Re: D correctly simulated by H cannot possibly halt --- templates and infinite sets

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.

(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.

 

>
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.

I'm not sure what you can define your computation system to be actually based on, and what its supposed use is, since your 'decider' and 'input' are so intertwined.
 

The whole purpose here is to get you to understand what
templates are and how they reference infinite sets.

And your supposed algorithm just doesn't work when you try to make you system "Turing Complete" by letting D have the ability to have a COPY of H, and being able to make copies of its input, like real Turing machines can.
 

The whole purpose here is to get you to understand what
templates are and how they reference infinite sets.
All the other issues are for another different post.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer