Introduction to weighted automata theory Lectures given at the - PowerPoint PPT Presentation

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 � B ∗ � ◮ N � � weighted transducers ◮ P ( F ( B )) pushdown automata

Another example 0 a 1 a 0 0 � A ∗ � L 1 ∈ Z min � � L 1 p q 1 b 0 b

Another example 0 a 1 a 0 0 � A ∗ � L 1 ∈ Z min � � L 1 p q 1 b 0 b 1 b 0 a 1 b 0 0 − − → p − − − → p − − − → p − − − → p − − → 0 b 1 a 0 b 0 0 − − → q − − − → q − − − → q − − − → q − − →

Another example 0 a 1 a 0 0 � A ∗ � L 1 ∈ Z min � � L 1 p q 1 b 0 b 1 b 0 a 1 b 0 0 − − → p − − − → p − − − → p − − − → p − − → 0 b 1 a 0 b 0 0 − − → q − − − → q − − − → q − − − → q − − → ◮ 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 label w .

Another example 0 a 1 a 0 0 � A ∗ � L 1 ∈ Z min � � L 1 p q 1 b 0 b 1 b 0 a 1 b 0 0 − − → p − − − → p − − − → p − − − → p − − → 0 b 1 a 0 b 0 0 − − → q − − − → q − − − → q − − − → q − − → ◮ 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 label w . L 1 : A ∗ − b ab �− → min(1 + 0 + 1 , 0 + 1 + 0) = 1 → Z min

Another example 0 a 1 a 0 0 � A ∗ � L 1 ∈ Z min � � L 1 p q 1 b 0 b 1 b 0 a 1 b 0 0 − − → p − − − → p − − − → p − − − → p − − → 0 b 1 a 0 b 0 0 − − → q − − − → q − − − → q − − − → q − − → ◮ 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 label w . C 1 = 01 A ∗ + 0 a + 0 b + 1 ab + 1 b a + 0 b b + · · · + 1 b ab + · · ·

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

Weighted automata theory is linear algebra 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. Recognisability 3. Reduction and equivalence 4. Morphisms of automata

Lecture II Rationality

Outline of Lecture II ◮ The set of series K � � A ∗ � � is a K -algebra. ◮ Automata are (essentially) matrices: A = � I , E , T � ◮ Computing the behaviour of an automaton boils 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 by an infinite sum Id + E + E 2 + E 3 + · · · : the star of E ◮ What can be computed by a finite automaton is exactly what can be computed by the star operation (together with the algebra operations)

� A ∗ � The semiring K � � A ∗ free monoid K semiring s : A ∗ → K � A ∗ � s ∈ K � � s : w �− → � s , w � � � s , w � w s = w ∈ A ∗ Point-wise addition � s + t , w � = � s , w � + � t , w � � Cauchy product � s t , w � = � s , u �� t , v � u v = w { ( u , v ) | u v = w } finite ⇒ = Cauchy product well-defined � A ∗ � K � � is a semiring

The semiring K � � M � � K semiring M monoid s ∈ K � � M � � s : M → K s : m �− → � s , m � � � s , w � w s = m ∈ M Point-wise addition � s + t , m � = � s , m � + � t , m � � Cauchy product � s t , m � = � s , x �� t , y � x y = m ∀ m { ( x , y ) | x y = m } finite ⇒ = Cauchy product well-defined

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

The algebra K � � M � � K semiring M f.g. graded monoid s : A ∗ → K � A ∗ � s ∈ K � � s : w �− → � s , w � � � s , w � w s = w ∈ A ∗ Point-wise addition � s + t , m � = � s , m � + � t , m � � Cauchy product � s t , m � = � s , x �� t , y � x y = m External multiplication � k s , m � = k � s , m � K � � M � � is an algebra

The star operation � t ∗ = t n t ∈ K n ∈ N How to define infinite sums ? One possible solution Topology on K Definition of summable families and of their sum t ∗ defined { t n } n ∈ N summable if Other possible solutions axiomatic definition of star, equational definition of star

The star operation � t ∗ = t n t ∈ K n ∈ N

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

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

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

Rational series K � A ∗ � ⊆ K � � A ∗ � � subalgebra of polynomials K Rat A ∗ K � A ∗ � closure of under ◮ sum ◮ product ◮ exterior multiplication ◮ and star K Rat A ∗ ⊆ K � � A ∗ � � subalgebra of rational series

Fundamental theorem of finite automata Theorem s ∈ K Rat A ∗ ∃A ∈ WA ( A ∗ ) ⇐ ⇒ s = | | |A| | |

Fundamental theorem of finite automata Theorem s ∈ K Rat A ∗ ∃A ∈ WA ( A ∗ ) ⇐ ⇒ s = | | |A| | | Kleene theorem ?

Fundamental theorem of finite automata Theorem s ∈ K Rat A ∗ ∃A ∈ WA ( A ∗ ) ⇐ ⇒ s = | | |A| | | Kleene theorem ? Theorem M finitely generated graded monoid s ∈ K Rat M ⇐ ⇒ ∃A ∈ WA ( M ) s = | | |A| | |

Automata are matrices a 2 a b C 1 p q b 2 b �� � a + b � � 0 �� � b C 1 = � I 1 , E 1 , T 1 � = 1 0 . , , 0 2 a + 2 b 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 label of x In our model, e transition ⇒ wl ( e ) = k a

Automata are matrices A = � I , E , T � E = incidence matrix Notation wl ( x ) = weighted label of x In our model, e transition ⇒ wl ( e ) = k a � 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 label of x In our model, e transition ⇒ wl ( e ) = k a � E p , q = { wl ( e ) | e transition from p to q } Lemma � En 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 ∗ = E n n ∈ 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 ∗ = E n n ∈ 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 Q × Q � � Q × Q K � � M � is isomorphic to � M � � E ∗ defined � Q × Q E ∈ K � � M � E proper = ⇒

Automata are matrices K semiring M graded monoid K Q × Q � � Q × Q K � � M � is isomorphic to � M � � E ∗ defined � Q × Q E ∈ K � � M � E proper = ⇒ 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 Q × Q � � Q × Q K � � M � is isomorphic to � M � � E ∗ defined � Q × Q E ∈ K � � M � E proper = ⇒ Theorem The entries of E ∗ are in the rational closure of the entries of E Theorem The family of behaviours 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 above isomorphism, K Rat A ∗ × B ∗ corresponds to [ K Rat B ∗ ] Rat A ∗

Lecture III Recognisability

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

Recognisable series A ∗ free monoid K semiring

Recognisable series A ∗ free monoid K semiring K -representation µ : A ∗ → K Q × Q Q finite morphism µ : A ∗ → K Q × I ∈ K 1 × Q Q T ∈ K Q × 1 ( I , µ, T )

Recognisable series A ∗ free monoid K semiring K -representation µ : A ∗ → K Q × Q Q finite morphism µ : A ∗ → K Q × I ∈ K 1 × Q Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � A ∗ � � ( I , µ, T ) realises (recognises) ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T

Recognisable series A ∗ free monoid K semiring K -representation µ : A ∗ → K Q × Q Q finite morphism µ : A ∗ → K Q × I ∈ K 1 × Q Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � A ∗ � � ( I , µ, T ) realises (recognises) ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T s ∈ K � � A ∗ � � recognisable if s realised by a K -representation

Recognisable series A ∗ free monoid K semiring K -representation µ : A ∗ → K Q × Q Q finite morphism µ : A ∗ → K Q × I ∈ K 1 × Q Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � A ∗ � � ( I , µ, T ) realises (recognises) ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T s ∈ K � � A ∗ � � recognisable if s realised by a K -representation K Rec A ∗ ⊆ K � � A ∗ � � submodule of recognisable series

Recognisable series A ∗ free monoid K semiring K -representation µ : A ∗ → K Q × Q Q finite morphism µ : A ∗ → K Q × I ∈ K 1 × Q Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � A ∗ � � ( I , µ, T ) realises (recognises) ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T Example � 1 � � 1 � � 0 � � � 0 1 I = 1 0 µ ( a ) = µ ( b ) = T = , , , 0 1 0 1 1 � ∈ K Rec A ∗ | w | b w ( I , µ, T ) realises w ∈ A ∗

Recognisable series K semiring M monoid K -representation µ : A ∗ → K Q × Q Q finite morphism µ : A ∗ → K Q × I ∈ K 1 × Q Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � A ∗ � � ( I , µ, T ) realises (recognises) ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T

Recognisable series K semiring M monoid K -representation µ : M → K Q × Q Q finite morphism I ∈ K 1 × Q µ : M → K Q × Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � A ∗ � � ( I , µ, T ) realises (recognises) ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T

Recognisable series K semiring M monoid K -representation µ : M → K Q × Q Q finite morphism I ∈ K 1 × Q µ : M → K Q × Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � M � � ( I , µ, T ) realises (recognises) ∀ m ∈ M � s , m � = I · µ ( m ) · T

Recognisable series K semiring M monoid K -representation µ : M → K Q × Q Q finite morphism I ∈ K 1 × Q µ : M → K Q × Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � M � � ( I , µ, T ) realises (recognises) ∀ m ∈ M � s , m � = I · µ ( m ) · T s ∈ K � � M � � recognisable if s realised by a K -representation

Recognisable series K semiring M monoid K -representation µ : M → K Q × Q Q finite morphism I ∈ K 1 × Q µ : M → K Q × Q T ∈ K Q × 1 ( I , µ, T ) s ∈ K � � M � � ( I , µ, T ) realises (recognises) ∀ m ∈ M � s , m � = I · µ ( m ) · T s ∈ K � � M � � recognisable if s realised by a K -representation K Rec M ⊆ K � � M � � submodule of recognisable series

The key lemma A ∗ free monoid K semiring

The key lemma A ∗ free monoid K semiring µ : A ∗ → K Q × Q { µ ( a ) } a ∈ A defined by

The key lemma K semiring M monoid µ : A ∗ → K Q × Q { µ ( a ) } a ∈ A defined by

The key lemma K semiring M monoid ? µ : M → K Q × Q defined by

The key lemma A ∗ free monoid K semiring µ : A ∗ → K Q × Q { µ ( a ) } a ∈ A defined by