XPost: comp.theory, sci.logic, sci.math

I have reformulated my system so that it correctly handles this.

// Full Linz Ĥ as C/x86
void P(ptr x)
{
ptr y = copy(x);
if (H(x, y))
HERE: goto HERE;
}

Because it no longer relies on machine addresses H1(P,P) == H(P,P).

[Olcott 2021 generic halt deciding principle] Whenever the pure
simulation of the input to simulating halt decider H(x,y) never stops
running unless H aborts its simulation H correctly aborts this
simulation and returns 0 for not halting.


Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
If the UTM simulation of the input to Ĥ.qx ⟨Ĥ⟩ applied to ⟨Ĥ⟩ reaches
its own final state.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
If the pure simulation of the input to Ĥqx ⟨Ĥ⟩ ⟨Ĥ⟩ would never reach its
final state (whether or not this simulation is aborted) then it is
necessarily true that Ĥqx transitions to Ĥ.qn correctly.



Halting problem undecidability and infinitely nested simulation V2

https://www.researchgate.net/publication/356105750_Halting_problem_undecidability_and_infinitely_nested_simulation_V2

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

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XPost: comp.theory, sci.logic, sci.math

On 12/8/2021 4:16 PM, Ben Bacarisse wrote:

olcott <NoOne@NoWhere.com> writes:

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
If the UTM simulation...

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
If the pure simulation...

No. As you know, the correct annotations for those lines are:

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
if Ĥ applied to ⟨Ĥ⟩ halts, and

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
if Ĥ applied to ⟨Ĥ⟩ does not halt

and the usual conclusion follows immediately form these facts.

Actually even Linz got that incorrectly.
Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ is not deciding whether or not Ĥ (and thus itself) halts it
is deciding whether or not ⟨Ĥ⟩ applied to ⟨Ĥ⟩ would halt.

When we further hypothesize the the original H was a simulating halt
decider then we get:

Ĥ applied to ⟨Ĥ⟩ essentially specifies infinite recursion that is
aborted after the first invocation of this infinite recursion.

Because it is not aborted at the first invocation we get the confusing
case where Ĥ halts yet its own simulation of itself must be aborted.

We don't have this issue with H. H applied to ⟨Ĥ⟩ ⟨Ĥ⟩ aborts this infinite recursion at its first invocation.

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer

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