Re: DDD specifies recursive emulation to HHH and halting to HHH1

On 3/31/25 11:10 PM, olcott wrote:

On 3/31/2025 10:02 PM, dbush wrote:

On 3/31/2025 10:57 PM, olcott wrote:

On 3/31/2025 9:48 PM, dbush wrote:

On 3/31/2025 10:38 PM, olcott wrote:

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HHH must report on the behavior that its input
actually specifies.

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And the input specifies an algorithm that halts when executed directly.
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Which is NOT how it is executed.

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But IS what it is required to report on.
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It IS executed with recursive emulation.

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Finitely recursive emulation, which is part of the algorithm DDD that halts.
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This seems way too difficult
for people that can only spout off words that
they learned by rote, with no actual understanding
of the underlying principles involved.
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Every actual computation can only transform input
finite strings into outputs. HHH does do this.

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But not as per the requirements:
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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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The "requirements" that you mindlessly spout off

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Which would make all true statements provable if they could be met.
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violate this foundational principle of functions
computed by Turing machines.

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Which just says that no Turing machine satisfies those requirements, as Linz and others proved.
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int sum(int x, int y)
{
   return 5;
}
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sum(2,3) does not compute the sum of 2 + 3.
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It absolutely does.  There are *NO* requirements on the implementation, only the result.

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Unless and algorithm transforms its inputs
into its outputs it is not a Turing computable
function.
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The only requirements of an algorithm are to generate the results of a mapping.  How that is accomplished is irrelevant.

 It may have been historically considered irrelevant.
When it comes down to determining that mere guesses
are not any sort of: "computing the mapping" then it
does become relevant.
 

Nope, it isn't a case of "Show your work" with partial credit for trying to do the right thing.
Hard Reality of either right answer or wrong answer.
You are computing the mapping, if you can get EVERY answer right, not just one.
Your problem is you think you can prove something correctly computing the mapping by showing just one example, but that is just a logical fallicy.
You can prove an algorithm to be WRONG with one example, an input whose answer given doesn't match the answer required, but proving correct can be harder.
And it is possible for a machine to be correct and not provable that it is, as we have an infinite number of inputs to test, so if we can't find an induction principle on the algorithm (a real one, not one of your fakes) then its correctness is unprovable.
This is the issue with Godel's G, there is no number that meets the relationship, so the statement that says that is true, but it can not be proven, and the only way to verify the fact is to actually test every number, which isn't a finite operation, so not a proof.
Sorry, you are just showing your ignorance of what you talk about.