Re: Turing computable functions

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.

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Correct, algorithms can only compute computable mathematical function.
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A Turing machine halt decider

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Does not exist because the required mapping is not computable:
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Given any algorithm (i.e. a fixed immutable sequence of instructions) X described as <X> with input Y:
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A solution to the halting problem is an algorithm H that computes the following mapping:
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(<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
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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.

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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.