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From: Richard Heathfield <rjh@cpax.org.uk>
Newsgroups: comp.theory
Subject: Re: Turing Machine computable functions apply finite string
 transformations to inputs VERIFIED FACT
Date: Tue, 29 Apr 2025 08:19:03 +0100
Organization: Fix this later
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On 29/04/2025 05:57, olcott wrote:
> On 4/28/2025 10:00 PM, dbush wrote:
>> On 4/28/2025 10:50 PM, olcott wrote:
>>> On 4/28/2025 3:13 PM, Richard Heathfield wrote:
>>>> On 28/04/2025 19:30, olcott wrote:
>>>>> On 4/28/2025 11:38 AM, Richard Heathfield wrote:
>>>>>> On 28/04/2025 16:01, olcott wrote:
>>>>>>> On 4/28/2025 2:33 AM, Richard Heathfield wrote:
>>>>>>>> On 28/04/2025 07:46, Fred. Zwarts wrote:
>>>>>>>>
>>>>>>>> <snip>
>>>>>>>>
>>>>>>>>> So we agree that no algorithm exists that can determine 
>>>>>>>>> for all possible inputs whether the input specifies a 
>>>>>>>>> program that (according to the semantics of the machine 
>>>>>>>>> language) halts when directly executed.
>>>>>>>>> Correct?
>>>>>>>>
>>>>>>>> Correct. We can, however, construct such an algorithm 
>>>>>>>> just as long as we can ignore any input we don't like the 
>>>>>>>> look of.
>>>>>>>>
>>>>>>>
>>>>>>> The behavior of the direct execution of DD cannot be derived
>>>>>>> by applying the finite string transformation rules specified
>>>>>>> by the x86 language to the input to HHH(DD). This proves that
>>>>>>> this is the wrong behavior to measure.
>>>>>>>
>>>>>>> It is the behavior THAT IS derived by applying the finite
>>>>>>> string transformation rules specified by the x86 language
>>>>>>> to the input to HHH(DD) proves that THE EMULATED DD NEVER 
>>>>>>> HALTS.
>>>>>>
>>>>>> The x86 language is neither here nor there. 
>>>>>
>>>>> Computable functions are the formalized analogue
>>>>> of the intuitive notion of algorithms, in the sense
>>>>> that a function is computable if there exists an
>>>>> algorithm that can do the job of the function, i.e.
>>>>> *given an input of the function domain it*
>>>>> *can return the corresponding output*
>>>>> https://en.wikipedia.org/wiki/Computable_function
>>>>>
>>>>> *Outputs must correspond to inputs*
>>>>>
>>>>> *This stipulates how outputs must be derived*
>>>>> Every Turing Machine computable function is
>>>>> only allowed to derive outputs by applying
>>>>> finite string transformation rules to its inputs.
>>>>
>>>> In your reply to my article, you forgot to address what I 
>>>> actually wrote. I'm not sure you understand what 'reply' means.
>>>>
>>>> Still, I'm prepared to give you another crack at it. Here's 
>>>> what I wrote before:
>>>>
>>>> What matters is whether a TM can be constructed that can 
>>>> accept an arbitrary TM tape P and an arbitrary input tape D 
>>>> and correctly calculate whether, given D as input, P would 
>>>> halt. Turing proved that such a TM cannot be constructed.
>>>>
>>>> This is what we call the Halting Problem.
>>>>
>>>
>>> Yet it is H(P,D) and NOT P(D) that must be measured.
>>
>> Not if it's the behavior of P(D) we want to know about, which 
>> it is.
>>
> 
> Wanting the square root of a rotten egg would do as well.
> When P defines a pathological relationship with H it
> is stupidly incorrect to expect the behavior of P to remain the 
> same.

Or, to put it more simply, wanting a TM that can compute an 
incomputable function is futile.

Indeed it is.

Conclusion: Turing was right.

-- 
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within