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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 10:45:57 -0400
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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.

>>>
>>> The math halting function is free to "abstract away" key
>>> details that change everything. That is why I have never
>>> been talking about the pure math and have always been
>>> referring to its implementation in a model of computation.
>>>
>>
>> There are no details abstracted away.  The halting function is simply 
>> uncomputable.
>>
> 
> When the measure of the behavior specified by a
> finite string input DD is when correctly emulated
> by HHH then DD can't reach its own final halt state
> then not-halting is decidable.
> 

That's not non-halting, that's not-PO-halting.  Halting, as currently 
defined, is not decidable.

When the measure of the behavior specified by a
finite string input DD is when directly executed then DD halts and HHH 
doesn't report that.

>>> A halt decider cannot exist 
>>
>> So again, you explicitly agree with the Linz proof and all other 
>> proofs of the halting function.
>>
>>> because the halting problem is defined incorrectly 
>>
>> There's nothing incorrect about wanting something that would solve the 
>> Goldbach conjecture and make unknowable truths knowable.  Your 
>> alternate definition won't provide that.
>>
>> There's no requirement that a function be computable.
> 
>