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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable functions
Date: Tue, 25 Mar 2025 11:22:18 +0200
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On 2025-03-25 03:29:06 +0000, olcott said:

> 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>

There are computable functions that take Turing machines as arguments.
For example, the number of states of a Turing machine.

The computability of a function requires that the domain can be mapped
to finite strings.

-- 
Mikko