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

Blank symbol head Input string Remark: the input string is never empty

11 States & Transitions Read Write Move Left Move Right

12 Example: Time 1 current state

Time Time 2

Time Time 2 Example:

Time Time 2 Example:

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

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

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

19 Example: No possible transition HALT!!!

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

21 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

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

23 Time 0

24 Time 1

25 Time 2

26 Time 3

27 Time 4 Halt & Accept

28 Rejection Example Time 0

29 Time 1 No possible Transition Halt & Reject

30 Infinite Loop Example A Turing machine for language

31 Time 0

32 Time 1

33 Time 2

34 Time 2 Time 3 Time 4 Time 5 Infinite loop

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

36 Formal Definitions for Turing Machines

37 Transition Function

38 Transition Function

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

40 Configuration Instantaneous description:

41 Time 4Time 5 A Move:

42 Time 4Time 5 Time 6Time 7

43 Equivalent notation:

44 Initial configuration: Input string

45 The Accepted Language For any Turing Machine Initial stateFinal state

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