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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: Turing Machine computable functions apply finite string
 transformations to inputs VERIFIED FACT
Date: Wed, 30 Apr 2025 19:37:59 -0400
Organization: i2pn2 (i2pn.org)
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On 4/30/25 10:58 AM, olcott wrote:
> On 4/30/2025 1:05 AM, Richard Heathfield wrote:
>> On 30/04/2025 03:45, Richard Damon wrote:
>>> On 4/29/25 3:57 PM, olcott wrote:
>>>> On 4/29/2025 10:33 AM, Fred. Zwarts wrote:
>>>>> Op 29.apr.2025 om 15:11 schreef olcott:
>>
>> <snip>
>>
>>>>>> No H can possibly see the behavior of P(D)
>>>>>> when-so-ever D has defined a pathological
>>>>>> relationship with H this 
>>>>>
>>>>> makes it impossible for H to see the behaviour of P(D).
>>>>> The behaviour of P(D) does not change, but H does not see it.
>>>>
>>>> H MUST REPORT ON THE BEHAVIOR THAT IT DOES SEE
>>>>
>>>
>>> No, it must report on the behavior that exists.
>>>
>>> It is only ABLE to correctly report on behavior it can "see", but 
>>> there is no structural restriction that says we can't ask it about 
>>> something that it can't see.
>>
>> Nor is there any restriction that says it can't deduce behaviour it 
>> can't see, simply by reading the tapes.
>>
> 
> It is ONLY allowed to derive its output by
> applying finite string transformations to its input.
> For the HHH/DD pair these must be the finite string
> transformations specified by the x86 language.]

It is only ABLE to.

The correct answer comes from the Halting Function mapping which has no 
such restriction.

> 
> For the embedded_H / ⟨Ĥ⟩ ⟨Ĥ⟩ pair these
> transformations are specified by the Linz template.
> 
> When Ĥ is applied to ⟨Ĥ⟩
> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
> 
> Above adapted from bottom of page 2
> https://www.liarparadox.org/Linz_Proof.pdf
> 
> (a) Ĥ copies its input ⟨Ĥ⟩
> (b) Ĥ invokes embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩
> (c) embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩
> (d) simulated ⟨Ĥ⟩ copies its input ⟨Ĥ⟩
> (e) simulated ⟨Ĥ⟩ invokes simulated embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩
> (f) simulated embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩
> (g) goto (d) with one more level of simulation

And since H (Ĥ) (Ĥ) aborts its emulation and return to qn, then so must 
the embedded_H started in (c), and when that happens we end up unwinding 
  and then going do (d1) where embedded_H goes to Ĥ.qn and Ĥ halts.

> 
>> Mr Olcott seems unable to recognise this possibility. Having built his 
>> hammer, he is determined to see the Halting Problem as a nail that 
>> cannot withstand being pounded hard enough.
>>
>> Unfortunately for him, the problem is more like a 16 puzzle with two 
>> tiles swapped. If he plays by the rules there is no solution, no 
>> matter how hard he hits it.
>>
>> This is why he keeps trying to change the rules, with word salad like 
>> 'pathological self-reference', when self-reference is the whole reason 
>> the proof works.
>>
> 
>