Introduction to weighted automata theory

Introduction to weighted automata theory Lectures given at the 19th Estonian Winter School in Computer Science Jacques Sakarovitch CNRS / Telecom ParisTech

Based on Chapter III Chapter 4

The presentation is very much inspired y a joint work with entitled Marie-Pierre Béal (Univ. Paris-Est) and Sylvain Lomardy (Univ. Bordeaux) On the equivalence and conjugacy of weighted automata, a first version of which has een pulished in Proc. of CSR 2006 and whose final complete version is still in preparation.

Lecture I The model of (finite) weighted automata

A touch of general system theory Finite control p State a 1 a 2 a 3 a 4 a n $ k 1 k 2 k 3 k 4 k l $ Paradigm of a machine for the computer scientists

A touch of general system theory output input Paradigm of a machine for the rest of the world

A touch of general system theory y α( ) x y = α(x) Paradigm of a machine for the rest of the world

A touch of general system theory y α( ) x y = α(x) x R n, y R m Paradigm of a machine for the rest of the world

Getting ack to computer science y α( ) x

Getting ack to computer science y α( ) w A The input elongs to a free monoid A

Getting ack to computer science B k α( ) w A The input elongs to a free monoid A The output elongs to the Boolean semiring B

Getting ack to computer science B k L w A L A The input elongs to a free monoid A The output elongs to the Boolean semiring B The function realised is a language

Getting ack to computer science B k α( ) (u, v) A B The input elongs to a direct product of free monoids A B The output elongs to the Boolean semiring B

Getting ack to computer science B k R (u, v) A B R A B The input elongs to a direct product of free monoids A B The output elongs to the Boolean semiring B The function realised is a relation etween words

The simplest Turing Machine Finite control p State a 1 a 2 a 3 a 4 a n $ Direction of movement of the read head The 1 way 1 tape Turing Machine (1W1TM)

The simplest Turing Machine is equivalent to finite automata a a B 1 p q

The simplest Turing Machine is equivalent to finite automata a a B 1 p q a A

The simplest Turing Machine is equivalent to finite automata a a B 1 p q a A p p p q a p a q q q

The simplest Turing Machine is equivalent to finite automata a a B 1 p q L(B 1 ) A a A p p p q a p a q q q L(B 1 )={w A w A A } = {w A w 1}

Rational (or regular) languages Languages accepted (or recognized) y finite automata = Languages descried y rational (or regular) expressions = Languages defined y MSO formulae

Remarkale features of the finite automaton model Decidale equivalence (decidale inclusion) Closure under complement Canonical automaton (minimal deterministic automaton)

Automata versus languages B 1 a a p q L(B 1 ) A

Automata versus languages B 1 a a p q L(B 1 ) A B 1 a p a q L(B 1 ) A

Automata versus languages B 1 a a p q L(B 1 ) A B 1 a p a q L(B 1 ) A L(B 1 )=L(B 1 )={ w A w 1 }

Automata versus languages B 1 a a p q L(B 1 ) A B 1 a p a q L(B 1 ) A L(B 1 )=L(B 1 )={ w A w 1 } = A A

Amiguity: a preliminary to multiplicity Here, automaton stands for classical (Boolean) automaton. Definition A (trim) automaton A is unamiguous if no word is the lael of more than one successful computation of A.

Amiguity: a preliminary to multiplicity a a B 1 p q L(B 1 )=A A

Amiguity: a preliminary to multiplicity a a B 1 p q L(B 1 )=A A B 1 a p a q L(B 1 )=A A

Amiguity: a preliminary to multiplicity a a B 1 p q L(B 1 )=A A B 1 a p a q L(B 1 )=A A Counting the numer of successful computations B 1 : a 2 B 1 : a 1

Amiguity: a preliminary to multiplicity a a B 1 p q L(B 1 )=A A B 1 a p a q L(B 1 )=A A Counting the numer of successful computations B 1 : w w B 1 : w 1

A new automaton model N k α( ) w A The input elongs to a free monoid A The output elongs to the integer semiring N

A new automaton model N k s w A s : A N The input elongs to a free monoid A The output elongs to the integer semiring N The function realised is a function from A to N

A new automaton model N k s w A s : A N s N A The input elongs to a free monoid A The output elongs to the integer semiring N The function realised is a function from A to N we call it aseries

A new automaton model N k s w A s : A N s N A s 1 = + a+ a+2+ aa+ +2a+3+ The input elongs to a free monoid A The output elongs to the integer semiring N The function realised is a function from A to N we call it aseries

The weighted automaton model a a B 1 p q

The weighted automaton model C 1 a 2a p q 2

The weighted automaton model C 1 a 2a p q 2 1 p p 1 p q a p 2 a q q 1 2 q 1

The weighted automaton model C 1 a 2a p q 2 1 p p 1 p q a p 2 a q q 1 2 q 1 Weight of a path c: product of the weights of transitions in c Weight of a word w: sum of the weights of paths with. lael w

The weighted automaton model C 1 a 2a p q 2 1 p p 1 p q a p 2 a q q 1 2 q 1 Weight of a path c: product of the weights of transitions in c Weight of a word w: sum of the weights of paths with. lael w a 1+4=5

The weighted automaton model a 2a C 1 p q C 1 N A 2 1 p p 1 p q a p 2 a q q 1 2 q 1 Weight of a path c: product of the weights of transitions in c Weight of a word w: sum of the weights of paths with. lael w a 1+4=5 C 1 : A N

The weighted automaton model K k s w A s : A K s K A The input elongs to a free monoid A The output elongs to a semiring K The function realised is a function from A to K: aseriesin K A

Richness of the model of weighted automata B classic automata N usual counting Z, Q, R numerical multiplicity Z +, min, + Min-plus automata Z, min, max fuzzy automata P (B )=B B transducers N B weighted transducers P (F (B)) pushdown automata

Another example 0a 1a 0 p q 0 L 1 L 1 Zmin A 1 0

Another example 0a 1a 0 p q 0 L 1 L 1 Zmin A 1 0 0 p p 1 0 q q 0 0 a p 1 a q 1 p 0 0 q 0

Another example 0a 1a 0 p q 0 L 1 L 1 Zmin A 1 0 0 p p 1 0 q q 0 0 a p 1 a q 1 p 0 0 q 0 Weight of a path c: product, that is, the sum, of the weights of transitions in c Weight of a word w: sum, that is, the min of the weights of paths with lael w.

Series play the role of languages K A plays the role of P (A )

Weighted automata theory is linear algera of computer science

The Turing Machine equivalent to finite transducers Finite control p State a 1 a 2 a 3 a 4 a n $ k 1 k 2 k 3 k 4 k l $ Direction of movement of the k read heads The 1 way k tape Turing Machine (1WkTM)

Outline of the lectures 1. Rationality 2. Recognisaility 3. Reduction and equivalence 4. Morphisms of automata

Lecture II Rationality

Outline of Lecture II The set of series K A is a K-algera. Automata are (essentially) matrices: A = I, E, T Computing the ehaviour of an automaton oils down to solving a linear system X = E X + T (s) Solving the linear system (s) amounts to invert the matrix (Id E) (hence the name rational) The inversion of Id E is realised y an infinite sum Id + E + E 2 + E 3 + :thestar of E What can e computed y a finite automaton is exactly what can e computed y the star operation (together with the algera operations)

The semiring K A K semiring A free monoid s K A s : A K s : w s, w s = s, w w w A Point-wise addition s + t, w = s, w + t, w Cauchy product st, w = s, u t, v uv=w {(u, v) uv = w} finite = Cauchy product well-defined K A is a semiring

The semiring K M K semiring M monoid s K M s : M K s : m s, m s = s, w w m M Point-wise addition s + t, m = s, m + t, m Cauchy product st, m = s, x t, y xy=m m {(x, y) xy = m} finite = Cauchy product well-defined

The semiring K M Conditions for {(x, y) xy = m} finite for all m Definition M is graded if M equipped with a length function ϕ ϕ: M N ϕ(mm )=ϕ(m)+ϕ(m ) M f.g. and graded = K M is a semiring Examples M trace monoid, then K M is a semiring K A B is a semiring F (A), the free group on A, is not graded

The algera K M K semiring M f.g. graded monoid s K A s : A K s : w s, w s = s, w w w A Point-wise addition s + t, m = s, m + t, m Cauchy product st, m = s, x t, y xy=m External multiplication ks, m = k s, m K M is an algera

The star operation t K t = n N t n How to define infinite sums? Onepossilesolution Topology on K Definition of summale families and of their sum t defined if {t n } n N summale Other possile solutions axiomatic definition of star, equational definition of star

The star operation t K t = n N t n

The star operation t K t = n N t n K (0K ) =1 K K = N x 0 x not defined. K = N = N {+ } x 0 x =. K = Q ( 1 2 ) =2 with the natural topology, ( 1 2 ) is undefined with the discrete topology.

The star operation t K t = n N t n In any case t =1 K + tt If K is a ring Star has the same flavor as the inverse t (1 K t) =1 K 1 K 1 K t =1 K + t + t 2 + + t n +

Star of series s K A When is s = n N s n defined? Topology on K yields topology on K A s proper s 0 = s, 1 A =0 K s proper = s defined

Rational series K A K A sualgera of polynomials KRat A closure of K A under sum product exterior multiplication and star KRat A K A sualgera of rational series

Fundamental theorem of finite automata Theorem s KRat A A WA (A ) s = A

Fundamental theorem of finite automata Theorem s KRat A A WA (A ) s = A Kleene theorem?

Fundamental theorem of finite automata Theorem s KRat A A WA (A ) s = A Kleene theorem? Theorem M finitely generated graded monoid s KRat M A WA (M) s = A

Automata are matrices C 1 a 2a p q 2 C 1 = I 1, E 1, T 1 = ( ) ( ) (1 ) a + 0 0,, 0 2a +2 1.

Automata are matrices A = I, E, T E = incidence matrix

Automata are matrices A = I, E, T E = incidence matrix Notation wl(x) = weighted lael of x In our model, e transition wl(e) =ka

Automata are matrices A = I, E, T E = incidence matrix Notation wl(x) = weighted lael of x In our model, e transition wl(e) =ka E p,q = {wl(e) e transition from p to q}

Automata are matrices A = I, E, T E = incidence matrix Notation wl(x) = weighted lael of x In our model, e transition wl(e) =ka E p,q = {wl(e) e transition from p to q} Lemma E n p,q = {wl(c) c computation from p to q of length n}

Automata are matrices A = I, E, T E = incidence matrix E p,q = {wl(e) e transition from p to q}

Automata are matrices A = I, E, T E = incidence matrix E p,q = {wl(e) e transition from p to q} E = n N E n E p,q = {wl(c) c computation from p to q}

Automata are matrices A = I, E, T E = incidence matrix E p,q = {wl(e) e transition from p to q} E = n N E n E p,q = {wl(c) c computation from p to q} A = I E T

Automata are matrices K semiring M graded monoid K M Q Q is isomorphic to K Q Q M E K M Q Q E proper = E defined

Automata are matrices K semiring M graded monoid K M Q Q is isomorphic to K Q Q M E K M Q Q E proper = E defined Theorem The entries of E are in the rational closure of the entries of E

Fundamental theorem of finite automata K semiring M graded monoid K M Q Q is isomorphic to K Q Q M E K M Q Q E proper = E defined Theorem The entries of E are in the rational closure of the entries of E Theorem The family of ehaviours of weighted automata over M with coefficients in K is rationally closed.

The collect theorem K A B is isomorphic to [K B ] A Theorem Under the aove isomorphism, KRat A B corresponds to [KRat B ] Rat A

Lecture III Recognisaility

Outline of Lecture III Representation and recognisale series. Automata over free monoids are representations The notion of action and deterministic automata The reachaility space and the control morphism The notion of quotient and the minimal automaton The oservation morphism The representation theorem

Recognisale series K semiring A free monoid

Recognisale series K semiring A free monoid K-representation Q finite µ: A K Q Q morphism ( I,µ,T ) I K 1 Q µ: A K Q Q T K Q 1

Recognisale series K semiring A free monoid K-representation Q finite µ: A K Q Q morphism ( I,µ,T ) I K 1 Q µ: A K Q Q T K Q 1 ( I,µ,T ) realises (recognises) s K A w A s, w = I µ(w) T

Recognisale series K semiring A free monoid K-representation Q finite µ: A K Q Q morphism ( I,µ,T ) I K 1 Q µ: A K Q Q T K Q 1 ( I,µ,T ) realises (recognises) s K A w A s, w = I µ(w) T s K A recognisale if s realised y a K-representation

Recognisale series K semiring A free monoid K-representation Q finite µ: A K Q Q morphism ( I,µ,T ) I K 1 Q µ: A K Q Q T K Q 1 Example ( I,µ,T ) realises (recognises) s K A w A I = ( 1 0 ), µ(a) = ( I,µ,T ) realises s, w = I µ(w) T ( ) 1 0, µ() = 0 1 w A w w ( ) 1 1, T = 0 1 KRec A ( ) 0 1

Recognisale series K semiring M monoid K-representation Q finite µ: A K Q Q morphism ( I,µ,T ) I K 1 Q µ: A K Q Q T K Q 1 ( I,µ,T ) realises (recognises) s K A w A s, w = I µ(w) T

Recognisale series K semiring M monoid K-representation Q finite µ: M K Q Q morphism ( I,µ,T ) I K 1 Q µ: M K Q Q T K Q 1 ( I,µ,T ) realises (recognises) s K A w A s, w = I µ(w) T

Recognisale series K semiring M monoid K-representation Q finite µ: M K Q Q morphism ( I,µ,T ) I K 1 Q µ: M K Q Q T K Q 1 ( I,µ,T ) realises (recognises) s K M m M s, m = I µ(m) T

Recognisale series K semiring M monoid K-representation Q finite µ: M K Q Q morphism ( I,µ,T ) I K 1 Q µ: M K Q Q T K Q 1 ( I,µ,T ) realises (recognises) s K M m M s, m = I µ(m) T s K M recognisale if s realised y a K-representation

The key lemma K semiring A free monoid

The key lemma K semiring A free monoid µ: A K Q Q defined y {µ(a)} a A

The key lemma K semiring M monoid µ: A K Q Q defined y {µ(a)} a A

The key lemma K semiring M monoid µ: M K Q Q defined y?

The key lemma K semiring A free monoid µ: A K Q Q defined y {µ(a)} a A

The key lemma K semiring A free monoid µ: A K Q Q defined y {µ(a)} a A Lemma µ: A K Q Q X = a A µ(a)a w A X, w = µ(w)

Automata are matrices C 1 a 2a p q C 1 = I 1, E 1, T 1 = 2 ( ) ( ) (1 ) a + 0 0,, 0 2a +2 1.

Automata over free monoids are representations a 2a C 1 p q ( ) ( ) (1 ) a + 0 C 1 = I 1, E 1, T 1 = 0,, 0 2a +2 1 ( ) ( ) 1 0 1 1 E 1 = a + 0 2 0 2 2.

Automata over free monoids are representations C 1 a p ( ) ( ) (1 ) a + 0 C 1 = I 1, E 1, T 1 = 0,, 0 2a +2 1 ( ) ( ) 1 0 1 1 E 1 = a + 0 2 0 2 C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = 2a 2 q ( ) 1 0, µ 0 2 1 () =. ( ) 1 1 0 2

Automata over free monoids are representations C 1 a p ( ) ( ) (1 ) a + 0 C 1 = I 1, E 1, T 1 = 0,, 0 2a +2 1 ( ) ( ) 1 0 1 1 E 1 = a + 0 2 0 2 C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = C 1 = I 1 E 1 T 1 = 2a 2 q w A (I 1 µ 1 (w) T 1 )w ( ) 1 0, µ 0 2 1 () =. ( ) 1 1 0 2

Automata over free monoids are representations C 1 a p ( ) ( ) (1 ) a + 0 C 1 = I 1, E 1, T 1 = 0,, 0 2a +2 1 ( ) ( ) 1 0 1 1 E 1 = a + 0 2 0 2 C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = 2a 2 q ( ) 1 0, µ 0 2 1 () =. ( ) 1 1 0 2 C 1 = I 1 E 1 T 1 = w A (I 1 µ 1 (w) T 1 )w C 1 KRec A

Automata over free monoids are representations C 1 a p ( ) ( ) (1 ) a + 0 C 1 = I 1, E 1, T 1 = 0,, 0 2a +2 1 ( ) ( ) 1 0 1 1 E 1 = a + 0 2 0 2 C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = 2a 2 q ( ) 1 0, µ 0 2 1 () =. ( ) 1 1 0 2 Conversely, representations are automata

The Kleene-Schützenerger Theorem Fundamental Theorem of Finite Automata and Key Lemma yield Theorem A finite KRec A = KRat A

The reachaility set A =(I,µ,T )

The reachaility set A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A

The reachaility set A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A A acts on R A : (I µ(w)) a =(I µ(w)) µ(a) =I µ(wa)

The reachaility set A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A A acts on R A : (I µ(w)) a =(I µ(w)) µ(a) =I µ(wa) This action turns R A into a deterministic automaton Â (possily infinite)

The reachaility set C 1 =(I 1,µ 1, T 1 ) Ĉ 1 ( 1 3 ) 3 a ( 1 7 ) 7 ( 1 6 ) ( 1 1 ) 1 0 ( ) 1 0 1 a a ( 1 2 ) 2 a 6 ( 1 5 ) 5 ( 1 4 ) 4

The reachaility set A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A R A is turned into a deterministic automaton Â

The reachaility set A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A R A is turned into a deterministic automaton Â If K = B, Â is the (classical) determinisation of A If K is locally finite, R A and Â are finite.

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A Ψ A : K A K Q w A Ψ A (w) =I µ(w)

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A Ψ A : K A K Q w A Ψ A (w) =I µ(w) R A =Ψ A (A ) ImΨ A =Ψ A (K A )= R A

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A Ψ A : K A K Q w A Ψ A (w) =I µ(w) R A =Ψ A (A ) ImΨ A =Ψ A (K A )= R A K A Ψ A K Q The control morphism

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A Ψ A : K A K Q w A Ψ A (w) =I µ(w) R A =Ψ A (A ) ImΨ A =Ψ A (K A )= R A K A w Ψ A Ψ A K Q x The control morphism

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A Ψ A : K A K Q w A Ψ A (w) =I µ(w) R A =Ψ A (A ) ImΨ A =Ψ A (K A )= R A A K A K A w wa Ψ A Ψ A K Q x The control morphism

The control morphism A =(I,µ,T ) Reachaility set Reachaility space R A = {I µ(w) w A } R A K Q R A Ψ A : K A K Q w A Ψ A (w) =I µ(w) R A =Ψ A (A ) ImΨ A =Ψ A (K A )= R A A K A K A w wa Ψ A K Q A K Q Ψ A Ψ A x Ψ A x µ(a) The control morphism is a morphism of actions

A asic construct: the quotient of series a 1 a 2 a 3...a n s K A The input elongs to a free monoid A

A asic construct: the quotient of series a 1 a 2...a n s K A The input elongs to a free monoid A

A asic construct: the quotient of series s s K A The input elongs to a free monoid A

A asic construct: the quotient of series s, a 1...a n = k s K A The input elongs to a free monoid A

A asic construct: the quotient of series a 1 a 2 a 3...a n s K A

A asic construct: the quotient of series s a 1 a 2 s K A

A asic construct: the quotient of series s, a 1...a n = k s K A

A asic construct: the quotient of series a 1 a 2 a 3...a n s K A

A asic construct: the quotient of series s a 1 a 2 s K A

A asic construct: the quotient of series k a 1 a 2 k = s, a 3...a n = s, a 1 a 2 a 3...a n

A asic construct: the quotient of series k a 1 a 2 k = s, a 3...a n = s, a 1 a 2 a 3...a n s =[a 1 a 2 ] 1 s The series s is the quotient of s y a 1 a 2

A asic construct: the quotient of series uv s K A

A asic construct: the quotient of series u v

A asic construct: the quotient of series k = s, v = s, uv

A asic construct: the quotient of series k = s, v = s, uv s = u 1 s The series s is the quotient of s y u

The quotient operation s K A v A v 1 s = w A s, vw w

The quotient operation s K A v A v 1 s = w A s, vw w v 1 : K A K A endomorphism of K-modules

The quotient operation s K A v A v 1 s = w A s, vw w v 1 : K A K A endomorphism of K-modules v 1 (s + t) =v 1 s + v 1 t v 1 (ks)=k (v 1 s)

The quotient operation s K A v A v 1 s = w A s, vw w v 1 : K A K A endomorphism of K-modules A K A K A s v 1 s Quotient is a (right) action of A on K A

The quotient operation s K A v A v 1 s = w A s, vw w v 1 : K A K A endomorphism of K-modules A K A K A s v 1 s Quotient is a (right) action of A on K A (uv) 1 s = v 1 (u 1 s)

The minimal automaton s K A R s = { v 1 s v A }

The minimal automaton s K A R s = { v 1 s v A } Quotient turns R s into the minimal automaton A s of s (possily infinite)

The oservation morphism A =(I,µ,T ) Φ A : K Q K A Φ A (x) =( x,µ,t ) = w A (x µ(w) T )w

The oservation morphism A =(I,µ,T ) Φ A : K Q K A Φ A (x) =( x,µ,t ) = w A (x µ(w) T )w s = ( I,µ,T ) =Φ A (I ) w 1 s = ( I µ(w),µ,t )

The oservation morphism A =(I,µ,T ) Φ A : K Q K A Φ A (x) =( x,µ,t ) = w A (x µ(w) T )w s = ( I,µ,T ) =Φ A (I ) w 1 s = ( I µ(w),µ,t ) w 1 Φ A (x) =Φ A (x µ(w))

The oservation morphism A =(I,µ,T ) Φ A : K Q K A Φ A (x) =( x,µ,t ) = (x µ(w) T )w w A w 1 Φ A (x) =Φ A (x µ(w)) K Q x Φ A Φ A K A t

The oservation morphism A =(I,µ,T ) Φ A : K Q K A Φ A (x) =( x,µ,t ) = (x µ(w) T )w w A w 1 Φ A (x) =Φ A (x µ(w)) K Q A K Q x x µ(a) Φ A A Φ A K A K A Φ A t a 1 t Φ A The oservation morphism is a morphism of actions

The oservation morphism A =(I,µ,T ) Φ A : K Q K A Φ A (x) =( x,µ,t ) = (x µ(w) T )w w A w 1 Φ A (x) =Φ A (x µ(w)) A K A K A w wa Ψ A K Q A K Q Ψ A Ψ A x Ψ A x µ(a) Φ A A Φ A K A K A Φ A t a 1 t Φ A The oservation morphism is a morphism of actions

The representation theorem U K A sumodule U stale (y quotient) Theorem (Fliess 71, Jaco 74) s KRec A U stale finitely generated s U

The representation theorem U K A sumodule U stale (y quotient) Theorem (Fliess 71, Jaco 74) s KRec A U stale finitely generated s U A K A K A Ψ A K Q A K Q Ψ A Φ A A Φ A K A K A

The representation theorem U K A sumodule U stale (y quotient) Theorem (Fliess 71, Jaco 74) s KRec A = U stale finitely generated s U 1 A K A K A Ψ A I Im Ψ A K Q K Q Φ A s Φ A (Im Ψ A ) K A K A A A A Ψ A Φ A

The representation theorem U K A sumodule U stale (y quotient) Theorem (Fliess 71, Jaco 74) s KRec A = U stale finitely generated s U A K A K A Ψ A K Q A K Q Ψ A Φ A A Φ A K A K A

Lecture IV Reduction and morphisms

Outline of Lecture IV An appetizing theorem Reduction of automata with weights in fields The decidaility of equivalence prolem The notion of conjugacy of automata Out-morphisms and In-morphisms of automata

An appetizing result K semiring A free monoid Definition The Hadamard product of s, t K A is w A s t, w = s, w t, w

An appetizing result K semiring A free monoid Definition The Hadamard product of s, t K A is w A s t, w = s, w t, w Theorem If K is commutative, then KRec A is closed under Hadamard product

An appetizing result K semiring A free monoid Definition The Hadamard product of s, t K A is w A s t, w = s, w t, w Theorem If K is commutative, then KRec A is closed under Hadamard product ( I,µ,T ) ( J,κ,U ) = ( I J,µ κ, T U )

An appetizing result C 1 a 2a p q 2 C 2 a j 2a r C 2 = C 1 C 1 2 2a s 2 2 u 4a 2 4

Reduced representation A =(I,µ,T ) A is reduced if its dimension is minimal (among all equivalent representations) We suppose now that K is a (skew) field Proposition A is reduced iff Ψ A is surjective and Φ A injective Theorem A reduced representation of A is effectively computale (with cuic complexity) Corollary Equivalence of K-recognisale series is decidale

Equivalence of weighted automata Equivalence of weighted automata with weights in the Boolean semiring B a susemiring of a field (Z, min, +) Rat B NRat B decidale decidale undecidale undecidale decidale Equivalence of transducers undecidale transducers with multiplicity in N decidale functional transducers finitely amiguous (Z, min, +) decidale decidale

Conjugacy of automata Definition Let A = I, E, T and B = J, F, U e two K-automata. A is conjugate to B if X K-matrix IX = J, EX = XF, and T = XU

Conjugacy of automata Definition Let A = I, E, T and B = J, F, U e two K-automata. A is conjugate to B if X K-matrix IX = J, EX = XF, and T = XU This is denoted as A X = B.

Conjugacy of automata C = ( ) 0 z 0 0 1 0 0, 0 0 z, 1 0 0 2z 2 A = (1 ) 0, ( ) ( ) 0 z 0 2z, 0 1 ( 1 0 1 0 0) 0 1 = ( 1 0 ), 0 2 0 z 0 1 0 1 0 0 0 z 0 1 = 0 1 (0 ) z 0 2z, 0 0 2z 0 2 0 2 0 1 0 1 = 0 1 (0) 1 2 0 2 A 1z 2z 1 0 0 1 0 2 = 1z 1z 2z 2 C

Conjugacy of automata Definition Let A = I, E, T and B = J, F, U e two K-automata. A is conjugate to B if X K-matrix IX = J, EX = XF, and T = XU This is denoted as A X = B.

Conjugacy of automata Definition Let A = I, E, T and B = J, F, U e two K-automata. A is conjugate to B if X K-matrix IX = J, EX = XF, and T = XU This is denoted as A = X B. Conjugacy is a preorder (transitive and reflexive, ut not symmetric).

Morphisms of weighted automata Definition Amap ϕ: Q R defines a (Q R)-amalgamation matrix H ϕ 1 0 0 ϕ 2 : {j, r, s, u} {i, q, t} defines H ϕ2 = 0 1 0 0 1 0 0 0 1

Morphisms of weighted automata Definition A = I, E, T and B = J, F, U Amap ϕ: Q R defines K-automata of dimension Q and R. an Out-morphism ϕ: A B if A is conjugate to B y the matrix H ϕ : A Hϕ = B IH ϕ = J, EH ϕ = H ϕ F, T = H ϕ U B is a quotient of A

Morphisms of weighted automata a 2a C 2 j r 2 2a 2 4a 2 s u 2 4

ϕ 2 : {j, r, s, u} {i, q, t} Morphisms of weighted automata a 2a C 2 j r 2 2a 2 s 2 2 4 u 4a 1 0 0 H ϕ2 = 0 1 0 0 1 0 0 0 1

ϕ 2 : {j, r, s, u} {i, q, t} Morphisms of weighted automata a 2a C 2 j r 2 2a 2 s 2 2 4 u 4a 1 0 0 H ϕ2 = 0 1 0 0 1 0 0 0 1 V 2 a 2a i 2 q 2 t 4a 2 4 C 2 H ϕ2 = V2

Morphisms of weighted automata Definition A = I, E, T and B = J, F, U K-automata of dimension Q and R. Amap ϕ: Q R defines an In-morphism ϕ: A B if A is conjugate to B y the matrix H ϕ : A Hϕ = B IH ϕ = J, EH ϕ = H ϕ F, T = H ϕ U B is a quotient of A Directed notion Price to pay for the weight

Morphisms of weighted automata Definition A = I, E, T and B = J, F, U K-automata of dimension Q and R. Amap ϕ: Q R defines an In-morphism ϕ: A B if B is conjugate to A y the matrix t H ϕ : B t H ϕ = A J t H ϕ = I, F t H ϕ = t H ϕ E, U = t H ϕ T B is a co-quotient of A Directed notion Price to pay for the weight

ϕ 2 : {j, r, s, u} {i, q, t} Morphisms of weighted automata a 2a C 2 j r 2 2a 2 s 2 2 4 u 4a 1 0 0 H ϕ2 = 0 1 0 0 1 0 0 0 1 a 2a 4 i q t V 2 4a V 2 2 t H ϕ2 = C2 4

ϕ 2 : {j, r, s, u} {i, q, t} Morphisms of weighted automata a 2a C 2 j r 2 2a 2 s 2 2 4 u 4a 1 0 0 H ϕ2 = 0 1 0 0 1 0 0 0 1 V 2 a 2a i 2 q 2 t 4a a 2a 4 i q t V 2 4a 2 4 2 4 C 2 H ϕ2 = V2 V 2 t H ϕ2 = C2

Morphisms of weighted automata Definition A = I, E, T and B = J, F, U Amap ϕ: Q R defines K-automata of dimension Q and R. an Out-morphism ϕ: A B if A is conjugate to B y the matrix H ϕ : A Hϕ = B B is a quotient of A Theorem Every K-automaton has a minimal quotient that is effectively computale (y Moore algorithm).

Documents for these lectures To e found at http://www.telecom-paristech.fr/ jsaka/ewscs2014/ In particular, a set of instructions for downloading a α release of a pre-experimental version of the Vaucanson 2 platform implemented as a virtual machine interfaced with IPython