XPost: comp.theory, sci.logic, sci.math
Halting problem undecidability and infinitely nested simulation (V3)
We define Linz H to base its halt status decision on the behavior of its
pure simulation of N steps of its input. If the simulated input cannot
reach its own final state in any finite number of steps then H aborts
the simulation of this input and transitions to H.qn.
H determines this on the basis of matching an infinitely repeating
behavior pattern. The copy of H embedded in Ĥ computes the mapping from
its input ⟨Ĥ⟩ ⟨Ĥ⟩ to Ĥ.qn on the basis of the above criteria.
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
Because it is known that the UTM simulation of a machine is
computationally equivalent to the direct execution of this same machine
H can always form its halt status decision on the basis of what the
behavior of the UTM simulation of its inputs would be.
When embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩ these steps would keep repeating: Ĥ copies its input ⟨Ĥ⟩ to ⟨Ĥ⟩ then embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩...
computation that halts … the Turing machine will halt whenever it enters
a final state. (Linz:1990:234)
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.
if embedded_H does correctly recognize an infinitely repeating behavior
pattern in the behavior of its simulated input: ⟨Ĥ⟩ applied to ⟨Ĥ⟩ then
embedded_H is necessarily correct to abort the simulation of its input
and transition to Ĥ.qn.
A halt decider is a decider thus embedded_H is only accountable for
computing the mapping from ⟨Ĥ⟩ ⟨Ĥ⟩ to Ĥ.qy or Ĥ.qn on the basis of the
behavior specified by these inputs. embedded_H is not accountable for
any other behavior besides the behavior specified by its actual inputs.
Peter Linz Halting Problem Proof
https://www.liarparadox.org/Linz_Proof.pdf
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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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