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

On 3/25/2025 10:53 AM, olcott wrote:

On 3/25/2025 9:45 AM, dbush wrote:

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:

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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>
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False.  The mathematical function that counts the number of instructions in a turing machine is computable.
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 It is impossible for an actual Turing machine to
be input to any other TM.
 

But a description of a turing machine can be, for example in the form of source code or a binary.  And a turing machine by definition *always* behaves the same for a given input when executing directly.