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From: "Fred. Zwarts" <F.Zwarts@HetNet.nl>
Newsgroups: comp.theory,comp.lang.c++
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 10:19:10 +0100
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Op 25.mrt.2025 om 00:04 schreef olcott:
> On 3/24/2025 5:49 PM, André G. Isaak wrote:
>> 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
>>
>>
> 
> _III()
> [00002172] 55         push ebp      ; housekeeping
> [00002173] 8bec       mov  ebp,esp  ; housekeeping
> [00002175] 6872210000 push 00002172 ; push III
> [0000217a] e853f4ffff call 000015d2 ; call EEE(III)
> [0000217f] 83c404     add  esp,+04
> [00002182] 5d         pop  ebp
> [00002183] c3         ret
> Size in bytes:(0018) [00002183]
> 
> When III is emulated by pure emulator EEE for any finite
> number of steps of emulation according to the semantics
> of the x86 language it never reaches its own "ret"
> instruction final halt state 

If fails to complete the simulation of a program that halts when the 
exact same input is used for direct execution or other world-class 
simulators, so it cannot conclude:

> THUS DOES NOT HALT.

(Unable to reach the moon with an aeroplane does not mean that the moon 
does not exist.)

It can only correctly report: 'Unable to complete the simulation'.
That is how the result should be interpreted.