1 Turing Machines

2 The Language Hierarchy Regular Languages Context-Free Languages ? ?

3 Regular Languages Context-Free Languages Languages accepted by Turing Machines

4 A Turing Machine Tape Read-Write head Control Unit

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

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

Example: Time Time 1 1. Reads 2. Writes 3. Moves Left

Time Time 2 1. Reads 2. Writes 3. Moves Right

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

10 States & Transitions Read Write Move Left Move Right

11 Example: Time 1 current state

Time Time 2

Time Time 2 Example:

Time Time 2 Example:

15 Determinism Allowed Not Allowed No lambda transitions allowed Turing Machines are deterministic

16 Partial Transition Function Example: No transition for input symbol Allowed:

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

18 Example: No possible transition HALT!!!

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

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

21 Turing Machine Example A Turing machine that accepts the language:

22 Time 0

23 Time 1

24 Time 2

25 Time 3

26 Time 4 Halt & Accept

27 Rejection Example Time 0

28 Time 1 No possible Transition Halt & Reject

29 Infinite Loop Example Another Turing machine for language

30 Time 0

31 Time 1

32 Time 2

33 Time 2 Time 3 Time 4 Time 5... Infinite Loop

34 Because of the infinite loop: The final state cannot be reached The machine never halts The input is not accepted

35 Another Turing Machine Example Turing machine for the language

36 Time 0

37 Time 1

38 Time 2

39 Time 3

40 Time 4

41 Time 5

42 Time 6

43 Time 7

44 Time 8

45 Time 9

46 Time 10

47 Time 11

48 Time 12

49 Halt & Accept Time 13

50 If we modify the machine for the language we can easily construct a machine for the language Observation:

51 Formal Definitions for Turing Machines

52 Transition Function

53 Transition Function

54 Turing Machine: States Input alphabet Tape alphabet Transition function Initial state blank Final states

55 Configuration Instantaneous description:

56 Time 4Time 5 A Move:

57 Time 4Time 5 Time 6Time 7

58 Equivalent notation:

59 Initial configuration: Input string

60 The Accepted Language For any Turing Machine Initial stateFinal state

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