uniformisation of two way transducers

Uniformisation of Two-Way Transducers Rodrigo de Souza UFRPE - PowerPoint PPT Presentation

Uniformisation of Two-Way Transducers Rodrigo de Souza UFRPE Recife Brazil Supported by CAPES Foundation A two-way transducer a | a a | 1 a | a b | b | 1 p q r


  1. Uniformisation of Two-Way Transducers Rodrigo de Souza UFRPE – Recife – Brazil Supported by CAPES Foundation

  2. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b

  3. A two-way transducer − → ← − − → a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b An input word ⊲ aababaa ⊳

  4. A two-way transducer − → ← − − → a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b The initial configuration ⊲ paababaa ⊳

  5. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − → − b | b b | 1 b | b A computation − → a ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ The output : a

  6. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b A computation − → − → a a ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ − − → ⊲ aapbabaa ⊳ The output : aa

  7. A two-way transducer − → ← − → − a | a a | 1 a | a ← − − → b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − → − b | b b | 1 b | b A computation ← − − → − → a a b ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ − − → ⊲ aapbabaa ⊳ − − → ⊲ aqababaa ⊳ The output : aab

  8. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b A computation ← − − → − → a a b ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ − − → ⊲ aapbabaa ⊳ − − → ⊲ aqababaa ⊳ ← − ← − − → − → a a a ⊲ − − → ⊲ qaababaa ⊳ − − → q ⊲ aababaa ⊳ − − → ⊲ raababaa ⊳ − − → ⊲ arababaa ⊳ − → → − − → → − a b a b − − → ⊲ aarbabaa ⊳ − − → ⊲ aabrabaa ⊳ − − → ⊲ aabarbaa ⊳ − − → ⊲ aababraa ⊳ − → − → → − a a a − − → ⊲ aababara ⊳ − − → ⊲ aababaar ⊳ − − → ⊲ aababaa ⊳ r The output : aabaababaa

  9. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b Another computation – same input − → → − − → a a b ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ − − → ⊲ aapbabaa ⊳ − − → ⊲ aabpabaa ⊳ ← − ← − → − ← − a b a b − − → ⊲ aabapbaa ⊳ − − → ⊲ aabqabaa ⊳ − − → ⊲ aaqbabaa ⊳ − − → ⊲ aqababaa ⊳ → − ← − ← − → − a a a ⊲ − − → ⊲ qaababaa ⊳ − − → q ⊲ aababaa ⊳ − − → ⊲ raababaa ⊳ − − → ⊲ arababaa ⊳ − → → − − → − → a b a b − − → ⊲ aarbabaa ⊳ − − → ⊲ aabrabaa ⊳ − − → ⊲ aabarbaa ⊳ − − → ⊲ aababraa ⊳ − → − → − → a a ⊳ − − → ⊲ aababara ⊳ − − → ⊲ aababaar ⊳ − − → ⊲ aababaa ⊳ r

  10. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − → − b | b b | 1 b | b Another computation – same input − → − → − → a a b ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ − − → ⊲ aapbabaa ⊳ − − → ⊲ aabpabaa ⊳ ← − ← − → − ← − a b a b − − → ⊲ aabapbaa ⊳ − − → ⊲ aabqabaa ⊳ − − → ⊲ aaqbabaa ⊳ − − → ⊲ aqababaa ⊳ → − ← − ← − → − a a a ⊲ − − → ⊲ qaababaa ⊳ − − → q ⊲ aababaa ⊳ − − → ⊲ raababaa ⊳ − − → ⊲ arababaa ⊳ − → − → − → − → a b a b − − → ⊲ aarbabaa ⊳ − − → ⊲ aabrabaa ⊳ − − → ⊲ aabarbaa ⊳ − − → ⊲ aababraa ⊳ − → → − − → a a ⊳ − − → ⊲ aababara ⊳ − − → ⊲ aababaar ⊳ − − → ⊲ aababaa ⊳ r The output : aababaababaa

  11. A two-way transducer − → ← − → − a | a a | 1 a | a ← − → − b | b ⊲ | 1 ⊲ ⊳ p q r → − ← − − → b | b b | 1 b | b Another computation – same input − → − → − → a a b ⊲ paababaa ⊳ − − → ⊲ apababaa ⊳ − − → ⊲ aapbabaa ⊳ − − → ⊲ aabpabaa ⊳ ← − ← − → − ← − a b a b − − → ⊲ aabapbaa ⊳ − − → ⊲ aabqabaa ⊳ − − → ⊲ aaqbabaa ⊳ − − → ⊲ aqababaa ⊳ → − ← − ← − − → a a a ⊲ − − → ⊲ qaababaa ⊳ − − → q ⊲ aababaa ⊳ − − → ⊲ raababaa ⊳ − − → ⊲ arababaa ⊳ − → → − → − − → a b a b − − → ⊲ aarbabaa ⊳ − − → ⊲ aabrabaa ⊳ − − → ⊲ aabarbaa ⊳ − − → ⊲ aababraa ⊳ − → − → → − a a ⊳ − − → ⊲ aababara ⊳ − − → ⊲ aababaar ⊳ − − → ⊲ aababaa ⊳ r The output : aababaababaa The behaviour : u → { xbu | xb prefix of u }

  12. Sequential ? An (one-way) automaton... a c b p q b

  13. Sequential ? An (one-way) automaton... a c b p q b

  14. Sequential ? An (one-way) automaton... ... and a deterministic equivalent one a c a b c b b c p q 10 11 01 a b

  15. Sequential ? An (one-way) automaton... ... and a deterministic equivalent one a c a b c b b c p q 10 11 01 a b One-way automata (Rabin–Scott 1959) Subset construction Two-way automata (Rabin–Scott 1959, Shepherdson 1959) Crossing sequence construction

  16. Sequential ? A transducer for a function f ... a | a a | 1 b | 1 p q ( uba n ) f = u b | b

  17. Sequential ? A transducer for a function f ... a | a a | 1 b | 1 p q ( uba n ) f = u b | b

  18. Sequential ? A transducer for a function f ... ... which is not sequential a | a a | 1 || ba n , ba n b || = 1 b | 1 p q || ( ba n ) f , ( ba n b ) f || = n + 1 b | b Functional one-way transducers (Choffrut 1977) f sequential ⇔ f uniformly divergent

  19. Sequential ? Functional 2 -way transducers (Engelfriet–Hoogeboom 2001) Via MSO definable graph relations

  20. Sequential ? Functional 2 -way transducers (Engelfriet–Hoogeboom 2001) Via MSO definable graph relations Complexity ?

  21. Sequential ? Functional 2 -way transducers (Engelfriet–Hoogeboom 2001) Via MSO definable graph relations Complexity ? Functional 2 -way transducers (RdS 2013) Via structural constructions

  22. Sequential ? Functional 2 -way transducers (Engelfriet–Hoogeboom 2001) Via MSO definable graph relations Complexity ? Functional 2 -way transducers (RdS 2013) Via structural constructions 4-exp

  23. Our true result Theorem (Kobayashi 1969) r one-way transducer admits uniformisation f functional unambigu- ous transducer

  24. Our true result Theorem (Kobayashi 1969) r one-way transducer admits uniformisation f functional unambigu- ous transducer dom r = dom f , f ⊆ r

  25. Our true result Theorem (Kobayashi 1969) r one-way transducer admits uniformisation f functional unambigu- ous transducer dom r = dom f , f ⊆ r Theorem (RdS 2013) r two-way transducer admits uniformisation f sequential two-way transducer

  26. Our true result Theorem (Kobayashi 1969) r one-way transducer admits uniformisation f functional unambigu- ous transducer dom r = dom f , f ⊆ r Theorem (RdS 2013) r two-way transducer admits uniformisation f sequential two-way transducer uniformisation construction functional two-way equivalent sequential �

  27. Outline of the proof U = T ×L×R L R T A S B

  28. Outline of the proof U = T ×L×R L R T A S B underlying input automaton of T

  29. Outline of the proof U = T ×L×R L R T A S B folding of A

  30. Outline of the proof U = T ×L×R L R T A S B Proposition successful computations one-way S ↔ successful loop-free computations two-way A

  31. Outline of the proof U = T ×L×R L R T A S B lexicographic covering of S (JS & RdS 2007)

  32. Outline of the proof U = T ×L×R L R T A S B Proposition successful computations unambiguous one-way B ։ successful loop-free computations A

  33. Outline of the proof U = T ×L×R L R T A S B left (Hopcroft & Ullman 1967) / right pathfinders for B

  34. Outline of the proof U = T ×L×R L R T A S B T + L + R = sequential bidirectional reading folded computations

  35. Outline of the proof U = T ×L×R L R T A S B Main result sequential two-way U realises an uniformisation of T

  36. Outline of the proof U = T ×L×R L R T A S B The folding construction

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