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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: Wed, 26 Mar 2025 09:15:23 +0200
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On 2025-03-25 14:49:44 +0000, olcott said:

> On 3/25/2025 4:22 AM, Mikko wrote:
>> 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.
> 
> IT IS COUNTER-FACTUAL THAT A MACHINE DESCRIPTION SPECIFIES
> BEHAVIOR IDENTICAL TO THE DIRECTLY EXECUTED MACHINE.

Nothing counter-factual there. From the meanings of the words follows
that a machine description does not specify any other behaviour than
the behaviour of the directly executed machine. A specification of
any other behaviour is not a part of the description of the machine.

-- 
Mikko