1. 2. Turing’s Thesis Continued The Halting Problem
4/13/2019
2. Turing’s Thesis Continued The Halting Problem
Alexander M. Fedorec
CSN9902Turing

2. The spring swallows are perching side by side on a willow branch;
4/13/2019
The leisurely ants are struggling to carry away the wings of a dead dragonfly;
The spring swallows are perching side by side on a willow branch;
The silk-worm women, pale and tired, stand holding the baskets filled with mulberry leaves;
The village urchins are seen with stolen bamboo-shoots creeping through a broken fence.
Hakuin
CSN9902Turing

3. The Universal Turing Machine
4/13/2019
The Universal Turing Machine
Instead of the program quintuples being ‘hardwired’ into a Turing machine so that it can only do one task, Turing argued it would be possible to represent the quintuples as a string of ones and zeroes and write them onto the tape and thus have a general purpose machine that reads it’s instructions from the tape (i.e. given current state and what has just been input, what to output to the tape, where to move the read/write head and what the next state should be).
This is called the Universal Turing Machine.
The general purpose stored program digital von Neuman computers we know and love can be shown to be equivalent to a UTM (given enough time and memory).
CSN9902Turing

4. 4/13/2019
The Halting Problem
A reformulation of Hilbert’s entscheidungsproblem and generalization of Gödel’s theorem:
Given a program P for a Turing machine TM, and an input string s, does the computation halt or not?
Is this a solvable problem, i.e. is there an effective algorithm which when given a description of P and s will say “Yes, this computation will halt” or “No, this computation will not halt” ?
CSN9902Turing

5. Proof of the Halting Problem
4/13/2019
Proof of the Halting Problem
Assume that it can be proved.
i.e. Assume that a Turing machine D exists that accepts as input dt, the description (quintuples of the machine T) and D will then decide whether or not T applied to tape t will halt or not.
CSN9902Turing

6. t dt 2. Input to machine D: 3. Output from D:
4/13/2019
2. Input to machine D:
3. Output from D:
Output is the final state, yes or no
and then halt
t dt
A program for machine T
Inputs to the program
CSN9902Turing

7. 4/13/2019
4. If it can be done for any tape, then it can be done for it’s own quintuples (dd).
5. From machine D we can construct a machine E which takes dt, copies it and gives the answer:
Yes D halts when given dt then halts
No D does not halt when given dt then halts.
i.e. Machine E does not solve the general problem but solves the particular problem:
dt dt
CSN9902Turing

8. No T does not halt given dtde*
4/13/2019
6. From E we construct another machine E*
i.e. is exactly like E except does not halt given T halts
No T does not halt given dtde*
HALT E* de dt
Yes T halts given dtde*
add quintuples to loop don’t halt
CSN9902Turing

9. 7. What happens when E* is applied to de* ? (Reductio ad Absurdum)
4/13/2019
7. What happens when E* is applied to de* ? (Reductio ad Absurdum)
CSN9902Turing