An algebraic characterization of unary 2-way transducers

Transcription

1 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 work published in MFCS / 15

2 2-way automaton over Σ ( ) Q, q -, F, δ A transition set: Q Σ, { 1, 0, 1} Q left endmarker t h e i n p u t w o r d Automaton READ WRITE right endmarker 2 / 15

3 2-way automaton over Σ ( ) Q, q -, F, δ A transition set: Q Σ, { 1, 0, 1} Q left endmarker t h e i n p u t w o r d Automaton READ WRITE right endmarker 2 / 15

4 2-way transducer over Σ, Γ ( ) Q, q -, F, δ (A, φ) production function: δ Rat(Γ ) left endmarker t h e i n p u t w o r d Automaton READ right endmarker t h e o u t p u t WRITE 2 / 15

5 A simple example: SQUARE = {(w, ww) w Σ } a b a c c a b 3 / 15

6 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

7 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

8 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

9 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

10 Another example: UnaryMult = { (a n, a kn ) k, n N } a a a a 4 / 15

11 Another example: UnaryMult = { (a n, a kn ) k, n N } a a a a a a a a copy the input word 4 / 15

12 Another example: UnaryMult = { (a n, a kn ) k, n N } a a a a a a a a copy the input word rewind the input tape 4 / 15

13 Another example: UnaryMult = { (a n, a kn ) k, n N } a a a a a a a a a a a a copy the input word rewind the input tape 4 / 15

14 Another example: UnaryMult = { (a n, a kn ) k, n N } a a a a a a a a a a a a copy the input word rewind the input tape 4 / 15

15 Another example: UnaryMult = { (a n, a kn ) k, n N } 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

16 Another example: UnaryMult = { (a n, a kn ) k, n N } 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

17 Another example: UnaryMult = { (a n, a kn ) k, n N } 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

18 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

19 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

20 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 ) 1-way transducers == the class of rational relations. 5 / 15

21 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

22 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

23 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

24 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} Example: UnaryMult = { } (a n, a kn ) k, n N = Identity H a a a a copy the input word rewind the input tape accept and halt with nondeterminism a a a a a a a a a a a a 6 / 15

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

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

27 Main result When Σ = {a} and Γ = {a}: Theorem (Elgot, Mezei ) This talk 1-way transducers == the class of rational relations. 2-way transducers H-Rat relations 8 / 15

28 Main result When Σ = {a} and Γ = {a}: Theorem (Elgot, Mezei ) This talk 1-way transducers == the class of rational relations. 2-way transducers H-Rat relations Proof : easy : difficult part 8 / 15

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

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

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

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

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

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

35 From H-Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is closed under, H and H. Proof. R 1 R 2 : simulate T1 or T 2 10 / 15

36 From H-Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is closed under, H and H. Proof. R 1 R 2 : simulate T1 or T 2 R 1 H R 2 : simulate T1 rewind the input tape simulate T 2 10 / 15

37 From H-Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is closed under, H and H. Proof. R 1 R 2 : simulate T1 or T 2 R 1 H R 2 : simulate T1 rewind the input tape simulate T 2 R H : repeat an arbitrary number of times: simulate T rewind the input tape reach the right endmarker and accept 10 / 15

38 From H-Rat to 2-way transducers (unary case) Property The family of relations accepted by 2-way transducers is closed under, H and H. Corollary H-Rat accepted by 2-way transducers 0 i n A i H B H i 10 / 15

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

40 From 2-way transducers to H-Rat (unary case) We fix a transducer T. Consider border to border run segments; u q 1 q 2 q 4 q 3 12 / 15

41 From 2-way transducers to H-Rat (unary case) We fix a transducer T. Consider border to border run segments; u q 1 q 2 R 1 = {(u, v 1 )} q 4 q 3 R 2 = {(u, v 2 )} R 3 = {(u, v 3 )} 12 / 15

42 From 2-way transducers to H-Rat (unary case) We fix a transducer T. Consider border to border run segments; Compose border to border segments; u q 1 q 2 R 1 = {(u, v 1 )} q 4 q 3 R 2 = {(u, v 2 )} R 3 = {(u, v 3 )} 12 / 15

43 From 2-way transducers to H-Rat (unary case) We fix a transducer T. Consider border to border run segments; Compose border to border segments; u q 1 q 2 R 1 = {(u, v 1 )} q 4 q 3 R 2 = {(u, v 2 )} R 3 = {(u, v 3 )} R 1 H R 2 H R 3 = {(u, v 1 v 2 v 3 )} 12 / 15

44 From 2-way transducers to H-Rat (unary case) We fix a transducer T. Consider border to border run segments; Compose border to border segments; Conclude using the closure properties of H-Rat. u q 1 q 2 R 1 = {(u, v 1 )} q 4 q 3 R 2 = {(u, v 2 )} R 3 = {(u, v 3 )} R 1 H R 2 H R 3 = {(u, v 1 v 2 v 3 )} 12 / 15

45 From 2-way transducers to H-Rat (unary case) u q 1 q 2 define a relation R bi, b j 13 / 15

46 From 2-way transducers to H-Rat (unary case) u q 1 q 2 define a relation R bi, b j Q {, } 13 / 15

47 From 2-way transducers to H-Rat (unary case) u q 1 q 2 define a relation R bi, b j 2 Q Q {, } HIT = R 0,0 R 0,1 R 0,2 R 1,0 R 1,1 R 1,2 R i,j R k,0 R k,1 R k,k 2 Q 13 / 15

48 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. 14 / 15

49 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. Third ingredient: Lemma Each entry R b1,b 2 of the matrix HIT is rational ( constructible ). 14 / 15

50 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. Third ingredient: Lemma Each entry R b1,b 2 of the matrix HIT is rational ( constructible ). a a a a a a a a a a q q output 14 / 15

51 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. Third ingredient: Lemma Each entry R b1,b 2 of the matrix HIT is rational ( constructible ). a a a a a a a a a a q rational output 14 / 15

52 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. Third ingredient: Lemma Each entry R b1,b 2 of the matrix HIT is rational ( constructible ). By closure property: Corollary Each entry of HIT H is in H-Rat. 14 / 15

53 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. Third ingredient: Lemma Each entry R b1,b 2 of the matrix HIT is rational ( constructible ). By closure property: Corollary Each entry of HIT H is in H-Rat. Remark The relation accepted by T is a union of entries of HIT H. 14 / 15

54 From 2-way transducers to H-Rat (unary case) Second ingredient: The behavior of T is given by the matrix HIT H. Third ingredient: Lemma Each entry R b1,b 2 of the matrix HIT is rational ( constructible ). By closure property: Corollary Each entry of HIT H is in H-Rat. Remark The relation accepted by T is a union of entries of HIT H. Corollary accepted by 2-way transducers H-Rat 14 / 15

55 Conclusion Theorem When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. 15 / 15

56 Conclusion Theorem When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows: 2-way transducers can be made sweeping. 15 / 15

57 Conclusion Theorem When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows: 2-way transducers can be made sweeping. With only Γ = {a}: 2-way deterministic unambiguous functional accept rational relations. 15 / 15

58 Conclusion Theorem When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows: 2-way transducers can be made sweeping. With only Γ = {a}: 2-way deterministic unambiguous functional accept rational relations. 2-way transducers are uniformizable by 1-way transducers. 15 / 15

59 Conclusion Theorem When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows: 2-way transducers can be made sweeping. With only Γ = {a}: 2-way deterministic unambiguous functional accept rational relations. 2-way transducers are uniformizable by 1-way transducers. Every thing is constructible. 15 / 15

60 Conclusion Theorem When Γ = {a} and Σ = {a}: 2-way transducers accept exactly the H-Rat relations. From our construction follows: 2-way transducers can be made sweeping. With only Γ = {a}: 2-way deterministic unambiguous functional accept rational relations. 2-way transducers are uniformizable by 1-way transducers. Every thing is constructible. Thank you for your attention. 15 / 15