An algebraic characterization of unary 2-way transducers Christian - PowerPoint PPT Presentation

An algebraic characterization of unary 2-way transducers Christian Choffrut 1 and Bruno Guillon 1,2 1 LIAFA - Université Paris-Diderot, Paris 7 2 Dipartimento di Informatica - Università degli studi di Milano Septembre 17, 2014 ICTCS - Perugia - 2014 work published in MFCS 2014 1 / 15

2-way automaton over Σ A � � Q , q - , F , δ transition set: ⊂ Q × Σ ⊲,⊳ × {− 1 , 0 , 1 } × Q t h e i n p u t w o r d ⊳ ⊲ ← → READ right endmarker left endmarker Automaton WRITE → 2 / 15

2-way automaton over Σ A � � Q , q - , F , δ transition set: ⊂ Q × Σ ⊲,⊳ × {− 1 , 0 , 1 } × Q t h e i n p u t w o r d ⊳ ⊲ ← → READ right endmarker left endmarker Automaton WRITE → 2 / 15

2-way transducer over Σ , Γ ( A , φ ) production function: δ → Rat (Γ ∗ ) � � Q , q - , F , δ t h e i n p u t w o r d ⊳ ⊲ ← → READ right endmarker left endmarker Automaton WRITE → t h e o u t p u t 2 / 15

A simple example: SQUARE = { ( w , ww ) | w ∈ Σ ∗ } a b a c c a b ⊲ ⊳ 3 / 15

A simple example: SQUARE = { ( w , ww ) | w ∈ Σ ∗ } a b a c c a b ⊲ ⊳ a b a c c a b ◮ copy the input word 3 / 15

A simple example: SQUARE = { ( w , ww ) | w ∈ Σ ∗ } a b a c c a b ⊲ ⊳ a b a c c a b ◮ copy the input word ◮ rewind the input tape 3 / 15

A simple example: SQUARE = { ( w , ww ) | w ∈ Σ ∗ } a b a c c a b ⊲ ⊳ a b a c c a b a b a c c a b ◮ copy the input word ◮ rewind the input tape ◮ append a copy of the input word 3 / 15

A simple example: SQUARE = { ( w , ww ) | w ∈ Σ ∗ } a b a c c a b ⊲ ⊳ a b a c c a b a b a c c a b ◮ copy the input word ◮ rewind the input tape ◮ append a copy of the input word 3 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a copy the input word 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a copy the input word rewind the input tape 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a a a a a copy the input word rewind the input tape 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a a a a a copy the input word rewind the input tape 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a a a a a a a a a copy the input word rewind the input tape 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a a a a a a a a a copy the input word rewind the input tape accept and halt with nondeterminism 4 / 15

� ( a n , a kn ) | k , n ∈ N � Another example: UnaryMult = ⊲ a a a a ⊳ a a a a a a a a a a a a copy the input word rewind the input tape accept and halt with nondeterminism 4 / 15

Rational operations ◮ Union R 1 ∪ R 2 ◮ Componentwise concatenation R 1 · R 2 = { ( u 1 u 2 , v 1 v 2 ) | ( u 1 , v 1 ) ∈ R 1 and ( u 2 , v 2 ) ∈ R 2 } ◮ Kleene star R ∗ = { ( u 1 u 2 · · · u k , v 1 v 2 · · · v k ) | ∀ i ( u i , v i ) ∈ R } 5 / 15

Rational operations ◮ Union R 1 ∪ R 2 ◮ Componentwise concatenation R 1 · R 2 = { ( u 1 u 2 , v 1 v 2 ) | ( u 1 , v 1 ) ∈ R 1 and ( u 2 , v 2 ) ∈ R 2 } ◮ Kleene star R ∗ = { ( u 1 u 2 · · · u k , v 1 v 2 · · · v k ) | ∀ i ( u i , v i ) ∈ R } Definition ( Rat (Σ ∗ × Γ ∗ ) ) The class of rational relations is the smallest class: ◮ that contains finite relations ◮ and which is closed under rational operations 5 / 15

Rational operations ◮ Union R 1 ∪ R 2 ◮ Componentwise concatenation R 1 · R 2 = { ( u 1 u 2 , v 1 v 2 ) | ( u 1 , v 1 ) ∈ R 1 and ( u 2 , v 2 ) ∈ R 2 } ◮ Kleene star R ∗ = { ( u 1 u 2 · · · u k , v 1 v 2 · · · v k ) | ∀ i ( u i , v i ) ∈ R } Definition ( Rat (Σ ∗ × Γ ∗ ) ) The class of rational relations is the smallest class: ◮ that contains finite relations ◮ and which is closed under rational operations Theorem (Elgot, Mezei - 1965) 1-way transducers = = the class of rational relations. 5 / 15

Hadamard operations ◮ H-product R 1 H R 2 = { ( u , v 1 v 2 ) | ( u , v 1 ) ∈ R 1 and ( u , v 2 ) ∈ R 2 } 6 / 15

Hadamard operations ◮ H-product R 1 H R 2 = { ( u , v 1 v 2 ) | ( u , v 1 ) ∈ R 1 and ( u , v 2 ) ∈ R 2 } Example: SQUARE = { ( w , ww ) | w ∈ Σ ∗ } = Identity H Identity ⊲ a b a c c a b ⊳ a b a c c a b a b a c c a b ◮ copy the input word ◮ rewind the input tape ◮ append a copy of the input word 6 / 15

Hadamard operations ◮ H-product R 1 H R 2 = { ( u , v 1 v 2 ) | ( u , v 1 ) ∈ R 1 and ( u , v 2 ) ∈ R 2 } ◮ H-star R H ⋆ = { ( u , v 1 v 2 · · · v k ) | ∀ i ( u , v i ) ∈ R } 6 / 15

Hadamard operations ◮ H-product R 1 H R 2 = { ( u , v 1 v 2 ) | ( u , v 1 ) ∈ R 1 and ( u , v 2 ) ∈ R 2 } ◮ H-star R H ⋆ = { ( u , v 1 v 2 · · · v k ) | ∀ i ( u , v i ) ∈ R } � � = Identity H ⋆ Example: UnaryMult = ( a n , a kn ) | k , n ∈ N copy the input word rewind the input tape ⊲ a a a a ⊳ accept and halt with nondeterminism a a a a a a a a a a a a 6 / 15

H - Rat relations Definition A relation R is in H - Rat (Σ ∗ × Γ ∗ ) if A i H B H ⋆ � R = i 0 ≤ i ≤ n where for each i , A i and B i are rational relations. 7 / 15

Main result When Σ = { a } and Γ = { a } : Theorem (Elgot, Mezei - 1965) 1-way transducers = = the class of rational relations . 8 / 15

Main result When Σ = { a } and Γ = { a } : k a l Theorem (Elgot, Mezei - 1965) t h i s T s e r u c n s n s d i o r a l a t 1-way transducers = y t = the class of rational relations . r e w a a t 2 - - R H 8 / 15

Main result When Σ = { a } and Γ = { a } : k a l Theorem (Elgot, Mezei - 1965) t h i s T s e r u c n s n s d i o r a l a t 1-way transducers = y t = the class of rational relations . r e w a a t 2 - - R H Proof ◮ ⊇ : easy ◮ ⊆ : difficult part 8 / 15

Known results ◮ 2-way functional = = MSO definable functions [Engelfriet, Hoogeboom - 2001] 9 / 15

Known results ◮ 2-way functional = = MSO definable functions [Engelfriet, Hoogeboom - 2001] ◮ 2-way general incomparable MSO definable relations [Engelfriet, Hoogeboom - 2001] 9 / 15

Known results ◮ 2-way functional = = MSO definable functions [Engelfriet, Hoogeboom - 2001] ◮ 2-way general incomparable MSO definable relations [Engelfriet, Hoogeboom - 2001] ◮ 1-way simulation of 2-way functional transducer: decidable and constructible [Filiot et al. - 2013] 9 / 15

Known results ◮ 2-way functional = = MSO definable functions [Engelfriet, Hoogeboom - 2001] ◮ 2-way general incomparable MSO definable relations [Engelfriet, Hoogeboom - 2001] ◮ 1-way simulation of 2-way functional transducer: decidable and constructible [Filiot et al. - 2013] When Γ = { a } : ◮ 2-way unambiguous − → 1-way [Anselmo - 1990] 9 / 15

Known results ◮ 2-way functional = = MSO definable functions [Engelfriet, Hoogeboom - 2001] ◮ 2-way general incomparable MSO definable relations [Engelfriet, Hoogeboom - 2001] ◮ 1-way simulation of 2-way functional transducer: decidable and constructible [Filiot et al. - 2013] When Γ = { a } : ◮ 2-way unambiguous − → 1-way [Anselmo - 1990] ◮ 2-way unambiguous = = 2-way deterministic [Carnino, Lombardy - 2014] 9 / 15

From H - Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is H and H ⋆ . closed under ∪ , 10 / 15

From H - Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is H and H ⋆ . closed under ∪ , Proof. ◮ R 1 ∪ R 2 : ◮ simulate T 1 or T 2 10 / 15

From H - Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is H and H ⋆ . closed under ∪ , Proof. ◮ R 1 ∪ R 2 : ◮ simulate T 1 or T 2 ◮ R 1 H R 2 : ◮ simulate T 1 ◮ rewind the input tape ◮ simulate T 2 10 / 15

From H - Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is H and H ⋆ . closed under ∪ , Proof. ◮ R H ⋆ : ◮ R 1 ∪ R 2 : ◮ repeat an arbitrary ◮ simulate T 1 or T 2 number of times: ◮ R 1 H R 2 : ◮ simulate T ◮ rewind the input tape ◮ simulate T 1 ◮ rewind the input tape ◮ reach the right endmarker ◮ simulate T 2 and accept 10 / 15

From H - Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is H and H ⋆ . closed under ∪ , Corollary H-Rat ⊆ accepted by 2-way transducers   A i H B H ⋆  � i  0 ≤ i ≤ n 10 / 15

From 2-way transducers to H - Rat (unary case) A first ingredient, a preliminary result: Lemma With arbitrary Σ and Γ = { a } : H and H ⋆ . H-Rat is closed under ∪ , Proof. Tedious formal computations. . . 11 / 15