Introduction to weighted automata theory

Transcription

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

2 Based on Chapter III Chapter 4

3 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.

4 Lecture I The model of (finite) weighted automata

5 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

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

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

8 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

9 Getting ack to computer science y α( ) x

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

11 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

12 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

13 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

14 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

15 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)

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

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

18 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

19 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}

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

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

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

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

24 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 }

25 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

26 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.

27 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. Theorem It is decidale whether an automaton is amiguous or not.

28 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. Theorem It is decidale whether an automaton is amiguous or not. Proof?

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

30 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

31 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

32 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

33 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

34 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

35 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

36 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

37 The weighted automaton model a a B 1 p q

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

39 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

40 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

41 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

42 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 = 101 2

43 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

44 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 C 1 = + a+2a+3+ aa+2aa+ +5a+

45 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

46 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

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

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

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

50 Another example 0a 1a 0 p q 0 L 1 L 1 Zmin A 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. a min( , 0+1+0)=1 L 1 : A Zmin

51 Another example 0a 1a 0 p q 0 L 1 L 1 Zmin A 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. C 1 =01 A +0a +0 +1a +1a+0+ +1a+

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

53 Weighted automata theory is linear algera of computer science

54 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)

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

56 Lecture II Rationality

57 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)

58 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

59 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

60 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

61 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

62 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

63 The star operation t K t = n N t n

64 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.

65 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 t n +

66 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

67 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

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

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

70 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

71 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.

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

73 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

74 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}

75 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}

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

77 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}

78 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

79 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

80 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

81 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.

82 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

83 Lecture III Recognisaility

84 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

85 Recognisale series K semiring A free monoid

86 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

87 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

88 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

89 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 KRec A K A sumodule of recognisale series

90 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

91 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

92 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

93 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

94 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

95 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 KRec M K M sumodule of recognisale series

96 The key lemma K semiring A free monoid

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

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

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

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

101 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)

102 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.

103 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 ( ) ( ) E 1 = a

104 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 ( ) ( ) E 1 = a C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = 2a 2 q ( ) 1 0, µ () =. ( )

105 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 ( ) ( ) E 1 = a 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 () =. ( )

106 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 ( ) ( ) E 1 = a C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = 2a 2 q ( ) 1 0, µ () =. ( ) C 1 = I 1 E 1 T 1 = w A (I 1 µ 1 (w) T 1 )w C 1 KRec A

107 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 ( ) ( ) E 1 = a C 1 =(I 1,µ 1, T 1 ) µ 1 (a) = 2a 2 q ( ) 1 0, µ () =. ( ) Conversely, representations are automata

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

109 The reachaility set A =(I,µ,T )

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

111 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)

112 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  (possily infinite)

113 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 ( ) a a ( 1 2 ) 2 a 6 ( 1 5 ) 5 ( 1 4 ) 4

114 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

115 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  If K = B,  is the (classical) determinisation of A

116 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  If K = B,  is the (classical) determinisation of A If K is locally finite, R A and  are finite.

117 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  If K = B,  is the (classical) determinisation of A If K is locally finite, R A and  are finite. Counting in a locally finite semiring is not really counting

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

119 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)

120 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

121 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

122 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

123 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

124 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

125 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

126 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

127 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

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

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

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

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

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

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

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

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

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

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

138 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

139 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

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

141 A asic construct: the quotient of series u v

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

143 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

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

145 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

146 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)

147 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

148 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)

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

150 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)

151 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

152 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 )

153 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))

154 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

155 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

156 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

157 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

158 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

159 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

160 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

161 Lecture IV Reduction and morphisms

162 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

163 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

164 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

165 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 )

166 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

167 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

168 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

169 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

170 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

171 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.

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

173 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.

174 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).

175 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). A X = B implies that A and B are equivalent.

176 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). A X = B implies that A and B are equivalent. IEET

177 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). A X = B implies that A and B are equivalent. IEET = IEEXU

178 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). A X = B implies that A and B are equivalent. IEET = IEEXU= IEXFU

179 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). A X = B implies that A and B are equivalent. IEET = IEEXU= IEXFU= IXFFU

180 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). A X = B implies that A and B are equivalent. IEET = IEEXU= IEXFU= IXFFU= JFFU

181 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). A X = B implies that A and B are equivalent. IEET = IEEXU= IEXFU= IXFFU= JFFU and then IE T = JF U

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

183 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

184 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 Directed notion

185 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 Directed notion Price to pay for the weight

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

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

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

189 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 Directed notion Price to pay for the weight

190 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

191 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

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

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

194 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

195 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).

196 Documents for these lectures To e found at 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