Re: The philosophy of computation reformulates existing ideas on a new basis ---

On 10/27/24 6:01 PM, olcott wrote:

On 10/27/2024 12:48 PM, Richard Damon wrote:

On 10/27/24 10:17 AM, olcott wrote:

I am keeping this post in both sci.logic and comp.theory
because it focuses on a similar idea to the Curry/Howard
correspondence between formal systems and computation.
>
Computation and all of the mathematical and logical operations
of mathematical logic can be construed as finite string
transformation rules applied to finite strings.
>
The semantics associated with finite string tokens can
be directly accessible to expression in the formal language.
It is basically an enriched type hierarchy called a knowledge
ontology.
>
A computation can be construed as the tape input to a
Turing machine and its tape output. All of the cases
where the output was construed as a set of final machine
states can be written to the tape.
>
I am not sure but I think that this may broaden the scope
of a computable function, or not.

>
Except that nothing you described related to what a "computabe function"

 I intend to reply to other aspects of your reply later
on as long as your reply to this reply is not lame.
 When a Turing machine transforms the contents of its
input tape into the contents of its output tape this
seems to necessarily always be a computable function
no matter what the TM does in-between.
 

Yes, a Turing Machine will always be computing the mapping from some computable function.
It is NOT the "Computable Function" itself, as that is a thing of a different ty[pe.
It just computed the mapping definied by that function.
Note, the mapping of the function might not be defined in terms of "finite-strings", but will be something that can be described by a finite string if we want to talk about it being computable.
For instance, the Halting Function, that the Halting problem is about, is defined with Turing Machines as its input (not finite strings).
The key point here is that different implementation of a attempted Turing Machines to try to compute this might use different ways of representing the machines, so the function can't just be thought of as taking the string.
We can look at the equivalent mapping based on the encoding of the given decider, if the encoding has the required property that a given finite string can only represent one Turing Machine by the rules of that decider.
Note, This is one spot your HHH/DDD pairing fails, as what you want to claim as the input reprenting DDD does NOT have that property, as the finite string does not represent a specific computation, as it depends on what HHH it is being pair with.