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

Halting problem undecidability and infinitely nested simulation (V3)

A simulating halt decider H bases its halt status decision on what the
behavior of its input would be if H was a UTM.

H.q0 Wm W ⊢* H.qy if UTM Wm W halts
H.q0 Wm W ⊢* H.qn if UTM Wm W does not halt

H determines that its simulated input would never reach its final state
on the basis of matching infinite behavior patterns. In this case H
aborts its simulation and transitions to its final reject state.
Otherwise H transitions to its accept state when its simulation ends.

The following simplifies the syntax for the definition of the Linz
Turing machine Ĥ, it is now a single machine with a single start state.
A copy of Linz H is embedded at Ĥ.qx.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

When Ĥ is applied to ⟨Ĥ⟩
Ĥ copies its input ⟨Ĥ1⟩ to ⟨Ĥ2⟩ then embedded_H simulates ⟨Ĥ1⟩ ⟨Ĥ2⟩

Then these steps would keep repeating:
Ĥ1 copies its input ⟨Ĥ2⟩ to ⟨Ĥ3⟩ then embedded_H simulates ⟨Ĥ2⟩ ⟨Ĥ3⟩
Ĥ2 copies its input ⟨Ĥ3⟩ to ⟨Ĥ4⟩ then embedded_H simulates ⟨Ĥ3⟩ ⟨Ĥ4⟩
Ĥ3 copies its input ⟨Ĥ4⟩ to ⟨Ĥ5⟩ then embedded_H simulates ⟨Ĥ4⟩ ⟨Ĥ5⟩...

This shows that the simulated input to embedded_H ⟨Ĥ⟩ ⟨Ĥ⟩ would never reach its final state conclusively proving that this simulated input
never halts. This enables embedded_H to abort the simulation of its
input and correctly transition to Ĥ.qn.

Because all simulating halt deciders are deciders they are only
accountable for computing the mapping from their input finite strings to
an accept or reject state on the basis of whether or not their correct simulation of this input could ever reach its final state.

embedded_H is only accountable for the behavior of its input ⟨Ĥ⟩ applied to ⟨Ĥ⟩. embedded_H is not accountable for the behavior of the
computation that it is contained within: Ĥ applied to ⟨Ĥ⟩.

Halting problem undecidability and infinitely nested simulation (V3) 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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