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From: dbush <dbush.mobile@gmail.com>
Newsgroups: comp.theory
Subject: Re: Halting Problem: What Constitutes Pathological Input
Date: Mon, 5 May 2025 15:29:21 -0400
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In-Reply-To: <vvb329$15u5b$1@dont-email.me>

On 5/5/2025 3:20 PM, olcott wrote:
> On 5/5/2025 1:14 PM, dbush wrote:
>> On 5/5/2025 2:09 PM, olcott wrote:
>>> On 5/5/2025 12:45 PM, dbush wrote:
>>>> On 5/5/2025 1:37 PM, olcott wrote:
>>>>> On 5/5/2025 11:13 AM, 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.
>>>>>>>>>
>>>>>>>>> /Flibble
>>>>>>>>
>>>>>>>> I prefer to look at it as a counter-example that refutes all of the
>>>>>>>> halting problem proofs.
>>>>>>>
>>>>>>> Which start with the assumption that the following mapping is 
>>>>>>> computable
>>>>>>> and that (in this case) HHH computes it:
>>>>>>>
>>>>>>>
>>>>>>> Given any algorithm (i.e. a fixed immutable sequence of 
>>>>>>> instructions) X
>>>>>>> described as <X> with input Y:
>>>>>>>
>>>>>>> A solution to the halting problem is an algorithm H that computes 
>>>>>>> the
>>>>>>> following mapping:
>>>>>>>
>>>>>>> (<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
>>>>>>> (<X>,Y) maps to 0 if and only if X(Y) does not halt when executed
>>>>>>> directly
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>> int DD()
>>>>>>>> {
>>>>>>>>     int Halt_Status = HHH(DD);
>>>>>>>>     if (Halt_Status)
>>>>>>>>       HERE: goto HERE;
>>>>>>>>     return Halt_Status;
>>>>>>>> }
>>>>>>>>
>>>>>>>> https://github.com/plolcott/x86utm
>>>>>>>>
>>>>>>>> The x86utm operating system includes fully operational HHH and DD.
>>>>>>>> https://github.com/plolcott/x86utm/blob/master/Halt7.c
>>>>>>>>
>>>>>>>> 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.
>>>>>>>>
>>>>>>>>
>>>>>>>
>>>>>>> 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.
>>>>>>
>>>>>> /Flibble
>>>>>
>>>>> The above example is category error because it asks
>>>>> HHH(DD) to report on the direct execution of DD() and
>>>>> the input to HHH specifies a different sequence of steps.
>>>>>
>>>>
>>>> In other words, you're demonstrating that you don't understand proof 
>>>> by contradiction, a concept taught to and understood by high school 
>>>> students more than 50 years your junior.
>>>>
>>>
>>> Self-contradiction is semantically ill-formed and has
>>> nothing to do with proof by contradiction.
>>>
>>
>> The contradiction comes about as a result of the assumption that an 
>> algorithm exists that computes the following mapping, 
> 
> NOT AT ALL.
> The self-contradiction comes when an input D
> to a termination analyzer H 

i.e. we assume the halting function is computable and that H computes it.


> is actually able
> to do the opposite of whatever value that H
> returns. In this case BOTH Boolean RETURN
> VALUES ARE THE WRONG ANSWER.
> 

And we reach a contradiction.  Therefore the assumption that there 
exists an algorithm to compute the halting function is proven false, as 
shown by Linz and other and as you have *explicitly* agreed is correct.