From Newsgroup: comp.lang.c++

On 12/11/2025 3:09 AM, Mikko wrote:

olcott kirjoitti 11.12.2025 klo 4.00:

*It has take me 21 years to boil it down to this*

When the halting problem requires a halt decider
to report on the behavior of a Turing machine this
is always a category error.

No, that is not a category error. If the question to be answered
is something other than "does this computation halt" then there
is not point to call the decider a "halting decider".

The corrected halting problem requires a Turing
machine decider to report in the behavior that
its finite string input specifies.

The usual defintion does the same. But usually the requirement is
that the solution to the problem includes encoding rules that
specify what the input shall be in order to specify the behaviour
asked about.

No this has always been the error of conflating the
behavior of the machine with the behavior specified
by the input finite string. In every case besides
pathological self-reference this makes no difference.

Turing machine deciders compute functions from finite
strings to {accept, reject}.

int DD()
{
int Halt_Status = HHH(DD);
if (Halt_Status)
HERE: goto HERE;
return Halt_Status;
}

It is a verified fact that N steps of DD simulated
by HHH according to the semantics of the C programming
do prove a behavior pattern that cannot possibly reach
the "return" statement final halt state of DD in any
number of steps.

It is possible to pose the problem so that encoding rules are a
part of the problem specification. For example, the problem nay
present a particular unversal Turing machine and require that
the halting decider can be given the same input as that universal
Turing machine, which then is required to accept if that universal
Turing mahine halts with the same input and to reject otherwise.

Even that is not the actual behavior actually specified
by its actual input even though most of the time it is
equivalent behavior.

The corrected halting problem requires a Turing machine
decider to report in the behavior that its finite string
input specifies.

This requires the halt decider be based on an augmented
UTM that watches the behavior of its simulated finite
string input.

*From the bottom of page 319 has been adapted to this* https://www.liarparadox.org/Peter_Linz_HP_317-320.pdf

-n.q0 rf?-nrf- reo* -n.embedded_H rf?-nrf- rf?-nrf- reo* -n.reR, // accept state
-n.q0 rf?-nrf- reo* -n.embedded_H rf?-nrf- rf?-nrf- reo* -n.qn // reject state

*Keeps repeating unless aborted*
(a) -n copies its input rf?-nrf-
(b) -n invokes embedded_H rf?-nrf- rf?-nrf-
(c) embedded_H simulates rf?-nrf- rf?-nrf-
--
Copyright 2025 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
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