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From: "Fred. Zwarts" <F.Zwarts@HetNet.nl>
Newsgroups: comp.theory
Subject: Re: Halting Problem: What Constitutes Pathological Input
Date: Tue, 6 May 2025 10:32:31 +0200
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Op 05.mei.2025 om 20:09 schreef olcott:
> 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.
> 

There is no self-contradiction. Only a HHH that contradicts the correct 
answer.