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From: =?UTF-8?B?QW5kcsOpIEcuIElzYWFr?= <agisaak@gm.invalid>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Mon, 24 Mar 2025 16:49:34 -0600
Organization: Christians and Atheists United Against Creeping Agnosticism
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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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