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Turing machines
- 1.
- 2. Invented by AlanTuring in 1936. A simple mathematical model of a general purpose computer. It is capable of performing any calculation which can be performed by any computing machine.
- 3. The Language Hierarchy n n n ? a b c ww ? Context-Free Languages n n R a b ww Regular Languages a* a *b *
- 4. Languages accepted by Turing Machines n n n a b c ww Context-Free Languages n n R a b ww NDPA Regular Languages Finite a * a *b * Automata
- 5. A Turing Machine Tape ...... ...... Read-Write head Control Unit
- 6. The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right
- 7. ...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right
- 8. Example: Time 0 ...... a b a c ...... Time 1 ...... a b k c ...... 1. Reads a 2. Writes k 3. Moves Left
- 9. Time 1 ...... a b k c ...... Time 2 ...... a f k c ...... 1. Reads b 2. Writes f 3. Moves Right
- 10. The Input String Input string Blank symbol ...... a b a c ...... head Head starts at the leftmost position of the input string Are treated as left and right brackets for the input written on the tape.
- 11. States & Transitions Read Write Move Left q1 a b, L q2 Move Right q1 a b, R q2
- 12. Example: Time 1 ...... a b a c ...... q1 current state q1 a b, R q2
- 13. Time 1 ...... a b a c ...... q1 Time 2 ...... a b b c ...... q2 q1 a b, R q2
- 14. Example: Time 1 ...... a b a c ...... q1 Time 2 ...... a b b c ...... q2 q1 a b, L q2
- 15. Example: Time 1 ...... a b a c ...... q1 Time 2 ...... a b b c g ...... q2 q1 g, R q2
- 16. Determinism Turing Machines are deterministic Allowed Not Allowed a b, R q2 a b, R q2 q1 q1 q3 a d, L q3 b d, L No lambda transitions allowed
- 17. Partial Transition Function Example: ...... a b a c ...... q1 a b, R q2 Allowed: q1 No transition for input symbol c b d, L q3
- 18. Halting The machine haltsif there are no possible transitions to follow
- 19. Example: ...... a b a c ...... q1 a b, R q2 No possible transition q1 HALT!!! b d, L q3
- 20. Final States q1 q2 Allowed q1 q2 Not Allowed • Final states have no outgoing transitions • In a final state the machine halts
- 21. Acceptance If machine halts Accept Input in a final state If machine halts in a non-final state Reject Input or If machine enters an infinite loop
- 22. Turing Machine Example ATuring machine that accepts the language: aa * a a, R ,L q0 q1
- 23. Time 0 a a a q0 a a, R ,L q0 q1
- 24. Time 1 a a a q0 a a, R ,L q0 q1
- 25. Time 2 a a a q0 a a, R ,L q0 q1
- 26. Time 3 a a a q0 a a, R ,L q0 q1
- 27. Time 4 a a a q1 a a, R Halt & Accept ,L q0 q1
- 28. Rejection Example Time 0 a b a q0 a a, R ,L q0 q1
- 29. Time 1 a b a q0 No possible Transition a a, R Halt & Reject ,L q0 q1
- 30. Infinite Loop Example b b, L a a, R ,L q0 q1
- 31. Time 0 a b a q0 b b, L a a, R ,L q0 q1
- 32. Time 1 a b a q0 b b, L a a, R ,L q0 q1
- 33. Time 2 a b a q0 b b, L a a, R ,L q0 q1
- 34. Time 2 a b a q0 Time 3 a b a q0 Time 4 a b a q0 Time 5 a b a q0 ... Infinite Loop
- 35. Because of theinfinite loop: •The final state cannot be reached •The machine never halts •The input is not accepted
- 36. Another Turing Machine Example n n Turing machine for the language {a b } q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 37. Time 0 a a b b q0 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 38. Time 1 x a b b q1 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 39. Time 2 x a b b q1 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 40. Time 3 x a y b q2 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 41. Time 4 x a y b q2 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 42. Time 5 x a y b q0 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 43. Time 6 x x y b q1 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 44. Time 7 x x y b q1 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 45. Time 8 x x y y q2 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 46. Time 9 x x y y q2 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 47. Time 10 x x y y q0 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 48. Time 11 x x y y q3 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 49. Time 12 x x y y q3 q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 50. Time 13 x x y y q4 Halt & Accept q4 y y, R y y, L y y, R a a, R a a, L ,L y y, R a x, R b y, L q3 q0 q1 q2 x x, R
- 51. Observation: Ifwe modify the machine for the language n n {a b } we can easily construct n n n a machine for the language {a b c }
- 52. Formal Definitions for Turing Machines
- 53. Transition Function q1 c d, L q2 (q1, c) (q2 , d , L)
- 54. Turing Machine: Input Tape alphabet alphabet States M (Q, , , , q0 , , F ) A partial Final Transition states function Initial Blank : a special symbol state Of
- 55. Configuration c a b a q1 Instantaneous description: ca q1 ba
- 56. Time 4 Time 5 x a y b x a y b q2 q0 A Move: q2 xayb x q0 ayb
- 57. Time 4 Time 5 x a y b x a y b q2 q0 Time 6 Time 7 x x y b x x y b q1 q1 q2 xayb x q0 ayb xx q1 yb xxy q1 b
- 58. q2 xayb x q0 ayb xx q1 yb xxy q1 b Equivalent notation: q2 xayb xxy q1 b
- 59. Initial configuration: q0 w Input string w a a b b q0
- 60. The Accepted Language Forany Turing Machine M L( M ) {w : q0 w x1 q f x2 } Initial state Final state
- 61. Standard Turing Machine Themachine we described is the standard: • Deterministic • Infinite tape in both directions •Tape is the input/output file
- 62. Design a Turingmachine to recognize all strings in which 010 is present as a substring. 0,0,R 0,0,R 1,1,R 0,0,R q0 q1 q2 H 1,1,R 1,1, R
- 63. DFA for theprevious language 0 0 1 0 q0 q1 q2 1 1 0,1
- 64. Turing machine forodd no of 1’s 1, 1 , R 1, b , R 1, b , R
- 65. Recursively Enumerable and Recursive Languages
- 66. Definition: A language is recursively enumerable if some Turing machine accepts it
- 67. Let L be a recursively enumerable language and M the Turing Machine that accepts it For string w: if w L then M halts in a final state if w L then M halts in a non-final state or loops forever
- 68. Definition: A language is recursive if some Turing machine accepts it and halts on any input string In other words: A language is recursive if there is a membership algorithm for it
- 69. Let L be a recursive language and M the Turing Machine that accepts it For string w: if w L then M halts in a final state if w L then M halts in a non-final state
- 70. We will prove: 1. There is a specific language which is not recursively enumerable (not accepted by any Turing Machine) 2. There is a specific language which is recursively enumerable but not recursive
- 71. Non Recursively Enumerable Recursively Enumerable Recursive
- 72. We will firstprove: • If a language is recursive then there is an enumeration procedure for it • A language is recursively enumerable if and only if there is an enumeration procedure for it
- 73.
- 74. Unrestricted Grammars: Productions u v String of variables String of variables and terminals and terminals
- 75. Example unrestricted grammar: S aBc aB cA Ac d
- 76. Theorem: A languageL is recursively enumerable if and only if L is generated by an unrestricted grammar
- 77. Context-Sensitive Grammars: Productions u v String of variables String of variables and terminals and terminals and: |u| |v|
- 78. The language n n n {a b c } is context-sensitive: S abc | aAbc Ab bA Ac Bbcc bB Bb aB aa | aaA
- 79. The language n n n {a b c } is context-sensitive: S abc | aAbc Ab bA Ac Bbcc bB Bb aB aa | aaA
- 80. Theorem: A language L is context sensistive if and only if L is accepted by a Linear-Bounded automaton
- 81. Observation: There is alanguage which is context-sensitive but not recursive
- 82. The Chomsky Hierarchy Non-recursivelyenumerable Recursively-enumerable Recursive Context-sensitive Context-free Regular