Re: Turing Machine computable functions apply finite string transformations to inputs VERIFIED FACT

Richard Heathfield <rjh@cpax.org.uk> writes:

On 30/04/2025 19:30, Mike Terry wrote:

On 30/04/2025 16:46, Richard Heathfield wrote:

On 30/04/2025 16:15, olcott wrote:

On 4/29/2025 5:03 PM, Richard Heathfield wrote:

On 29/04/2025 22:38, olcott wrote:
>
<snip>
>

>
int DD()
{
   int Halt_Status = HHH(DD);
   if (Halt_Status)
     HERE: goto HERE;
   return Halt_Status;
}
>
HHH is correct DD as non-halting BECAUSE THAT IS
WHAT THE INPUT TO HHH(DD) SPECIFIES.

>
You're going round the same loop again.
>
Either your HHH() is a universal termination analyser or it isn't.

>
The domain of HHH is DD.

>
Then it is attacking not the Halting Problem but the Olcott Problem,
which is of interest to nobody but you.

It would be (if correct) attacking the common proof for HP theorem as it
occurs for instance in the Linz book which PO links to from time to time.

>
Yes. That's what I call the Olcott Problem.
>
De gustibus non est disputandum, but I venture to suggest that (correctly)
overturning Turing's proof would be of cosmos-rocking interest to the world
of computer science, compared to which pointing out a minor flaw in a
minor[1] proof would, even if correct, have no more effect on our field
than lobbing a pebble into the swash at high tide.
>
I suspect that the only reason we bother to argue with Mr Olcott so much is
because (even if he does so unwittingly) he manages to convey the
appearance of attacking the Halting Problem, and arguing about the Halting
Problem is a lot more fun than arguing about the Olcott Problem.
>
To be of any interest, solving the Olcott Problem would have to have
important consequences. But does it? Let's see.
>
Dr Linz Theorem 12.1 (Halting Problem is Undecidable): There does not exist
any Turing machine H that behaves as required by Linz Definition 12.1. Thus
the halting problem is undecidable.
>
Dr Linz has a proof for this claim, which can be found here:
<https://john.cs.olemiss.edu/~hcc/csci311/notes/chap12/ch12.pdf>
>
If the proof is flawless, the conclusion stands and Mr Olcott is simply
wrong.

There is peculiar irony here.  The proof is /not/ flawless.  It has, in
fact, a flaw that PO pointed out (although in passing).  PO does not
care about the flaw because it is easily fixed, but it's there none the
less[1].

Anyway, Linz only gives this argument because it is of historical
interest.  His "real" proof immediately follows this argument (in the
book) and is a trivial corollary of the fact, proved in chapter 11, that
not all recursively enumerable languages are recursive.  But no crank
ever addresses that proof.  I wonder why...

If the proof is flawed through some error of reasoning, *either* it merely
fails to correctly support its conclusion *or* a duly corrected proof
/overturns/ the conclusion.

... or a duly corrected proof /supports/ the conclusion.  Maybe you
intended this possibility to be include in your first "merely fails to
support" alternative.

The latter would be /extremely/ interesting, but it would also mean that we
have two proofs proving opposite things, and so it would effectively be a
cataclysmic sideways attack on Turing's reasoning.

It would be a cataclysmic attack on all of conventional mathematics and
logic as two proofs, one of T and another of ~T, makes the system of
proof oneself inconsistent (and everything becomes a theorem).

[1] It centres on the naming of states in H and the derived machines H'
and H^.  Details available if you really care.

--
Ben.