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

On 2025-05-05 02:49:46 +0000, olcott said:

On 5/4/2025 7:21 PM, Richard Damon wrote:

On 5/4/25 6:15 PM, olcott wrote:

On 5/4/2025 2:21 PM, Richard Heathfield wrote:

On 04/05/2025 18:55, olcott wrote:

Changing my words then rebutting these changed
words is dishonest.
 Functions computed by Turing Machines require INPUTS
and produce OUTPUTS DERIVED FROM THESE INPUTS.

 Counter-example: a Turing Machine can calculate pi without any input whatsoever.
 As Mikko rightly said: a Turing machine does not need to require an input.
 

 IT IS NOT COMPUTING FUNCTION THEN
IT IS NOT COMPUTING FUNCTION THEN
IT IS NOT COMPUTING FUNCTION THEN
IT IS NOT COMPUTING FUNCTION THEN

 Right, not all Turing Machine compute Functions, they all do perform Computations.
 

 Even those that know this pretend that they don't.
 

 Computable functions are the basic objects of study in computability theory. 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
 given an input of the function domain it can return the corresponding output.

 Right, and the input to a Halt Decider is the representation of a Program,

 Not exactly. It is a 100% specific precise sequence of encoded steps.
Not at all the same as a mere description.

The definition of the halting problem requires a description of the
program and a description of the input. How a program and an input
shall be presented must be specified as a part of the solution. One
way to specify it is that the solution includes an universal Turing
machine and specifies that the input description must be so that
that universal Turing machine reproduces the correct behaviour.
Whatever description language the solution rquires it must be a
complete language for the purpose: the encoding rules must be
applicable to every Turing machine and every input to that machine.
--
Mikko