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From: dbush <dbush.mobile@gmail.com>
Newsgroups: comp.theory
Subject: Re: Turing computable functions
Date: Tue, 25 Mar 2025 16:05:31 -0400
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On 3/25/2025 3:47 PM, olcott wrote:
> On 3/25/2025 2:32 PM, dbush wrote:
>> On 3/25/2025 3:24 PM, olcott wrote:
>>> Cannot possibly derive any outputs not computed from
>>> their inputs.
>>
>> Correct, algorithms can only compute computable mathematical function.
>>
>>>
>>> A Turing machine halt decider 
>>
>> Does not exist because the required mapping is not computable:
>>
>>
>> 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
>>
>>
>>
>>> cannot possibly report
>>> on the behavior of any directly executing process.
>>> No Turing machine can every do this. This has always
>>> been beyond what any Turing machine can ever do.
>>
>>
>> Strawman: reporting on an executing process is not a requirement. 
> 
> YOU JUST SAID THAT IT WAS
> YOU KEEP MINDLESSLY REPEATING THAT IT IS
> 
> On 3/25/2025 2:32 PM, dbush wrote:
>  > (<X>,Y) maps to 1 if and only if X(Y) halts when executed directly
> 
> 

I never said it had to actually watch an executing process, only report 
what would happen if it did run.

For example, the following correctly reports the halt status of all 
Turing machines that halt when executed directly.

int H(uintptr_t *p)
{
     return 1;
}


Remember, because a Turing machine will always behave *exactly* the same 
for a given input, a complete description of that Turing machine fully 
specifies what would happen in the hypothetical case that such a machine 
was executed directly.