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From: Mikko <mikko.levanto@iki.fi>
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Date: Sun, 10 Mar 2024 15:21:49 +0200
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On 2024-03-09 22:15:20 +0000, olcott said:

> On 3/9/2024 3:49 PM, immibis wrote:
>> On 9/03/24 22:30, 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
>>>> 
>>> Yes. (Notice that I am agreeing with you, yet never do that with me)
>>> 
>>>> It DOESN'T unless its algorithm says it does,
>>>> 
>>> Yes. (Notice that I am agreeing with you, yet never do that with me)
>>> 
>>>> If it just fails to answer, then it has failed to be a correct Halt Decider.
>>>> 
>>> Yes. (Notice that I am agreeing with you, yet never do that with me)
>>> 
>>>> The fact that you reach this conflict in actions, is the reason Halt 
>>>> Deciding is uncomputable.
>>> 
>>> *No. We know that Ĥ ⟨Ĥ⟩ either halts or fails to halt*
>>> 
>>> If it halts then Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ transitioned to Ĥ.Hqn.
>> 
>> If Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ transitioned to Ĥ.Hqn then H ⟨Ĥ⟩ transitioned to H.qn or 
>> else Ĥ is the wrong Ĥ or you can't read instructions.
>> 
> 
> I generally agree that a pair of identical machines
> must have the same behavior on the same input.

If two machines do not behave the same on the same input
they are not identical because that is what the word
means.

> This may not apply when these machines having identical
> states and identical inputs:

Meanings of the words do apply.

-- 
Mikko