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

On 4/28/2025 12:38 PM, olcott wrote:

On 4/28/2025 10:41 AM, dbush wrote:

On 4/28/2025 11:35 AM, olcott wrote:

On 4/28/2025 10:18 AM, dbush wrote:

On 4/28/2025 11:01 AM, 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

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The assumption that an H exists that meets the below requirements is false, as shown by Linz and others:
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I have just proved that those requirements are stupidly wrong

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Category error.  The mapping exists

 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.

i.e. a computable function is a mathematical mapping for which an algorithm exists to compute in.
And the halting function below is not a computable function:
Given any algorithm (i.e. a fixed immutable sequence of instructions) X described as <X> with input Y:
A solution to the halting problem is an algorithm H that computes the following mapping:
(<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

*given an input of the function domain*
*it can return the corresponding output*
https://en.wikipedia.org/wiki/Computable_function
 

In other words, the algorithm, given an input in the domain of the mathematical function, returns the corresponding output of the mathematical function.

There is a mapping from the input to HHH(DD) by applying
the finite string transformation rules specified by the
x86 language to this DD input that derives:
*no correctly emulated DD ever reaches its final halt state*
 

In other words, the mathematical function that algorithm HHH is mapping is not the halting function, and therefore isn't a solution to the halting problem.