IDT Open Seminar ALAN TURING AND HIS LEGACY 100 Years Turing celebration

1. IDT Open Seminar ALAN TURING AND HIS LEGACY 100 Years Turing celebration Gordana Dodig Crnkovic, Computer Science and Network Department Mälardalen University March 8th 2012 http://www.mrtc.mdh.se/~gdc/work/TuringCentenary.pdf http://www.mrtc.mdh.se/~gdc/work/TuringMachine.pdf

2. Chomsky Language Hyerarchy Turing Machines Push-down Automata Finite Automata

3. TURING MACHINES “Turing’s "Machines". These machines are humans who calculate.” (Wittgenstein) “A man provided with paper, pencil, and rubber, and subject to strict discipline, is in effect a universal machine.” (Turing)

4. Turing Machine Tape ...... ...... Read-Write head Control Unit

5. The Tape No boundaries -- infinite length ...... ...... Read-Write head The head moves Left or Right

6. ...... ...... Read-Write head The head at each time step: 1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

7. The Input String Input string Blank symbol ...... ...... head Head starts at the leftmost position of the input string

8. Determinism Turing Machines are deterministic Not Allowed Allowed No lambda transitions allowed in TM!

9. Determinism Note the difference between stateindeterminismwhen not even possible future states are known in advance. and choice indeterminismwhen possible future states are known,but we do not know which state will be taken.

10. Halting The machine halts if there are no possible transitions to follow

11. Example ...... ...... No possible transition HALT!

12. Allowed Not Allowed Final States • Final states have no outgoing transitions • In a final state the machine halts

13. Acceptance If machine halts in a final state Accept Input If machine halts in a non-final state or If machine enters an infinite loop Reject Input

14. Formal Definitions for Turing Machines

15. Transition Function

16. Transition Function

17. Turing Machine Input alphabet Tape alphabet States Final states Transition function Initial state blank

18. The Accepted Language For any Turing Machine Initial state Final state

19. Standard Turing Machine The machine we described is the standard: • Deterministic • Infinite tape in both directions • Tape is the input/output file

20. Computing FunctionswithTuring Machines

21. A function is computable if there is a Turing Machine such that Initial Configuration Final Configuration For all Domain

22. is computable The function are integers Example (Addition) Turing Machine: Input string: unary Output string: unary

23. Start initial state Finish final state

24. Turing machine for function

25. Execution Example: Time 0 (2) (2) Final Result

26. Time 0

27. Time 1

28. Time 2

29. Time 3

30. Time 4

31. Time 5

32. Time 6

33. Time 7

34. Time 8

35. Time 9

36. Time 10

37. Time 11

38. Time 12 HALT & accept

39. Universal Turing Machine

40. Turing Machines are “hardwired” they execute only one program A limitation of Turing Machines:

41. Solution: Universal Turing Machine Characteristics: • Reprogrammable machine • Simulates any other Turing Machine

42. Universal Turing Machine simulates any other Turing Machine Input to Universal Turing Machine: • Description of transitions of • Initial tape contents of

43. Tape 1 Three tapes Description of Tape 2 Universal Turing Machine Tape Contents of Tape 3 State of

44. Tape 1 Description of We describe Turing machine as a string of symbols: We encode as a string of symbols

45. Alphabet Encoding Symbols: Encoding:

46. States: Encoding: Move: Encoding: State Encoding Head Move Encoding

47. Transition Encoding Transition: Encoding: separator

48. Machine Encoding Transitions: Encoding: separator

49. Tape 1 contents of Universal Turing Machine: encoding of the simulated machine as a binary string of 0’s and 1’s

50. As Turing Machine is described with a binary string of 0’s and 1’s the set of Turing machines forms a language: Each string of the language is the binary encoding of a Turing Machine.