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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: Halting Problem: What Constitutes Pathological Input
Date: Wed, 7 May 2025 11:31:04 +0300
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On 2025-05-06 15:23:18 +0000, olcott said:

> On 5/6/2025 4:30 AM, Mikko wrote:
>> On 2025-05-05 19:27:18 +0000, olcott said:
>> 
>>> 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.
>>>>>>>>>>>> 
>>>>>>>>>>>> /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
>>>>>>>> 
>>>>>>>> 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.
>>>>>> 
>>>>>> 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.
>>>> 
>>>> 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.
>> 
>> It is well defined. There are computations that halt and computations that
>> do not. Nothing else is in the scope of the halting problem.
> 
> It is incorrectly defined when-so-ever it is not specified
> that a specific sequence of steps must be applied to the
> input to derive the output.

Apparently you don't know shat the word "problem" means. For a rough
idea see
    https://en.wiktionary.org/wiki/problem

A problem specification specifies what is and what is not acceptable
as a solution of that problem. There is no general rule that a
specific sequence of steps must be applied to the input to derive the
output (or even that there must be an input and an output). The
halting problem requires that the solution specifies those steps and
presents them as a Turing machine.

-- 
Mikko