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From: dbush <dbush.mobile@gmail.com>
Newsgroups: comp.theory
Subject: Re: Turing Machine computable functions apply finite string
 transformations to inputs +++
Date: Mon, 28 Apr 2025 15:58:41 -0400
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On 4/28/2025 3:45 PM, olcott wrote:
> On 4/28/2025 2:07 PM, dbush wrote:
>> On 4/28/2025 2:58 PM, olcott wrote:
>>> On 4/28/2025 1:46 PM, dbush wrote:
>>>> On 4/28/2025 2:30 PM, 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.
>>>>>
>>>>
>>>>
>>>> And no turing machine exists that can derive the following mapping 
>>>> (i.e. the mapping is not a computable function), as proven by Linz 
>>>> and others:
>>>>
>>>
>>> Because the theory of computation was never previously
>>> elaborated to make it clear that Turing computable
>>> functions are required to derive their output by applying
>>> finite string transformations to their input finite strings.
>>>
>>
>> And no such algorithm can derive this 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
>>
>>
>>> *When we do this then a mapping suddenly appears*
>>
>> And that mapping is not the halting mapping, therefore the algorithm 
>> is not a halt decider.
>>
>>> DD emulated by HHH according to the finite string
>>> transformation rules of the x86 language DOES NOT HALT.
>>
>> In other words, HHH doesn't map the halting function.
>>
>>>
>>> *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
>>
>> And the halting function is not a computable function:
>>
> 
> Maybe you have ADD like Richard and can only
> pay attention to a point when it is repeated many times
> 
> I just proved that your halting function is incorrect.
>

Category error.  The halting function below is fully defined, and this 
mapping is not computable *as you have explicitly admitted*.


Given any algorithm (i.e. a fixed immutable sequence of instructions) X 
described as <X> with input Y:

(<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