Re: Halting Problem: What Constitutes Pathological Input

Am Mon, 05 May 2025 14:27:18 -0500 schrieb olcott:

On 5/5/2025 2:12 PM, dbush wrote:

On 5/5/2025 2:47 PM, olcott wrote:

On 5/5/2025 1:21 PM, dbush wrote:

On 5/5/2025 2:14 PM, olcott wrote:

On 5/5/2025 11:16 AM, dbush wrote:

On 5/5/2025 12:13 PM, Mr Flibble wrote:

On Mon, 05 May 2025 11:58:50 -0400, dbush wrote:

On 5/5/2025 11:51 AM, 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.

What categories are being confused?

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.

Incorrectly, because the HHH that DD calls does in fact contain an abort.

Thus the Halting Problem's "impossible" input is correctly
determined to be non-halting.
>

Which is a contradiction.  Therefore the assumption that the
above mapping is computable is proven false, as Linz and others
have proved and as you have *explicitly* agreed is correct.

>
The category (type) error manifests in all extant halting problem
proofs including Linz.  It is impossible to prove something which
is ill-formed in the first place.

>
All algorithms either halt or do not halt when executed directly.
Therefore the problem is not ill formed.
>

When BOTH Boolean RETURN VALUES are the wrong answer THEN THE
PROBLEM IS ILL-FORMED. Self-contradiction must be screened out as
semantically incorrect.

Including the supposed halting decider HHH.

In other words, you're claiming that there exists an algorithm, i.e.
a fixed immutable sequence of instructions, that neither halts nor
does not halt when executed directly.
>

That is not what I said.

You said that both return values of HHH(DD) are incorrect.

Then there's no category error, and the halting function is well
defined.  It's just that no algorithm can compute it.

 
It is insufficiently defined thus causing it to be incoherently defined.

The mathematical association of programs to their halting state (when
directly executed or correctly simulated by a UTM) is perfectly well-
defined.

Compute the mapping FROM INPUTS.

There is indeed no concrete implementation of an algorithm that does that.

The details of this cannot be as easily seen with the somewhat vague
abstraction of Turing Machines that do not even have a standard language
definition.

There are many equivalent definitions.

--
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.