Re: Correcting the definition of the halting problem --- Computable functions

On 2025-03-24 16:43, olcott wrote:

Computable functions don't have inputs. They have domains. Turing machines have inputs.p
>

 Maybe when pure math objects. In every model of
computation they seem to always have inputs.
https://en.wikipedia.org/wiki/Computable_function

Computable functions *are* pure math objects. You seem to want to conflate them with C functions, but that is not the case.
The crucial point is that the domains of computable functions are *not* restricted to strings, even if the inputs to Turing Machines are.

While the inputs to TMs are restricted to strings, there is no such such restriction on computable functions.

 

The vast majority of computable functions of interest do *not* have strings as their domains, yet they remain computable functions (a simple example would be the parity function which maps NATURAL NUMBERS (not strings) to yes/no values.)
>

 Since there is a bijection between natural numbers
and strings of decimal digits your qualification
seems vacuous.

There is not a bijection between natural numbers and strings. There is a one-to-many mapping from natural numbers to strings, just as there is a one-to-many mapping from computations (i.e. turing machine/input string pairs, i.e. actual Turing machines directly running on their inputs) to strings.
André
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