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

On 01/05/2025 21:15, olcott wrote:
<snip>

What I am explaining is that a halt decider
must compute the mapping FROM THE INPUTS ONLY
by applying a specific set of finite string
transformations to the inputs.

I don't think that's in dispute. Were a successful universal halt decider to be possible (which of course it isn't), it would have no information available to it except what's on the tapes, which consist of finite strings of symbols, and it must use such information as it has in order to correctly report on the nature of the input program. Yes, it's a mapping. Which transformations are to be applied will undoubtedly depend heavily on the input program's internal structure, but transformations there must be, even if they're as simple as turning "return 0" into "it halts".

int DD()
{
   int Halt_Status = HHH(DD);
   if (Halt_Status)
     HERE: goto HERE;
   return Halt_Status;
}
 Everyone assumes that it is correct to ignore
the required finite string transformations
that mandate DD correctly emulated by HHH
cannot possibly reach its own final halt state
no matter what of an infinite set of HHH's does.

Word salads do not a proof overturn.
HHH can't emulate DD correctly because it can only ever get as far as the first line of the emulation before it recurses. It never reaches the if() because the runaway recursion will crash the program before it gets that far.
Since HHH fails to report its result, it is not a working halt-decider.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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