1. More Theory of Computing
Computer Science 101
More Theory of Computing

2. Another Example: Odd Parity
A parity bit is added to the right end of a string of bits if it makes the total number of 1’s odd; otherwise, add a 0
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3. Design Strategy Three states
Even parity: if we read a blank here, we have an even number of 1s, so we write a 1 and goto the halt state
Odd parity: if we read a blank here, we have an odd number of 1s, so we write a 0 and goto the halt state
Halt

4. Design Strategy Even parity Odd parity
We stay here while we keep reading 0s
Otherwise, we goto odd parity
Odd parity
Otherwise, we goto even parity

5. State Diagram for Odd Parity TM
Even parity
Odd parity
Halt

6. Tuples for Transitions
The Turing machine program
(1,1,1,2,R) Even parity state reading 1, change state
(1,0,0,1,R) Even parity state reading 0, don’t change state
(2,1,1,1,R) Odd parity state reading 1, change state
(2,0,0,2,R) Odd parity state reading 0, don’t change state
(1,b,1,3,R) End of string in even parity state, write 1 and go to state 3
(2,b,0,3,R) End of string in odd parity state, write 0 and go to state 3

7. The Church-Turing Thesis
If there exists an algorithm to do a symbol manipulation task, then there exists a Turing machine to do that task
Thesis
Statement advanced for consideration and maintained by argument
Theorem
Ideas that can be proved in a formal, mathematical way

8. Emulating an Algorithm on a TM

9. A Universal TM Represent the algorithm as symbols on the tape
Put the algorithm’s inputs on the tape as well
Run the universal TM to run the algorithm
The universal TM is like a stored program computer (von Neumann machine)

10. Unsolvable Problems Problem to investigate
Decide, given any collection of Turing machine instructions together with any initial tape contents, whether that Turing machine will ever halt if started on that tape

11. The Proposed Solution T* = TM being tested t = T’s input
P = The testing TM

12. A Proposed Counterexample
T* = TM being tested
t = T’s input
Q = The testing TM

13. An Actual Counterexample
S* = TM being tested
S*bS* = S’s input
S = The testing TM

14. Other Unsolvable Problems
Unsolvable problems, related to the halting problem, have the following consequences
No program can be written to decide whether any given program always stops eventually, no matter what the input
No program can be written to decide whether any two programs are equivalent
No program can be written to decide whether any given program run on any given input will ever produce some specific output