Duality and Automata Theory 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 . f g The initial state is 1, the final states are 1 and 2. The alphabet is A = a; b The transitions are f g 1 a =22 a =33 a =3 · · · 1 b =32 b =13 b =3 · · · Duality and Automata Theory Elements of automata theory a Recognition by automata 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 f j jg 2 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 f g where a = 1; 4; 5 and b = 2; 3; 6 . f g f g Duality and Automata Theory Connection to logic on words Some examples The formula φ = x ax interprets as: 9 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 9 9 ^ ^ A∗aA∗bA∗. The formula x y [(x < y) (x = y)] ax defines the language 9 8 _ ^ 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 8 6 y y x means x = u 8 6 j j x < y z (x < z y z) means y = x + 1 ^ 8 ! 6 Let φ = X (1 = X u X x (x X x + 1 = X)) 9 2 ^ j j 2 ^ 8 2 $ 2 Then 1 = X, 2 X, 3 = X, 4 X,..., u X. Thus 2 2 2 2 j j 2 2 L(φ) = u u is even = (A )∗ f j j j g 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: [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. 2 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 2 ∗ ⊆ ∗ 1 x− L= u A∗ xu L (= x L) f 2 j 2 g f gn and 1 Ly− = u A∗ uy L (= L= y ) f 2 j 2 g f g Given a language L A , let (L) be the Boolean algebra of ⊆ ∗ B languages generated by 1 1 x− Ly− x; y A∗ j 2 NB! (L) is closed under quotients since the quotient operations B commute with the Boolean operations. Duality and Automata Theory Duality and automata I a Quotients of a recognizable language 1 2 b b a 3 L( ) = (ab) (ab) a A ∗ [ ∗ 1 a− L= u A∗ au L = (ba)∗b (ba)∗ f 2 j 2 g [ a, b 1 La− = u A∗ ua L = (ab)∗ f 2 j 2 g 1 b− L= u A∗ bu L = f 2 j 2 g ; 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 ) 2 B 2 P ∗ 1 S K = u− K (L) n 2 B u S \2 1 K=S = Ku− (L) 2 B u S \2 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 ) 2 B 2 P ∗ 1 S K = u− K (L) n 2 B u S \2 1 K=S = Ku− (L) 2 B u S \2 Theorem: For a recognizable language L, the dual space of the algebra ( (L); ; ; ()c; 0; 1; ; =) is the syntactic monoid of L. B \ [ n { 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 \ [ n ⊆ 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 \ [ n ⊆ 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 \ [ n { 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 P OS |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= .