XPost: comp.theory, sci.logic, sci.math
Proving that embedded_H at Ĥ.qx correctly maps its inputs ⟨Ĥ⟩ ⟨Ĥ⟩ to Ĥ.qn on the basis of the behavior of the UTM simulation of these inputs.
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
*My criterion measure with Ben's notational conventions*
H.q0 wM w ⊢* H.qy iff UTM(wM, w) halts
H.q0 wM w ⊢* H.qn iff UTM(wM, w) does not halt
We know that H would correctly decide the halt status of its input on
the basis of correctly deciding the halt status of the UTM simulation of
its input.
We know this because a UTM simulation of the Turing machine description
is computationally equivalent to the direct execution of this same
Turing machine.
HERE IS WHERE PEOPLE GET STUCK
HERE IS WHERE PEOPLE GET STUCK
HERE IS WHERE PEOPLE GET STUCK
We know that the copy of H is at Ĥ.qx (AKA embedded_H) applies this same criterion measure to every instance of embedded_H to any recursive depth.
If the UTM simulation of the input to any embedded_H would never stop
running without being aborted by this embedded_H then this embedded_H
has met its non-halting criteria.
https://www.liarparadox.org/Peter_Linz_HP_315-320.pdf
Linz, Peter 1990. An Introduction to Formal Languages and Automata. Lexington/Toronto: D. C. Heath and Company. (315-320)
--
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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