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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory,sci.logic
Subject: Re: D correctly simulated by H cannot possibly halt --- templates and
 infinite sets
Date: Wed, 29 May 2024 20:09:10 -0400
Organization: i2pn2 (i2pn.org)
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On 5/29/24 8:01 PM, olcott wrote:
> On 5/29/2024 6:47 PM, Richard Damon wrote:
>> On 5/29/24 9:24 AM, olcott wrote:
>>> On 5/29/2024 4:14 AM, Mikko wrote:
>>>> On 2024-05-29 03:49:02 +0000, olcott said:
>>>>
>>>>> On 5/28/2024 10:38 PM, Richard Damon wrote:
>>>>>> On 5/28/24 10:23 PM, olcott wrote:
>>>>>>> On 5/28/2024 9:04 PM, Richard Damon wrote:
>>>>>>>> On 5/28/24 12:16 PM, olcott wrote:
>>>>>>>>> typedef int (*ptr)();  // ptr is pointer to int function in C
>>>>>>>>> 00       int H(ptr p, ptr i);
>>>>>>>>> 01       int D(ptr p)
>>>>>>>>> 02       {
>>>>>>>>> 03         int Halt_Status = H(p, p);
>>>>>>>>> 04         if (Halt_Status)
>>>>>>>>> 05           HERE: goto HERE;
>>>>>>>>> 06         return Halt_Status;
>>>>>>>>> 07       }
>>>>>>>>> 08
>>>>>>>>> 09       int main()
>>>>>>>>> 10       {
>>>>>>>>> 11         H(D,D);
>>>>>>>>> 12         return 0;
>>>>>>>>> 13       }
>>>>>>>>>
>>>>>>>>> When Ĥ is applied to ⟨Ĥ⟩
>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
>>>>>>>>> Ĥ.q0 ⟨Ĥ⟩ ⊢* embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
>>>>>>>>>
>>>>>>>>> *Formalizing the Linz Proof structure*
>>>>>>>>> ∃H  ∈ Turing_Machines
>>>>>>>>> ∀x  ∈ Turing_Machines_Descriptions
>>>>>>>>> ∀y  ∈ Finite_Strings
>>>>>>>>> such that H(x,y) = Halts(x,x)
>>>>>>>>
>>>>>>>> But since for x being the description of the H^ built from that 
>>>>>>>> H and y being the same, it turns out that no matter what answer 
>>>>>>>> H gives, it will be wrong.
>>>>>>>>
>>>>>>>
>>>>>>> We have not gotten to that point yet this post is so that
>>>>>>> you can fully understand what templates are and how they work.
>>>>>>
>>>>>> But note, x, being a Turing Machine, is NOT a "template"
>>>>>>
>>>>>> And H, isn't a "set of Turing Machines", but an arbitrary member 
>>>>>> of that set, so all we need to do is find a single x, y, possible 
>>>>>> determined as a function of H (so, BUILT from a template, but not 
>>>>>> a template themselves) that shows that particular H was wrong.
>>>>>>
>>>>>>
>>>>>> That is basically what Linz does.
>>>>>>
>>>>>> Given a SPECIFIC (but arbitary) H, we can construct a specific H^ 
>>>>>> built from a template from H, that that H can not get right.
>>>>>>
>>>>>> All the other H's might get this input right, but we don't care, 
>>>>>> we have shown that for every H we
>>>>>>
>>>>>>>
>>>>>>>> (And I think you have an error in your reference to Halts, I 
>>>>>>>> think you mean Halts(x,y) not Halts(x,x)
>>>>>>>>
>>>>>>>
>>>>>>> Yes good catch. I was trying to model embedded_H / ⟨Ĥ⟩
>>>>>>> and then changed my mind to make it more general.
>>>>>>>
>>>>>>>>>
>>>>>>>>> *Here is the same thing applied to H/D pairs*
>>>>>>>>> ∃H ∈ C_Functions
>>>>>>>>> ∀D ∈ x86_Machine_Code_of_C_Functions
>>>>>>>>> such that H(D,D) = Halts(D,D)
>>>>>>>>
>>>>>>>> Not the same thing.
>>>>>>>> ∃H ∈ C_Functions
>>>>>>>> is not equivalent to
>>>>>>>> ∃H  ∈ Turing_Machines
>>>>>>>>
>>>>>>>> as there are many C_Functions that are not the equivalent of 
>>>>>>>> Turing Machines.
>>>>>>>>
>>>>>>>
>>>>>>> The whole purpose here is to get you to understand what
>>>>>>> templates are and how they reference infinite sets.
>>>>>>>
>>>>>>
>>>>>> But the problem is that even in your formulation, H and D are, 
>>>>>> when doing the test, SPECIFIC PROGRAMS and not "templates" as 
>>>>>> Halts is defined on the domain of PROGRAMS.
>>>>>>
>>>>>> Similarly, a "Template" doesn't have a specific set of 
>>>>>> x86_Machine_Code_of_C_function, at least not one with defined 
>>>>>> behavior since if it tries to reference code outside of itself, 
>>>>>> then Halts of that just isn't defined, only Halts of that code + 
>>>>>> the specific machine deciding it.
>>>>>>
>>>>>>>>
>>>>>>>>>
>>>>>>>>> In both cases infinite sets are examined to see
>>>>>>>>> if any H exists with the required properties.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Yes, but the logic of Turing Machines looks at them one at a 
>>>>>>>> time, and the input is a FULL INDEPENDENT PROGRAM.
>>>>>>>>
>>>>>>>
>>>>>>> ∃H  ∈ Turing_Machines
>>>>>>> That does not look at one machine it looks as an infinite set of
>>>>>>> machines. I am very happy to find out that you were not playing head
>>>>>>> games. Linz actually used the words that you referred to.
>>>>>>
>>>>>> while the ∃H part can create a set of machines, each element of 
>>>>>> that set is INDIVIDUALLY TESTED in the following conditions, so, 
>>>>>> when we get to your test  H(x,y) = Halts(x,x), each of H, x, y are 
>>>>>> individual members of the set, and we THEN collect the set of all 
>>>>>> of them.
>>>>>>
>>>>>> If we try to say
>>>>>> ∃x ∈ Natural Numbers, such that  x+x = 3
>>>>>> we can't say that x is both 1 and 2 and thus as a set meet the 
>>>>>> requirement. For the conditions, each qualifier select a single 
>>>>>> prospective element, and those are tested to see if that meet the 
>>>>>> requirement.
>>>>>>
>>>>>
>>>>> So it never was about any specific machine as Linz misleading words
>>>>> seemed to indicate. It was always about examining each element of an
>>>>> infinite set.
>>>>>
>>>>> Likewise: ∃H ∈ C_Functions is about examining each element
>>>>> of an infinite set. A program template specifies a set of programs
>>>>> the same way that an axiom schema specifies a set of axioms.
>>>>>
>>>>> I am very happy that the issue was the misleading words of Linz
>>>>> and not you playing head games.
>>>>
>>>> In an inderect proof of an unversal claim the counter-hypothesis must
>>>> be about one example. Then the proof is about that specific example
>>>> until a contradiction is derived.
>>>>
>>>
>>> Does there exist at least one example of this when the
>>> infinite set of Turing_Machines have been examined?
>>>
>>>
>>> Of the infinite set of Turing_Machines does there exist
>>> at least one H that always gets this H(x,y) = Halts(x,y)
>>> correctly for every {x,y} pair of the infinite set of {x,y} pairs?
>>
>> Then why was Linz able to create, for any specific H, an H^ that it 
>> get wrong?
>>
>>
>>>
>>> *Formalizing the Linz Proof structure*
>>> ∃H  ∈ Turing_Machines
>>> ∀x  ∈ Turing_Machines_Descriptions
>>> ∀y  ∈ Finite_Strings
>>> such that H(x,y) = Halts(x,y)
>>>
>>
>> And since NO H, can get right the H^ built to contradict IT, that 
>> claim is proven false.
>>
> 
> YOU KEEP TRYING TO GET AWAY WITH CHANGING THE SUBJECT
> THE ABOVE FORMALIZATION IS CORRECT
> 

How?

Since for EVERY H, Linz showed we can create an H^ (which creates the 
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