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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: DDD specifies recursive emulation to HHH and halting to HHH1
Date: Mon, 31 Mar 2025 22:10:07 -0500
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On 3/31/2025 10:02 PM, dbush wrote:
> On 3/31/2025 10:57 PM, olcott wrote:
>> On 3/31/2025 9:48 PM, dbush wrote:
>>> On 3/31/2025 10:38 PM, olcott wrote:
>>>>
>>>> HHH must report on the behavior that its input
>>>> actually specifies. 
>>>
>>> And the input specifies an algorithm that halts when executed directly.
>>>
>>
>> Which is NOT how it is executed.
> 
> But IS what it is required to report on.
> 
>> It IS executed with recursive emulation.
> 
> Finitely recursive emulation, which is part of the algorithm DDD that 
> halts.
> 
>>
>>>> This seems way too difficult
>>>> for people that can only spout off words that
>>>> they learned by rote, with no actual understanding
>>>> of the underlying principles involved.
>>>>
>>>> Every actual computation can only transform input
>>>> finite strings into outputs. HHH does do this.
>>>
>>> But not as per the requirements:
>>>
>>
>>>
>>> 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
>>>
>>>
>>>>
>>>> The "requirements" that you mindlessly spout off
>>>
>>> Which would make all true statements provable if they could be met.
>>>
>>>> violate this foundational principle of functions
>>>> computed by Turing machines.
>>>
>>> Which just says that no Turing machine satisfies those requirements, 
>>> as Linz and others proved.
>>>
>>>>
>>>> int sum(int x, int y)
>>>> {
>>>>    return 5;
>>>> }
>>>>
>>>> sum(2,3) does not compute the sum of 2 + 3.
>>>>
>>>
>>>
>>> It absolutely does.  There are *NO* requirements on the 
>>> implementation, only the result.
>>
>> Unless and algorithm transforms its inputs
>> into its outputs it is not a Turing computable
>> function.
>>
> 
> The only requirements of an algorithm are to generate the results of a 
> mapping.  How that is accomplished is irrelevant.

It may have been historically considered irrelevant.
When it comes down to determining that mere guesses
are not any sort of: "computing the mapping" then it
does become relevant.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer