XPost: comp.theory, sci.logic, sci.math
On 12/30/2021 12:21 PM, André G. Isaak wrote:
On 2021-12-30 09:54, olcott wrote:
On 12/30/2021 10:11 AM, Richard Damon wrote:
On 12/30/21 10:58 AM, olcott wrote:
On 12/30/2021 9:48 AM, Richard Damon wrote:
We don't care if embedded_H is 'Correct', we care that it give the
same answer as H.
If it is correct and gives a different answer than H then it remains
correct even if it gives a different answer than H.
But since it IS a copy of H, then if it gives a different answer than
'H' does, it shows that H fails to be a Computation, and thus Can't
be a Decider, and thus can't be a Halt Decider.
H and embedded_H are able to give a different halt status
determinations on the basis that embedded_H had its copy of H
corrupted by the appended infinite loop.
This means that a finite string comparison performed by H will
determine that embedded_H is not identical to H.
And which two strings are being compared here? H hasn't been provided a string describing itself nor a string describing embedded_H as part of
its input. All it has is a string describing Ĥ. What does it compare
this string to?
André
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
We as humans can see that while embedded_H performs a pure simulation of
its input ⟨Ĥ⟩ ⟨Ĥ⟩ that this simulated input:
copies its input ⟨Ĥ⟩ to ⟨Ĥ⟩ and then invokes embedded_H with this input
over and over without ever stopping.
Since a pure simulation of the input never stops we know that the UTM simulation of this same input never stops because a UTM performs a pure simulation of its input.
Thus we know that the "does not halt" halt status criteria of embedded_H
has been met.
In order for embedded_H to see this it needs to see that an identical
sequence of configurations keeps repeating.
We as humans can see that the input to embedded_H is always identical.
We as humans can see that the copy of embedded_H is always identical.
We as humans can see that these two things can verified by finite string comparison.
That an identical copy of a function is called with identical inputs
proves infinite behavior.
When embedded_H is about to simulate embedded_H with identical inputs
it terminates this simulation as matching the infinitely nested
simulation infinite behavior pattern.
--
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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