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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 09:53:31 -0500
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On 3/25/2025 9:45 AM, dbush wrote:
> On 3/24/2025 11:29 PM, olcott wrote:
>> On 3/24/2025 10:12 PM, dbush wrote:
>>> On 3/24/2025 10:07 PM, olcott wrote:
>>>> On 3/24/2025 8:46 PM, André G. Isaak wrote:
>>>>> On 2025-03-24 19:33, olcott wrote:
>>>>>> On 3/24/2025 7:00 PM, André G. Isaak wrote:
>>>>>
>>>>>>> In the post you were responding to I pointed out that computable 
>>>>>>> functions are mathematical objects.
>>>>>>
>>>>>> https://en.wikipedia.org/wiki/Computable_function
>>>>>>
>>>>>> Computable functions implemented using models of computation
>>>>>> would seem to be more concrete than pure math functions.
>>>>>
>>>>> Those are called computations or algorithms, not computable functions.
>>>>>
>>>>
>>>> https://en.wikipedia.org/wiki/Pure_function
>>>> Is another way to look at computable functions implemented
>>>> by some concrete model of computation.
>>>>
>>>
>>> And not all mathematical functions are computable, such as the 
>>> halting function.
>>>
>>>>> The halting problems asks whether there *is* an algorithm which can 
>>>>> compute the halting function, but the halting function itself is a 
>>>>> purely mathematical object which exists prior to, and independent 
>>>>> of, any such algorithm (if one existed).
>>>>>
>>>>
>>>> None-the-less it only has specific elements of its domain
>>>> as its entire basis. For Turing machines this always means
>>>> a finite string that (for example) encodes a specific
>>>> sequence of moves.
>>>
>>> False.  *All* turing machine are the domain of the halting function, 
>>> and the existence of UTMs show that all turning machines can be 
>>> described by a finite string.
>>>
>>
>> You just aren't paying enough attention. Turing machines
>> are never in the domain of any computable function.
>> <snip>
>>
> 
> False.  The mathematical function that counts the number of instructions 
> in a turing machine is computable.
> 

It is impossible for an actual Turing machine to
be input to any other TM.

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