Re: Correcting the definition of the halting problem --- Computable functions

On 3/24/25 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:

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In the post you were responding to I pointed out that computable functions are mathematical objects.

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https://en.wikipedia.org/wiki/Computable_function
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Computable functions implemented using models of computation
would seem to be more concrete than pure math functions.

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Those are called computations or algorithms, not computable functions.
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https://en.wikipedia.org/wiki/Pure_function
Is another way to look at computable functions implemented
by some concrete model of computation.
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And not all mathematical functions are computable, such as the halting function.
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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).
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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.

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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.
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 You just aren't paying enough attention. Turing machines
are never in the domain of any computable function.
<snip>

Sure they are, you seem to try to conviently forget about the ability to represent "abstract" or "Mathenatical" constructs with language.
And the finite string input to a Turing Machine/Computable Function is to be interpreted by the language defined by that machine.
OR, you need to define that the addition of two natural numbers isn't a computable funcition, as a "number" isn't in the domain of a computable function, as it isn't a finite string, just something that can be represented (in various ways) as one.

 

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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.
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There are no details abstracted away.  The halting function is simply uncomputable.
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 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.

But that isn't the measure of the behavior specified by the finite string to a Halt Decider.
All you are doing is admitting to your strawman.
Sorry, you are just showing that you are so stupid that you openly admit it without realizing it.

 

A halt decider cannot exist

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So again, you explicitly agree with the Linz proof and all other proofs of the halting function.
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because the halting problem is defined incorrectly

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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.
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There's no requirement that a function be computable.