XPost: comp.theory, sci.logic, sci.math
Because halt deciders are deciders they are only accountable for
computing the mapping their actual input finite strings to an accept or
reject state on the basis of the actual behavior specified by these
actual inputs.
It is like you put a guard on the front door that is supposed to report
anyone coming in the front door (the actual inputs). Then someone comes
in the back door (non inputs) and the guard does not report this.
Since the guard is only supposed to report people coming in the front
door (actual inputs) it is incorrect to say that the guard made a
mistake by not reporting people that came in the back door (non inputs).
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
The copy of Linz H at Ĥ.qx (embedded_H) determines the halt status of
its input on the basis of whether or not the pure simulation of any
finite number of steps of this input can possibly ever reach a final
state of this simulated input.
When embedded_H correctly determines that its simulated input would
never reach its final state it aborts its input and transitions to Ĥ.qn.
When this causes Ĥ applied to ⟨Ĥ⟩ to halt that makes no difference because the guard is only accountable for watching the front door.
https://www.researchgate.net/publication/358009319_Halting_problem_undecidability_and_infinitely_nested_simulation_V3
--
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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