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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Halting Problem: What Constitutes Pathological Input
Date: Mon, 5 May 2025 13:09:36 -0500
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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.

-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer