Rationality & Recognisability An introduction to weighted - PowerPoint PPT Presentation

Rationality & Recognisability An introduction to weighted automata theory Tutorial given at post-WATA 2014 Workshop Jacques Sakarovitch CNRS / Telecom ParisTech

Part I The model of weighted automata

Part II Rationality

Part III Recognisability

Outline of Part III ◮ Representation and recognisable series. • KS Theorem ◮ The reachability space and the control morphism • The notion of action ◮ The observation morphism • The notion of quotient and the minimal automaton • The representation theorem ◮ The reduced representation • The exploration procedure • Decidability of equivalence for weighted automata

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 ) � A ∗ � ( I , µ, T ) realises (recognises) s ∈ K � � ∀ 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 ) � A ∗ � ( I , µ, T ) realises (recognises) s ∈ K � � ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T � A ∗ � s ∈ K � � 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 ) � A ∗ � ( I , µ, T ) realises (recognises) s ∈ K � � ∀ w ∈ A ∗ � s , w � = I · µ ( w ) · T � A ∗ � s ∈ K � � 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 ) � A ∗ � ( I , µ, T ) realises (recognises) s ∈ K � � ∀ 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 ∗ ( I , µ, T ) realises | w | b w 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 ) � A ∗ � ( I , µ, T ) realises (recognises) s ∈ K � � ∀ 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 ) � A ∗ � ( I , µ, T ) realises (recognises) s ∈ K � � ∀ 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 ) ( I , µ, T ) realises (recognises) s ∈ K � � M � � ∀ 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 ) ( I , µ, T ) realises (recognises) s ∈ K � � M � � ∀ 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 ) ( I , µ, T ) realises (recognises) s ∈ K � � M � � ∀ 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 defined by { µ ( a ) } a ∈ A arbitrary

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

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 defined by { µ ( a ) } a ∈ A

The key lemma A ∗ free monoid K semiring µ : A ∗ → K Q × Q defined by { µ ( a ) } a ∈ A Lemma � µ : A ∗ → K Q × Q X = µ ( a ) a a ∈ A ∀ w ∈ A ∗ � X ∗ , w � = µ ( w )

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 over free monoids are representations 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 � 1 � � 1 � 0 1 E 1 = a + b 0 2 0 2

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

Automata over free monoids are representations 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 � 1 � � 1 � 0 1 E 1 = a + b 0 2 0 2 � 1 � � 1 � 0 1 C 1 = ( I 1 , µ 1 , T 1 ) µ 1 ( a ) = µ 1 ( b ) = , 0 2 0 2 � C 1 = I 1 · E 1 ∗ · T 1 = ( I 1 · µ 1 ( w ) · T 1 ) w w ∈ A ∗

Automata over free monoids are representations 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 � 1 � � 1 � 0 1 E 1 = a + b 0 2 0 2 � 1 � � 1 � 0 1 C 1 = ( I 1 , µ 1 , T 1 ) µ 1 ( a ) = µ 1 ( b ) = , 0 2 0 2 � C 1 = I 1 · E 1 ∗ · T 1 = C 1 ∈ K Rec A ∗ ( I 1 · µ 1 ( w ) · T 1 ) w w ∈ A ∗

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

The Kleene-Sch¨ utzenberger Theorem Fundamental Theorem of Finite Automata and Key Lemma yield

The Kleene-Sch¨ utzenberger Theorem Fundamental Theorem of Finite Automata and Key Lemma yield Theorem K Rec A ∗ = K Rat A ∗ A finite ⇒

The Kleene-Sch¨ utzenberger Theorem Fundamental Theorem of Finite Automata and Key Lemma yield Theorem K Rec A ∗ = K Rat A ∗ A finite ⇒ standard key lemma K WA ( A ∗ ) K Rat A ∗ K RatE A ∗ K Rec A ∗ elimination

Action of a monoid on a set

The reachability set A = ( I , µ, T )

The reachability set A = ( I , µ, T ) Reachability set Reachability space R A = { I · µ ( w ) | w ∈ A ∗ } R A ⊆ K Q � � � R A � � �

The reachability set A = ( I , µ, T ) Reachability set Reachability 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 ( w a )

The reachability set A = ( I , µ, T ) Reachability set Reachability 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 ( w a ) This action turns � R A into a deterministic automaton A (possibly infinite)

The reachability set C 1 = ( I 1 , µ 1 , T 1 ) � � 1 7 b � C 1 � � 7 1 3 b � � 3 a 1 6 � � 6 1 1 b 1 0 � � � � 1 1 0 1 5 b a � � 5 1 2 a � � 2 a 1 4 4

The reachability set A = ( I , µ, T ) Reachability set Reachability space R A = { I · µ ( w ) | w ∈ A ∗ } R A ⊆ K Q � � � R A � � � � R A is turned into a deterministic automaton A

The reachability set A = ( I , µ, T ) Reachability set Reachability space R A = { I · µ ( w ) | w ∈ A ∗ } R A ⊆ K Q � R A � � � � � � R A is turned into a deterministic automaton A � If K = B , A is the (classical) determinisation of A

The reachability set A = ( I , µ, T ) Reachability set Reachability space R A = { I · µ ( w ) | w ∈ A ∗ } R A ⊆ K Q � � � R A � � � � R A is turned into a deterministic automaton A � If K = B , A is the (classical) determinisation of A � If K is locally finite , R A and A are finite.