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

On 5/1/2025 10:25 PM, olcott wrote:

On 5/1/2025 9:10 PM, André G. Isaak wrote:

On 2025-05-01 19:55, olcott wrote:
>

Specify every single step of the mapping and you will
see that it has never been well defined. It has ONLY
ever been a leap to a conclusion.

>
Mappings don't HAVE steps. Again, you are confusing functions with algorithms.
>
André
>

 In other words people just guess that a mapping exists or not?

All mappings exist.  Some have an algorithm to compute them, some don't.   This one does not:
Given any algorithm (i.e. a fixed immutable sequence of instructions) X described as <X> with input Y:
(<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

THE MAPPING FROM INPUTS TO OUTPUTS IS BY FINITE STRING TRANSFORMATIONS.

Only if an algorithm exists to compute the mapping, and not all do, such as the one above as you have *explicitly* admitted.

 The mapping to compute the sum function is from integers
transformed by the steps of arithmetic to an integer OUTPUT.
 

And that particular mapping happens to be computable.

int sum(int x, int y) { return 5; }
The above mapping IS NOT for the sum function.
 

That's not a mapping, it's an algorithm, and it computes this function:
For all integers X and Y:
(X,Y) maps to 5