• Concise refutation of halting problem proofs V50 [ Linz Proof ]

    From olcott@21:1/5 to All on Sat Jan 22 09:01:06 2022
    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 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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