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From: joes <noreply@example.org>
Newsgroups: comp.theory
Subject: Re: Unpartial Halt Deciders --- category error
Date: Tue, 22 Apr 2025 13:02:32 -0000 (UTC)
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Am Sat, 19 Apr 2025 15:44:31 -0500 schrieb olcott:
> On 4/19/2025 1:06 PM, Mr Flibble wrote:
>> On Sat, 19 Apr 2025 13:34:40 -0400, Richard Damon wrote:
>>> On 4/19/25 8:05 AM, Mr Flibble wrote:
>>>> On Sat, 19 Apr 2025 07:55:55 -0400, Richard Damon wrote:
>>>>> On 4/18/25 11:52 PM, olcott wrote:
>>>>>> On 4/18/2025 2:32 PM, Keith Thompson wrote:
>>>>>>> Mr Flibble <flibble@red-dwarf.jmc.corp> writes:
>>>>>>>> On Fri, 18 Apr 2025 12:25:36 -0700, Keith Thompson wrote:
>>>>>>>>> Mr Flibble <flibble@red-dwarf.jmc.corp> writes:

>>>>>>>>>> I, aka Mr Flibble, have created a new computer science term,
>>>>>>>>>> the "Unpartial Halt Decider".  It is a Halt Decider over the
>>>>>>>>>> domain of all program-input pairs excluding pathological input
>>>>>>>>>> (a manifestation of the self referencial category error).
>>>>>>>>> [...]
>>>>>>>>>
>>>>>>>>> Do you have a rigorous definition of "pathological input"?
>>>>>>>>> Is there an algorithm to determine whether a given input is
>>>>>>>>> "pathological" or not?
>>>>>>>>> I could define an is_prime() function like this:
>>>>>>>>>
>>>>>>>>>        bool is_prime(int n) {
>>>>>>>>>            return n >= 3 && n % 2 == 1;
>>>>>>>>>            // returns true for odd numbers >= 3, false
>>>>>>>>>            for all others
>>>>>>>>>        }
>>>>>>>>> I'll just say that odd numbers that are not prime are
>>>>>>>>> pathological input, so I don't have to deal with them.
>>>>>>>>
>>>>>>>> Pathological input: Self-referencial to the decider.
>>>>>>>
>>>>>>> Do you have a *rigorous* definition of "pathological input"?
>>>>>>> Is there an algorithm to determine whether a given input is
>>>>>>> "pathological" or not?
>>>>>>>
>>>>>> int DD()
>>>>>> {
>>>>>>      int Halt_Status = HHH(DD);
>>>>>>      if (Halt_Status)
>>>>>>        HERE: goto HERE;
>>>>>>      return Halt_Status;
>>>>>> }
>>>>>> Patterns isomorphic to the above when simulated by HHH.
>>>>>>
>>>>> Examples are not definitions.
>>>>> And the problem is that the above example is itself a category error
>>>>> for the problem, as the DD provided above isn't a complete program,
>>>>> as it doesn't include the code for HHH as required, and when you
>>>>> include Halt7.c as part of the input, your HHH isn't a seperate
>>>>> program of its own, and thus doesn't have a Turing Complete range of
>>>>> inputs it can accept.
>>>>
>>>> Ah, the fundamental mistake you have been making all this time,
>>>> Damon!
>>>> The self-referencial category error doesn't magically disappear by
>>>> providing source code rather than a run-time function address to the
>>>> decider; you are simply transforming the same input without affecting
>>>> the result.
>>>
>>> And WHAT is the category error?  You stil can't show the difference in
>>> CATEGORY between what is allowed and what isn't, and in fact, you
>>> can't even precisely define what is and isn't allowed.
>>> Now, you also run into the issue that the "Olcott System" begins with
>>> an actual category error as we do not have the required two seperate
>>> programs of the "Decider" and the "Program to be decided on" given via
>>> representation as the input, as what you want to call that program to
>>> be decided isn't one without including the code of the decider it is
>>> using,
>>> and when you do include it, the arguments about no version of the
>>> decider being able to succeed is improper as it must always be that
>>> exact program that we started with, and thus it just FAILS to do a
>>> correct simulation, while a correct simulation of this exact input
>>> (which includes the ORIGINAL decider) will halt.
>> 
>> The category error is extant over the domain of pathological inputs, no
>> matter what form those inputs take.
> 
> The category error in the halting problem proof is to define an input D
> that is able to actually do the opposite of whatever value that H
> reports.
> Now the question: Does the input D halt becomes self-contradictory for
> H.
> So it is asking a yes/no question where both yes and no are the wrong
> answer that is the category error.
No, D either halts or doesn't depending on H (which must return a value).
And H can't turn around and return the other value, as that changes D.
There are other deciders that get D right, but the same construction
works on them: every supposed decider has a counterexample, none decides
every input.

-- 
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.