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From: "Fred. Zwarts" <F.Zwarts@HetNet.nl>
Newsgroups: comp.theory
Subject: Re: Turing Machine computable functions apply finite string
 transformations to inputs VERIFIED FACT
Date: Wed, 30 Apr 2025 11:04:11 +0200
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Op 29.apr.2025 om 21:57 schreef olcott:
> On 4/29/2025 10:33 AM, Fred. Zwarts wrote:
>> Op 29.apr.2025 om 15:11 schreef olcott:
>>> On 4/29/2025 2:10 AM, Richard Heathfield wrote:
>>>> On 29/04/2025 03:50, 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.
>>>>
>>>> Nothing /has/ to be measured. P's behaviour (halts, doesn't halt) 
>>>> when given D as input must be /established/. 
>>>
>>> No H can possibly see the behavior of P(D)
>>> when-so-ever D has defined a pathological
>>> relationship with H this 
>>
>> makes it impossible for H to see the behaviour of P(D).
>> The behaviour of P(D) does not change, but H does not see it.
> 
> H MUST REPORT ON THE BEHAVIOR THAT IT DOES SEE
> 

That is your error. H must report on the behaviour specified in the 
input. Bugs in H do not change the behaviour specified in the input.
H just fails to see it.