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From: olcott <polcott333@gmail.com>
Newsgroups: comp.theory
Subject: Re: Is Richard a Liar?
Date: Sun, 12 May 2024 10:58:02 -0500
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On 5/12/2024 10:21 AM, Mikko wrote:
> On 2024-05-12 11:34:17 +0000, Richard Damon said:
> 
>> On 5/12/24 5:19 AM, Mikko wrote:
>>> On 2024-05-11 16:26:30 +0000, olcott said:
>>>
>>>> I am working on providing an academic quality definition of this
>>>> term.
>>>
>>> The definition in Wikipedia is good enough.
>>>
>>
>> I think he means, he is working on a definition that redefines the 
>> field to allow him to claim what he wants.
> 
> Here one can claim whatever one wants anysay.
> In if one wants to present ones claims on some significant forum then
> it is better to stick to usual definitions as much as possible.
> 
>> Sort of like his new definition of H as an "unconventional" machine 
>> that some how both returns an answer but also keeps on running.
> 
> There are systems where that is possible but unsolvable problems are
> unsolvable even in those systems.
> 

When Ĥ is applied to ⟨Ĥ⟩
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

embedded_H is an actual UTM that has the extra feature of watching
every state transition of its simulated input so that it detects
the non-halting behavior pattern that we can all see.

Execution trace of  Ĥ applied to ⟨Ĥ⟩
(a) Ĥ.q0 The input ⟨Ĥ⟩ is copied then transitions to embedded_H
(b) embedded_H applied ⟨Ĥ⟩ ⟨Ĥ⟩ (input and copy) simulates ⟨Ĥ⟩ applied to 
⟨Ĥ⟩
(c) which begins at its own simulated ⟨Ĥ.q0⟩ to repeat the process

Simulation invariant: ⟨Ĥ⟩ correctly simulated by embedded_H never
reaches its own simulated final state of ⟨Ĥ.qn⟩.

(1) The directly executed embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ transitions to its own
internal state of embedded_H.qn indicating that the correct and
complete simulation of its input never halts. It never reaches Ĥ.qn.

(2) embedded_H continues to simulate its input after it transitions
to this internal state.

(3) Each simulated embedded_H does the same thing.

embedded_H is not a halt decider or a partial halt decider
because all deciders are required to halt. embedded_H is
not even a conventional termination analyzer for this same
reason. None-the-less embedded_H does derive the correct
halt status of its conventional HP input.





-- 
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer