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computers / comp.ai.philosophy / Concise refutation of halting problem proofs V50 [ Linz Proof ]

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o Concise refutation of halting problem proofs V50 [ Linz Proof ]olcott

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Subject: Concise refutation of halting problem proofs V50 [ Linz Proof ]
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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 infinitely repeating behavior patterns. 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.

Because halt decider is a decider 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. 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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