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From: dbush <dbush.mobile@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 10:59:49 -0400
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On 3/25/2025 9:14 AM, olcott wrote:
> On 3/25/2025 3:32 AM, joes wrote:
>> Am Mon, 24 Mar 2025 22:29:06 -0500 schrieb olcott:
>>> 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.
>>>>>>> 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>
> 
>> Fine, their descriptions are, and their behaviour is computable -
>> by running them.
>>
> 
> Halt deciders

Don't exist, because no H satisfies this requirement:


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

A solution to the halting problem is an algorithm H that computes the 
following 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