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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 08:01:14 -0500
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On 3/25/2025 3:47 AM, joes wrote:
> Am Mon, 24 Mar 2025 18:04:21 -0500 schrieb 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.
>>>> Maybe when pure math objects. In every model of computation they seem
>>>> to always have inputs.
>>> 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.
> 
>> 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.

> A pure simulator can not limit the number of steps. Also III doesn't
> halt in, say, 3 steps. Why should III call a different instance
> that doesn't abort, when it is being simulated?
> 

The fact that the same states in the program-under-test
keep repeating such that the program-under-test cannot
possibly reach its own final halt state proves that
program-under-test does not halt.

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