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learning regular languages over large alphabets

Learning Regular Languages over Large Alphabets Oded Maler Irini - PowerPoint PPT Presentation

Mar 15, 2024 β€’172 likes β€’1.62k views

Learning Symbolic Automata Symbolic Learning Conclusion Learning Regular Languages over Large Alphabets Oded Maler Irini Eleftheria Mens CNRS-VERIMAG University of Grenoble 2, avenue de Vignate 38610 Gieres France TACAS 2014 April 10,

  1. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a β€’ Ξ£ = { a, b } Η« βˆ’ + a + + β€’ S βŠ† Ξ£ βˆ— prefixes b βˆ’ βˆ’ βˆ’ βˆ’ β€’ R = S Β· Ξ£ \ S boundary ba + + aa β€’ E βŠ† Ξ£ βˆ— suffixes ab + + βˆ’ βˆ’ bb β€’ f : S βˆͺ R Γ— E β†’ { + , βˆ’} classification baa βˆ’ βˆ’ function βˆ’ βˆ’ bab

  2. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a β€’ Ξ£ = { a, b } Η« βˆ’ + a + + β€’ S βŠ† Ξ£ βˆ— prefixes b βˆ’ βˆ’ βˆ’ βˆ’ β€’ R = S Β· Ξ£ \ S boundary ba + + aa β€’ E βŠ† Ξ£ βˆ— suffixes ab + + βˆ’ βˆ’ bb β€’ f : S βˆͺ R Γ— E β†’ { + , βˆ’} classification baa βˆ’ βˆ’ function βˆ’ βˆ’ bab β€’ f s : E β†’ { + , βˆ’} for all s ∈ S βˆͺ R f s ( e ) = f ( s Β· e )

  3. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a β€’ Ξ£ = { a, b } Η« βˆ’ + a + + β€’ S βŠ† Ξ£ βˆ— prefixes b βˆ’ βˆ’ βˆ’ βˆ’ β€’ R = S Β· Ξ£ \ S boundary ba + + aa β€’ E βŠ† Ξ£ βˆ— suffixes ab + + βˆ’ βˆ’ bb β€’ f : S βˆͺ R Γ— E β†’ { + , βˆ’} classification baa βˆ’ βˆ’ function βˆ’ βˆ’ bab β€’ f s : E β†’ { + , βˆ’} for all s ∈ S βˆͺ R f s ( e ) = f ( s Β· e )

  4. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a β€’ Ξ£ = { a, b } Η« βˆ’ + a + + β€’ S βŠ† Ξ£ βˆ— prefixes b βˆ’ βˆ’ βˆ’ βˆ’ β€’ R = S Β· Ξ£ \ S boundary ba + + aa β€’ E βŠ† Ξ£ βˆ— suffixes ab + + βˆ’ βˆ’ bb β€’ f : S βˆͺ R Γ— E β†’ { + , βˆ’} classification baa βˆ’ βˆ’ function βˆ’ βˆ’ bab β€’ f s : E β†’ { + , βˆ’} for all s ∈ S βˆͺ R f s ( e ) = f ( s Β· e )

  5. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a β€’ Ξ£ = { a, b } Η« βˆ’ + a + + β€’ S βŠ† Ξ£ βˆ— prefixes b βˆ’ βˆ’ βˆ’ βˆ’ β€’ R = S Β· Ξ£ \ S boundary ba + + aa β€’ E βŠ† Ξ£ βˆ— suffixes ab + + βˆ’ βˆ’ bb β€’ f : S βˆͺ R Γ— E β†’ { + , βˆ’} classification baa βˆ’ βˆ’ function βˆ’ βˆ’ bab β€’ f s : E β†’ { + , βˆ’} for all s ∈ S βˆͺ R f s ( e ) = f ( s Β· e )

  6. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a Η« βˆ’ + a + + b βˆ’ βˆ’ βˆ’ βˆ’ ba + + aa ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  7. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a q 1 Η« βˆ’ + a + + b βˆ’ βˆ’ q 0 βˆ’ βˆ’ ba + + aa q 2 ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  8. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a q 1 Η« βˆ’ + a a + + b βˆ’ βˆ’ q 0 βˆ’ βˆ’ ba b + + aa q 2 ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  9. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a a, b q 1 Η« βˆ’ + a a + + b βˆ’ βˆ’ q 0 βˆ’ βˆ’ ba b + + aa q 2 ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  10. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) Η« a a, b q 1 Η« βˆ’ + a a + + b βˆ’ βˆ’ q 0 βˆ’ βˆ’ ba b + + aa a, b q 2 ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  11. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 b a + + q 2 a, b b βˆ’ βˆ’ βˆ’ βˆ’ ba + + aa ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  12. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 β€’ S βˆͺ R is prefix-closed b a + + q 2 a, b b βˆ’ βˆ’ βˆ’ βˆ’ ba + + aa ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  13. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 β€’ S βˆͺ R is prefix-closed b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba + + aa ab + + βˆ’ βˆ’ bb baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  14. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 β€’ S βˆͺ R is prefix-closed b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba + + aa ab + + β€’ T closed βˆ’ βˆ’ bb βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s baa βˆ’ βˆ’ βˆ’ βˆ’ bab

  15. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 β€’ S βˆͺ R is prefix-closed b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba + + aa ab + + β€’ T closed βˆ’ βˆ’ bb βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s baa βˆ’ βˆ’ + βˆ’ bab

  16. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 q 3 β€’ S βˆͺ R is prefix-closed b b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba bab + βˆ’ aa + + β€’ T closed + + ab βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s bb βˆ’ βˆ’ βˆ’ βˆ’ baa : : :

  17. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 q 3 β€’ S βˆͺ R is prefix-closed b b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba bab + βˆ’ aa + + β€’ T closed + + ab βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s bb βˆ’ βˆ’ β€’ T consistent βˆ’ βˆ’ baa βˆ€ s, s β€² ∈ S, βˆ€ a ∈ Ξ£, f s = f s β€² β‡’ f s Β· a = f s β€² Β· a : : :

  18. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 q 3 β€’ S βˆͺ R is prefix-closed b b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba bab + βˆ’ aa + + β€’ T closed + + ab βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s bb βˆ’ βˆ’ β€’ T consistent βˆ’ βˆ’ baa βˆ€ s, s β€² ∈ S, βˆ€ a ∈ Ξ£, f s = f s β€² β‡’ f s Β· a = f s β€² Β· a : : :

  19. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a a Η« βˆ’ + q 0 q 3 β€’ S βˆͺ R is prefix-closed b b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba bab + βˆ’ aa + + β€’ T closed + + ab βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s bb βˆ’ βˆ’ β€’ T consistent βˆ’ βˆ’ baa βˆ€ s, s β€² ∈ S, βˆ€ a ∈ Ξ£, f s = f s β€² β‡’ f s Β· a = f s β€² Β· a : : :

  20. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a b a Η« βˆ’ + q 0 q 3 β€’ S βˆͺ R is prefix-closed b b a + + q 2 a, b b βˆ’ βˆ’ β€’ E is suffix-closed βˆ’ βˆ’ ba bab + βˆ’ aa + + β€’ T closed + + ab βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s bb βˆ’ βˆ’ β€’ T consistent βˆ’ βˆ’ baa βˆ€ s, s β€² ∈ S, βˆ€ a ∈ Ξ£, f s = f s β€² β‡’ f s Β· a = f s β€² Β· a : : :

  21. Learning Symbolic Automata Symbolic Learning Conclusion Observation Table - T = (Ξ£ , S, R, E, f ) a, b q 1 Η« a b a a, b Η« βˆ’ + : q 0 q 4 ? β€’ S βˆͺ R is prefix-closed a + + : b b q 2 q 3 b βˆ’ βˆ’ βˆ’ a β€’ E is suffix-closed βˆ’ βˆ’ + ba a b bab + βˆ’ : aa + + : β€’ T closed + + : ab βˆ€ r ∈ R , βˆƒ s ∈ S, f r = f s bb βˆ’ βˆ’ : β€’ T consistent βˆ’ βˆ’ : baa βˆ€ s, s β€² ∈ S, βˆ€ a ∈ Ξ£, f s = f s β€² β‡’ f s Β· a = f s β€² Β· a : : : :

  22. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } )

  23. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a, b q 0 ? Η« βˆ’ a b

  24. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a, b q 0 ? Η« βˆ’ a + βˆ’ b

  25. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a, b q 0 ? Η« βˆ’ a + βˆ’ b

  26. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a, b q 0 q 1 ? Η« βˆ’ + a b βˆ’ b

  27. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a, b q 0 q 1 ? Η« βˆ’ + a b βˆ’ b aa ab

  28. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a, b q 0 q 1 ? Η« βˆ’ + a b βˆ’ b aa βˆ’ + ab

  29. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a, b q 0 q 1 ? Η« βˆ’ + a b βˆ’ b aa βˆ’ + ab

  30. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b βˆ’ b aa βˆ’ + ab

  31. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b Ask Equivalence Query: βˆ’ b counterexample: βˆ’ ba aa βˆ’ + ab

  32. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b ba βˆ’ counterexample: βˆ’ ba b βˆ’ βˆ’ aa ab +

  33. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b b βˆ’ βˆ’ ba counterexample: βˆ’ ba βˆ’ aa ab +

  34. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b b βˆ’ βˆ’ ba βˆ’ aa ab + bb baa bab

  35. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b b βˆ’ βˆ’ ba βˆ’ aa ab + bb + baa βˆ’ bab +

  36. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b b βˆ’ Table is not consistent! βˆ’ ba βˆ’ aa ab + bb + baa βˆ’ bab +

  37. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a q 0 q 1 b Η« βˆ’ a + a b b βˆ’ Table is not consistent! βˆ’ b a βˆ’ aa ab + bb + baa βˆ’ bab +

  38. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a q 0 q 1 b Η« βˆ’ a + a b b βˆ’ Table is not consistent! βˆ’ b a βˆ’ aa ab + bb + baa βˆ’ bab +

  39. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a b q 0 q 1 Η« βˆ’ + + βˆ’ b a b a b βˆ’ βˆ’ q 2 βˆ’ βˆ’ b a βˆ’ aa ab + bb + baa βˆ’ bab +

  40. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a b q 0 q 1 Η« βˆ’ + + βˆ’ b a b a b βˆ’ βˆ’ q 2 βˆ’ βˆ’ b a βˆ’ + aa ab + βˆ’ bb + βˆ’ baa βˆ’ βˆ’ bab + βˆ’

  41. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a b q 0 q 1 Η« βˆ’ + a + βˆ’ b a b a b βˆ’ βˆ’ q 2 βˆ’ βˆ’ ba βˆ’ + aa ab + βˆ’ bb + βˆ’ baa βˆ’ βˆ’ bab + βˆ’

  42. Learning Symbolic Automata Symbolic Learning Conclusion L βˆ— Example (Ξ£ = { a, b } ) observation table hypothesis automaton Η« a a b q 0 q 1 Η« βˆ’ + a + βˆ’ b a b a b βˆ’ βˆ’ q 2 βˆ’ βˆ’ ba Ask Equivalence Query: βˆ’ + aa True ab + βˆ’ bb + βˆ’ baa βˆ’ βˆ’ bab + βˆ’

  43. Learning Symbolic Automata Symbolic Learning Conclusion Outline Learning Regular Languages Learning regular languages L* Algorithm Queries Observation tables Symbolic Automata Symbolic automata Symbolic languages Symbolic Learning Evidences Symbolic Algorithm Example

  44. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Ξ£

  45. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Ξ£

  46. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Ξ£

  47. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Ξ£

  48. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Ξ£

  49. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N [1] [1 , 50] , [71 , 100] q 1 q 3 [73 , 100] [1 , 50] [51 , 70] q 0 [2 , 72] [51 , 100] [21 , 100] [1 , 20] q 2 q 4 Ξ£

  50. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 [1 , 50] a 12 [51 , 70] a 32 q 0 [2 , 72] [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Ξ£

  51. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] a 12 [51 , 70] a 32 q 0 [2 , 72] [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Ξ£

  52. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] Ξ£ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Ξ£ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Ξ£

  53. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] Ξ£ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Ξ£ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 οΏ½ a = Ξ£ a 21 a 22 [21 , 100] [1 , 20] a ∈ Ξ£ q q 2 q 4 a 41 Ξ£

  54. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] Ξ£ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Ξ£ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 οΏ½ a = Ξ£ a 21 a 22 [21 , 100] [1 , 20] a ∈ Ξ£ q q 2 q 4 ψ q : Ξ£ β†’ Ξ£ q , βˆ€ q ∈ Q a 41 Ξ£

  55. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] Ξ£ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Ξ£ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 οΏ½ a = Ξ£ a 21 a 22 [21 , 100] [1 , 20] a ∈ Ξ£ q q 2 q 4 ψ q : Ξ£ β†’ Ξ£ q , βˆ€ q ∈ Q a 41 Ξ£ ψ q 1 (10) = a 12

  56. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] Ξ£ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Ξ£ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 οΏ½ a = Ξ£ a 21 a 22 [21 , 100] [1 , 20] a ∈ Ξ£ q q 2 q 4 ψ q : Ξ£ β†’ Ξ£ q , βˆ€ q ∈ Q a 41 Ξ£ ψ q 1 (10) = a 12 Ξ΄ : Q Γ— Ξ£ β†’ Q

  57. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton Ξ£ = [1 , 100] βŠ† N a 01 = [1 , 50] a 02 = [51 , 100] a 11 [1] [1 , 50] , [71 , 100] a 11 = . . . a 31 , a 33 q 1 q 3 . . . [73 , 100] a 13 a 01 Ξ£ = { a 01 , a 02 , . . . } [1 , 50] Ξ£ q 0 = { a 01 , a 02 } a 12 [51 , 70] a 32 q 0 Ξ£ q 1 = { a 11 , a 12 , a 13 } [2 , 72] . . . [51 , 100] a 02 οΏ½ a = Ξ£ a 21 a 22 [21 , 100] [1 , 20] a ∈ Ξ£ q q 2 q 4 ψ q : Ξ£ β†’ Ξ£ q , βˆ€ q ∈ Q a 41 Ξ£ ψ q 1 (10) = a 12 Ξ΄ : Q Γ— Ξ£ β†’ Q Ξ΄ ( q, a ) = q β€² iff Ξ΄ ( q, a ) = q β€² , a ∈ a

  58. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Σ , Σ , ψ, Q, δ, q 0 , F )

  59. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Ξ£ , Ξ£ , ψ, Q, Ξ΄, q 0 , F ), where β€’ Ξ£ is the input alphabet, β€’ Q is a finite set of states, β€’ q 0 is the initial state, β€’ F is the set of accepting states,

  60. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Ξ£ , Ξ£ , ψ, Q, Ξ΄, q 0 , F ), where β€’ Ξ£ is the input alphabet, β€’ Q is a finite set of states, β€’ q 0 is the initial state, β€’ F is the set of accepting states, β€’ Ξ£ is a finite alphabet, decomposable into Ξ£ = οΏ½ q ∈ Q Ξ£ q ,

  61. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Ξ£ , Ξ£ , ψ, Q, Ξ΄, q 0 , F ), where β€’ Ξ£ is the input alphabet, β€’ Q is a finite set of states, β€’ q 0 is the initial state, β€’ F is the set of accepting states, β€’ Ξ£ is a finite alphabet, decomposable into Ξ£ = οΏ½ q ∈ Q Ξ£ q , β€’ ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Ξ£ β†’ Ξ£ q ,

  62. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Ξ£ , Ξ£ , ψ, Q, Ξ΄, q 0 , F ), where β€’ Ξ£ is the input alphabet, β€’ Q is a finite set of states, β€’ q 0 is the initial state, β€’ F is the set of accepting states, β€’ Ξ£ is a finite alphabet, decomposable into Ξ£ = οΏ½ q ∈ Q Ξ£ q , β€’ ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Ξ£ β†’ Ξ£ q , β€’ Ξ΄ : Q Γ— Ξ£ β†’ Q is a partial transition function decomposable into a family of total functions Ξ΄ q : { q } Γ— Ξ£ q β†’ Q ,

  63. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Ξ£ , Ξ£ , ψ, Q, Ξ΄, q 0 , F ), where β€’ Ξ£ is the input alphabet, β€’ Q is a finite set of states, β€’ q 0 is the initial state, β€’ F is the set of accepting states, β€’ Ξ£ is a finite alphabet, decomposable into Ξ£ = οΏ½ q ∈ Q Ξ£ q , β€’ ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Ξ£ β†’ Ξ£ q , β€’ Ξ΄ : Q Γ— Ξ£ β†’ Q is a partial transition function decomposable into a family of total functions Ξ΄ q : { q } Γ— Ξ£ q β†’ Q , w = a 1 a 2 . . . a n accepted Ξ΄ βˆ— ( q 0 , a 1 a 2 . . . a n ) = Ξ΄ βˆ— ( Ξ΄ ( q 0 , ψ q ( a 1 )) , a 2 . . . a n ) ∈ F

  64. Learning Symbolic Automata Symbolic Learning Conclusion Deterministic Symbolic Automaton A = (Ξ£ , Ξ£ , ψ, Q, Ξ΄, q 0 , F ), where β€’ Ξ£ is the input alphabet, β€’ Q is a finite set of states, β€’ q 0 is the initial state, β€’ F is the set of accepting states, β€’ Ξ£ is a finite alphabet, decomposable into Ξ£ = οΏ½ q ∈ Q Ξ£ q , β€’ ψ = { ψ q : q ∈ Q } is a family of total surjective functions ψ q : Ξ£ β†’ Ξ£ q , β€’ Ξ΄ : Q Γ— Ξ£ β†’ Q is a partial transition function decomposable into a family of total functions Ξ΄ q : { q } Γ— Ξ£ q β†’ Q , L ( A ) = { w ∈ Ξ£ βˆ— | Ξ΄ βˆ— ( q 0 , w ) ∈ F } w = a 1 a 2 . . . a n accepted Ξ΄ βˆ— ( q 0 , a 1 a 2 . . . a n ) = Ξ΄ βˆ— ( Ξ΄ ( q 0 , ψ q ( a 1 )) , a 2 . . . a n ) ∈ F

  65. Learning Symbolic Automata Symbolic Learning Conclusion Outline Learning Regular Languages Learning regular languages L* Algorithm Queries Observation tables Symbolic Automata Symbolic automata Symbolic languages Symbolic Learning Evidences Symbolic Algorithm Example

  66. Learning Symbolic Automata Symbolic Learning Conclusion Symbolic Algorithm Evidences a 11 [1] [1 , 50] , [71 , 100] a 31 , a 33 q 1 q 3 [73 , 100] a 13 a 01 [1 , 50] a 12 [51 , 70] a 32 q 0 [2 , 72] [51 , 100] a 02 a 21 a 22 [21 , 100] [1 , 20] q 2 q 4 a 41 Ξ£

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