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

On 29/04/2025 14:11, olcott wrote:

On 4/29/2025 2:10 AM, Richard Heathfield wrote:

On 29/04/2025 03:50, olcott wrote:

<snip>

Yet it is H(P,D) and NOT P(D) that must be measured.

>
Nothing /has/ to be measured. P's behaviour (halts, doesn't halt) when given D as input must be /established/.

 No H can possibly see the behavior of P(D)

It doesn't have to. It only has to be able to read two tapes, one containing P and the other containing D.

when-so-ever D has defined a pathological
relationship with H this changes the behavior
of P so that it is not the same as P(D).

D has no relationship with H. D is just a list of data symbols on a Turing Machine tape. P, too, is a list of symbols - a program. You don't have to run it to read it. H's job is not to run P but to analyse P to determine whether P would halt if fed D as input.
The behaviour that P would exhibit if fed D is determined only by the symbols on the P-tape and the D-tape. H is neither here nor there.
If H elects to run P as part of its job, fine, but it's not obligatory and does not give H licence to change P, change D, or reject either tape. H /must/ report, correctly, for /any/ P/D pair. And it can't, as you rightly conceded when you wrote: "Computing the actual behavior the direct execution of any input is ALWAYS IMPOSSIBLE".
<snip>
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Richard Heathfield
Email: rjh at cpax dot org dot uk
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