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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: Wed, 26 Mar 2025 10:30:26 +0100
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Op 25.mrt.2025 om 13:04 schreef olcott:
> On 3/25/2025 4:19 AM, Fred. Zwarts wrote:
>> 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 
> 
> It is stupid to ignore the pathological relationship
> that III defines with EEE.
> 


That relation is irrelevant, it fails to reach the end of the simulation 
anyhow.
It is not very interesting to know whether a simulator reports that it 
is unable to reach the end of the simulation of a program that halts in 
direct execution.
It is interesting to know:
'Is there an algorithm that can determine for all possible inputs 
whether the input specifies a program that (according to the semantics 
of the machine language) halts when directly executed?'
This question seems undecidable for Olcott.