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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 23:44:41 -0400
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In-Reply-To: <vvc088$20bmu$3@dont-email.me>

On 5/5/2025 11:38 PM, olcott wrote:
> On 5/5/2025 10:16 PM, dbush wrote:
>> On 5/5/2025 11:05 PM, olcott wrote:
>>> On 5/5/2025 9:56 PM, dbush wrote:
>>>> On 5/5/2025 10:51 PM, olcott wrote:
>>>>> On 5/5/2025 9:27 PM, dbush wrote:
>>>>>> On 5/5/2025 10:18 PM, olcott wrote:
>>>>>>> On 5/5/2025 8:59 PM, dbush wrote:
>>>>>>>> On 5/5/2025 8:57 PM, olcott wrote:
>>>>>>>>> On 5/5/2025 7:49 PM, dbush wrote:
>>>>>>>>>>
>>>>>>>>>> Which starts with the assumption that an algorithm exists that 
>>>>>>>>>> performs the following mapping:
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> 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
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>>> DO COMPUTE THAT THE INPUT IS NON-HALTING
>>>>>>>>>>> IFF (if and only if) the mapping FROM INPUTS
>>>>>>>>>>> IS COMPUTED.
>>>>>>>>>>
>>>>>>>>>> i.e. it is found to map something other than the above 
>>>>>>>>>> function which is a contradiction.
>>>>>>>>>>
>>>>>>>>>
>>>>>>>>> The above function VIOLATES COMPUTER SCIENCE.
>>>>>>>>> You make no attempt to show how my claim
>>>>>>>>> THAT IT VIOLATES COMPUTER SCIENCE IS INCORRECT
>>>>>>>>> you simply take that same quote from a computer
>>>>>>>>> science textbook as the infallible word-of-God.
>>>>>>>>
>>>>>>>> All you are doing is showing that you don't understand proof by 
>>>>>>>> contradiction, 
>>>>>>>
>>>>>>> Not at all. 
>>>>>>
>>>>>>
>>>>>> Yes.
>>>>>>
>>>>>> The mapping is well defined. 
>>>>>
>>>>> You don't even know that "well defined" means
>>>>> that all of the steps have been specified.
>>>>
>>>> A mapping doesn't *have* steps. It's simply an association between 
>>>> an input domain and an output domain.
>>>>
>>>
>>> Computing the mapping does > have 100% totally specific steps
>>
>>
>> So you're assuming an algorithm exists that can compute the below 
>> mapping.
>>
> 
> No the mapping below is stupidly wrong.

Not at all.

I want to know if any arbitrary algorithm X with input Y will halt when 
executed directly.

It would be *very* useful to me if I had an algorithm H that could tell 
me that in *all* possible cases.  If so, I could solve the Goldbach 
conjecture and make all truths knowable, among many other unsolved problems.

No algorithm H that can do this, as Linz and others have proved and you 
have *explicitly* agreed is correct.