Re: Halting Problem: What Constitutes Pathological Input

Am Mon, 05 May 2025 13:07:27 -0500 schrieb olcott:

On 5/5/2025 11:42 AM, Mr Flibble wrote:

On Mon, 05 May 2025 10:51:40 -0500, olcott wrote:

On 5/5/2025 10:17 AM, Mr Flibble wrote:

What constitutes halting problem pathological input:
Input that would cause infinite recursion when using a decider of the
simulating kind.
Such input forms a category error which results in the halting
problem being ill-formed as currently defined.

>
I prefer to look at it as a counter-example that refutes all of the
halting problem proofs.

When HHH computes the mapping from *its input* to the behavior of DD
emulated by HHH this includes HHH emulating itself emulating DD. This
matches the infinite recursion behavior pattern.
>
Thus the Halting Problem's "impossible" input is correctly determined
to be non-halting.

 
Disagree: infinite recursion maps to incomputable rather than
non-halting and I reject that: it is not incomputable as the problem
itself is ill- formed due to a category (type) error.

 
It is only ill-formed when construed in one of two ways:
(a) If D was actually able to do the opposite of whatever value that H
returns then it is ill-formed because it is self-contradictory.

D is actually able to be written assuming H exists. It does not contradict
itself but H.

(b) If the problem is defined to have HHH report on the direct execution
of DD() then it is ill-formed because HHH is required to report on
different behavior than its input DD actually specifies.
>
The call from the directly executed DD() to HHH(DD) returns. The call
from the correctly emulated DD() to HHH(DD)
CANNOT POSSIBLY RETURN. The directly executed HHH DOES RETURN.

Yes, HHH cannot simulate itself the same way as its direct execution.

--
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.