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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
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Date: Sat, 9 Mar 2024 16:19:29 -0800
Organization: i2pn2 (i2pn.org)
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On 3/9/24 2:22 PM, olcott wrote:
> On 3/9/2024 3:50 PM, immibis wrote:
>> On 9/03/24 22:34, olcott wrote:
>>> On 3/9/2024 3:17 PM, Richard Damon wrote:
>>>> On 3/9/24 10:33 AM, olcott wrote:
>>>>> *Verified fact that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ have different behavior*
>>>>>
>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts
>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn   // Ĥ applied to ⟨Ĥ⟩ does not halt
>>>>
>>>> Specifications, not actual behavior until the existance of such an H 
>>>> is shown.
>>>>
>>>> IF taken as actual behavior, then it is conditional on such an H 
>>>> existing.
>>>>
>>>>>
>>>>> Execution trace of Ĥ applied to ⟨Ĥ⟩
>>>>> (a) Ĥ.q0 The input ⟨Ĥ⟩ is copied then transitions to Ĥ.H
>>>>> (b) Ĥ.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
>>>>> *This proves that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ must abort its simulation*
>>>>
>>>> It NEEDS to in order to meet its specification
>>>>
>>>> It DOESN'T unless its algorithm says it does,
>>>>
>>>> If it just fails to answer, then it has failed to be a correct Halt 
>>>> Decider.
>>>>
>>>> The fact that you reach this conflict in actions, is the reason Halt 
>>>> Deciding is uncomputable.
>>>>
>>>>>
>>>>> *This is a verified fact*
>>>>> When simulating halt deciders always report on the behavior of
>>>>> their simulated input from their own POV then when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
>>>>> transitions to Ĥ.Hqn it is correct from its own POV.
>>>>
>>>> In other words, you are admitting to changing the question, and thus 
>>>> LYING that you are working on the actual original problem.
>>>
>>> This must just be over your head. It is very very difficult.
>>>
>>> What criteria would you use so that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ knows what
>>> wrong answer to provide?
>>>
>>
>> Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ is stipulated to use the exact same objective criteria 
>> that H ⟨Ĥ⟩ uses.
> 
> Simulating halt deciders must make sure that they themselves
> do not get stuck in infinite execution. This means that they
> must abort every simulation that cannot possibly otherwise halt.

Yes, but you can't just assume they do, you need to define on what 
condition they do that.

> 
> This requires Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ to abort its simulation and does not
> require H ⟨Ĥ⟩ ⟨Ĥ⟩ to abort its simulation when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ aborts
> its simulation.

Except that they are running the exact same code, so if the code says to 
abort, they will both abort.

If the code doesn't say it will abort, it won't and get stuck not returning.

> 
> Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ does simulate itself in recursive simulation H ⟨Ĥ⟩ ⟨Ĥ⟩
> does not simulate itself in recursive simulation.
> 

But because it is doing the exact same code on the exact same data will 
make the same decisions.

If they COULD tell themselves apart, you might be able justify that, but 
H^ specifically was define to use an EXACT copy, and Turing Machine 
enable this to be done.