1. Computing Functions with Turing Machines

2. A function
has:
Result Region:
Domain:

3. A function may have many parameters:
Example:
Addition function

4. Integer Domain
Decimal: 5
Binary:
101
We prefer unary representation:
easier to manipulate with Turing machines
Unary:
11111

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

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

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

8. Start
initial state
The 0 is the delimiter that
separates the two numbers

9. Start
initial state
Finish
final state

10. The 0 helps when we use
the result for other operations
Finish
final state

11. Turing machine for function

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

13. Time 0

14. Time 1

15. Time 2

16. Time 3

17. Time 4

18. Time 5

19. Time 6

20. Time 7

21. Time 8

22. Time 9

23. Time 10

24. Time 11

25. Time 12
HALT & accept

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

27. Start
initial state
Finish
final state

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

29. Turing Machine for

30. Example
Start
Finish

31. Another Example if
The function if
is computable

32. Turing Machine for if if
Input:
Output: or

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

34. Combining Turing Machines

35. Block Diagram
Turing
Machine
input
output

36. Example: if if
Adder
Comparer
Eraser

37. Turing’s Thesis

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

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

40. 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

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

42. Algorithms are Turing Machines
When we say:
There exists an algorithm
We mean:
There exists a Turing Machine
that executes the algorithm