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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Mon, 24 Mar 2025 18:42:50 -0500
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On 3/24/2025 6:05 PM, André G. Isaak wrote:
> On 2025-03-24 17:04, olcott wrote:
>> 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 THUS DOES NOT HALT.
>>
>> When III is directly executed calls an EEE instance
>> that only emulates finite number of steps then this
>> directly executed III always reaches its own "ret"
>> instruction final halt state THUS HALTS.
> 
> And that has what, exactly, to do with the post you are allegedly 
> responding to?
> 
> André
> 


THE INPUT FINITE STRING DOES SPECIFY RECURSIVE EMULATION.

The behavior specified by the finite string input to a
computable function implemented on a model of computation

does differ from the direct execution of this same finite
string because the direct execution avoids the pathological
self-reference that causes the recursive emulation.

THE INPUT FINITE STRING DOES SPECIFY RECURSIVE EMULATION.

-- 
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer