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

On 28/04/2025 16:01, olcott wrote:

On 4/28/2025 2:33 AM, Richard Heathfield wrote:

On 28/04/2025 07:46, Fred. Zwarts wrote:
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<snip>
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So we agree that no algorithm exists that can determine for all possible inputs whether the input specifies a program that (according to the semantics of the machine language) halts when directly executed.
Correct?

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Correct. We can, however, construct such an algorithm just as long as we can ignore any input we don't like the look of.
>

 The behavior of the direct execution of DD cannot be derived
by applying the finite string transformation rules specified
by the x86 language to the input to HHH(DD). This proves that
this is the wrong behavior to measure.
 It is the behavior THAT IS derived by applying the finite
string transformation rules specified by the x86 language
to the input to HHH(DD) proves that THE EMULATED DD NEVER HALTS.

The x86 language is neither here nor there. What matters is whether a TM can be constructed that can accept an arbitrary TM tape P and an arbitrary input tape D and correctly calculate whether, given D as input, P would halt. Turing proved that such a TM cannot be constructed.
This is what we call the Halting Problem.
Whatever you think you've proved, you haven't solved the Halting Problem. There are *no* solutions. We know this because there is a simple well-known proof. So the only way to devise a solution is to re-define the problem.
And that's fine. If that's what floats your boat, you can re-define it as much as you like. But any proofs you may devise apply not to the Halting Problem but to the Olcott problem.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
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