i `2, /mHBiv * = *QKT+i x, y (x y = [ U ClopDown(X)y U but x U]) ∀hP. = iQiHHv Q`/2`⇒ ∃ /Bb+QMM2+i2/∈ , ∈ 6∈ *QKTH2i2M2bb i?2Q`2Kb with continuous and order preserving maps t, v (t ! v 4 [ l *HQT.QrM(s)v l #mi t l]) ∀ ⇒ ∃ ∈ ∈ %∈ rBi? +QMiBMmQmb M/ Q`/2` T`2b2`pBM; KTb

Duality and Automata Theory Stone representation and Priestley duality

Priestley duality Let D be a DL, the Priestley dual of D is

(XD, πD, 6σ) where π = <η(a), (η(a))c a D> 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]) ∀ ⇒ ∃ ∈ ∈ 6∈ with continuous and order preserving maps

Duality and Automata Theory Stone representation and Priestley duality .mHBiv BM S`B2biH2v /mHBiv GQ;B+ G2+im`2 R PriestleyG2i . duality#2 .G- i?2 S`B2biH2v /mH Q7 . Bb Let D be a DL, the Priestley dual of D is GQ;B+ M/ HiiB+2b (s.,π., !σ) (XD, πD, 6σ) π = <η( ), (η( ))+ > .mHBiv r?2`2 . c . where πD = <η(a), (η(a)) a | D>∈ BMi`Q/m+iBQM h?2M i?2 BK;2 Q7 η Bb +?`+i2`Bx2/| ∈ b i?2 +HQT2M /QrMb2ib Then the image of η is characterized as the clopen downsets aiQM2 `2T`2@ *HQT2M.QrM b2MiiBQM

X 1H2K2Mib Q7 .G *hP. iQTQHQ;v S` aiQM2 M/ S`B2biH2v h?2 /mH +i2;Q`v `2 i?2 bT+2b i?i `2, /mHBiv * = *QKT+i hP. = iQiHHv Q`/2` /Bb+QMM2+i2/, *QKTH2i2M2bb i?2Q`2Kb t, v (t ! v 4 [ l *HQT.QrM(s)v l #mi t l]) ∀ ⇒ ∃ ∈ ∈ %∈ rBi? +QMiBMmQmb M/ Q`/2` T`2b2`pBM; KTb Duality and Automata Theory Stone representation and Priestley duality .mHBiv BM S`B2biH2v /mHBiv GQ;B+ G2+im`2 R PriestleyG2i . duality#2 .G- i?2 S`B2biH2v /mH Q7 . Bb Let D be a DL, the Priestley dual of D is GQ;B+ M/ HiiB+2b (s.,π., !σ) (XD, πD, 6σ) π = <η( ), (η( ))+ > .mHBiv r?2`2 . c . where πD = <η(a), (η(a)) a | D>∈ BMi`Q/m+iBQM h?2M i?2 BK;2 Q7 η Bb +?`+i2`Bx2/| ∈ b i?2 +HQT2M /QrMb2ib Then the image of η is characterized as the clopen downsets aiQM2 `2T`2@ *HQT2M.QrM b2MiiBQM

Priestley duality for additional operations

If f : Dn D preserves and 0 in each coordinate, then the dual → ∨ of f is R X Xn ⊆ × satisfying [n] I R ( ) 6 ◦ ◦ 6 I For each x X the set R[x] is closed in the product topology ∈ I For U ,...,U X clopen downsets, the set 1 n ⊆ R 1[U ... U ] is clopen − 1 × × n

NB! If f is unary, then R is a function if and only if f is a homomorphism 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 I

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 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, 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 I

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

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”. 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 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)] = z0 [R(y, z0) and z0 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

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

= (L) , (A∗) finite residuation ideal ⇒B → P = 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 ∗ { − | ∗ → ⊆ }

A∗ The inverse limit system FA∗ lim = A H FA∗ ∗ ←− F ! G

(A∗) P The direct limit system GA∗ (H) lim =Rec(A ) P (F ) GA∗ ∗ −→ (G) P P 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 thec residual operations is functional and continuous. 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 Boolean topological algebras as dual spaces

Duality results III

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 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. Duality and Automata Theory Boolean topological algebras as dual spaces

Duality results III

Corollary: Boolean topological quotients of a proﬁnite algebra are proﬁnite. Duality and Automata Theory Duality and automata III

Classes of languages

a class of recognizable languages closed under and C ∩ ∪

, Rec(A∗) , (A∗) C −→ −→ P DUALLY

X A∗ β(A∗) C − − c That is, is described dually by EQUATING elements of A . C ∗ This is Reiterman’s theorem in a very general form.c 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 c Closed under Equations Deﬁnition , u v ϕˆ(v) ϕ(L) ϕˆ(u) ϕ(L) ∪ ∩ → ∈ ⇒ ∈ quotienting u v for all x, y, xuy xvy 6 → 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 Eilenberg’s Varieties of Reiterman’s theorem finite monoids theorem

Duality and Automata TheoryVarieties Profinite Duality and automataof languages III identities Extended duality Lattices of regular Profinite equations languages Equational theory of lattices of languages

Varieties Profinite of languages Extended duality identities

Lattices of regular Profinite equations 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

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

I Every Priestley topological algebra is a dual space, and Boolean topological algebras are precisely the Boolean dual spaces whose relations are operations

I The Priestley topological algebra quotients of a Priestley topological algebra are in one-to-one correspondence with the residuation ideals of its dual

I A Boolean topological algebra is proﬁnite if and only if its dual is locally ﬁnite wrt residuation ideals

I In formal language theory, duality for sublattices generalizes Reiterman-Eilenberg theory