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From: joes <noreply@example.org>
Newsgroups: comp.theory
Subject: Re: Correcting the definition of the halting problem --- Computable
 functions
Date: Tue, 25 Mar 2025 08:47:59 -0000 (UTC)
Organization: i2pn2 (i2pn.org)
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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?

-- 
Am Sat, 20 Jul 2024 12:35:31 +0000 schrieb WM in sci.math:
It is not guaranteed that n+1 exists for every n.