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

On 2025-03-24 22:49:34 +0000, André G. Isaak said:

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 bijection between natural numbers and strings of any any finite
or countable string. For every such alphabeth the set of strings is
countable. Every countable set has a bijection with natural numbers.
--
Mikko