1 Computing Functions with Turing Machines

2 A function Domain Result Region has:

3 Integer Domain: Unary: Binary: Decimal: We prefer Unary representation: Easier to manipulate

4 A function may have many parameters: Example: Addition function are integers

5 Definition: A function is computable if there is a Turing Machine such that: Initial Configuration Final configuration Domain final state

6 Initial Configuration Final Configuration Domain A function is computable if there is a Turing Machine such that: In other words:

7 Example The function is computable Turing Machine: Input string: unary Output string:unary are integers

8 Start Finish final state

9 Turing machine for function

10 Execution Example: Time 0 Final Result (2)

11 Time 0Time 1

12 Time 2Time 3

13 Time 4Time 5

14 Time 6Time 7

15 Time 8Time 9

16 Time 10Time 11

17 Time 12 HALT & accept

18 Another Example The function is computable Turing Machine: Input string: unary Output string:unary is integer

19 Start Finish final state

20 Turing Machine Pseudocode for 1. Replace every 1 with $ Repeat: 2. Find rightmost $, replace it with 1 3. Go to right end, insert 1 Until no more $ remain

21 Turing Machine for

22 Example Start Finish

23 Another Example The function is computable if

24 Turing Machine for if Input: Output: or

25 Turing Machine Pseudocode: Match a 1 from with a 1 from 1. Repeat Until all or has been matched 2. If a 1 from is not matched erase tape, write 1 else erase tape, write 0

26 Combining Turing Machines

27 Block Diagram Turing Machine inputoutput

28 Example: if Comparer Adder Eraser

29 Turing ’ s Thesis

30 Do Turing machines have the same power with a digital computer?

31 Do Turing machines have the same power with a digital computer? Intuitive answer: Yes There is no formal answer

32 Turing ’ s thesis: Any computation carried out by mechanical means can be performed by Turing Machine (1930)

33 Computer Science Law: A computation is mechanical if and only if it can be performed by a Turing Machine There is no known model of computation more powerful than Turing Machines

34 Definition of Algorithm: An algorithm for function is a Turing Machine which computes

35 When we say: There exists an algorithm Algorithms are Turing Machines We mean: There exists a Turing Machine