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From: joes <noreply@example.org>
Newsgroups: comp.theory
Subject: Re: Halting Problem: What Constitutes Pathological Input
Date: Tue, 6 May 2025 10:01:32 -0000 (UTC)
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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.