Re: Turing Machine computable functions apply finite string transformations to inputs VERIFIED FACT

On 4/29/2025 12:57 AM, olcott wrote:

On 4/28/2025 10:00 PM, dbush wrote:

On 4/28/2025 10:50 PM, olcott wrote:

On 4/28/2025 3:13 PM, Richard Heathfield wrote:

On 28/04/2025 19:30, olcott wrote:

On 4/28/2025 11:38 AM, Richard Heathfield wrote:

On 28/04/2025 16:01, olcott wrote:

On 4/28/2025 2:33 AM, Richard Heathfield wrote:

On 28/04/2025 07:46, Fred. Zwarts wrote:
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So we agree that no algorithm exists that can determine for all possible inputs whether the input specifies a program that (according to the semantics of the machine language) halts when directly executed.
Correct?

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Correct. We can, however, construct such an algorithm just as long as we can ignore any input we don't like the look of.
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The behavior of the direct execution of DD cannot be derived
by applying the finite string transformation rules specified
by the x86 language to the input to HHH(DD). This proves that
this is the wrong behavior to measure.
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It is the behavior THAT IS derived by applying the finite
string transformation rules specified by the x86 language
to the input to HHH(DD) proves that THE EMULATED DD NEVER HALTS.

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The x86 language is neither here nor there.

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Computable functions are the formalized analogue
of the intuitive notion of algorithms, in the sense
that a function is computable if there exists an
algorithm that can do the job of the function, i.e.
*given an input of the function domain it*
*can return the corresponding output*
https://en.wikipedia.org/wiki/Computable_function
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*Outputs must correspond to inputs*
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*This stipulates how outputs must be derived*
Every Turing Machine computable function is
only allowed to derive outputs by applying
finite string transformation rules to its inputs.

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In your reply to my article, you forgot to address what I actually wrote. I'm not sure you understand what 'reply' means.
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Still, I'm prepared to give you another crack at it. Here's what I wrote before:
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What matters is whether a TM can be constructed that can accept an arbitrary TM tape P and an arbitrary input tape D and correctly calculate whether, given D as input, P would halt. Turing proved that such a TM cannot be constructed.
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This is what we call the Halting Problem.
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Yet it is H(P,D) and NOT P(D) that must be measured.

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Not if it's the behavior of P(D) we want to know about, which it is.
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 Wanting the square root of a rotten egg would do as well.

In other words, you're claiming that there's an algorithm, i.e. a fixed immutable sequence of instructions, that when given a particular input neither halts nor not halts when executed directly.
Show it.
Failure to do so will in your next reply, or within one hour of your next post on this newsgroup, be taken as you on-the-record admission that there is nothing incorrect about the halting question asking whether any arbitrary algorithm X with input Y will halt when executed directly.