• Concise refutation of halting problem proofs V53 [ Line Proof ]

    From olcott@21:1/5 to All on Tue Jan 25 09:53:55 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. N is either the number of steps
    that it takes for its simulated input to reach its final state or the
    number of steps required for H to match an infinite behavior pattern
    proving that its simulated input would never reach its own final state
    in any finite number of steps. In this case H aborts the simulation of
    this input and transitions to H.qn.

    Simulating halt deciders never determine whether or not their input
    stops running. Every input to a simulating halt decider always stops
    running either because it reached its final state or its simulation was aborted. Simulating halt deciders determine whether or not the Linz
    criteria can possibly be met … the Turing machine will halt whenever it enters a final state. (Linz:1990:234)

    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 Ĥ applied to ⟨Ĥ⟩ has embedded_H simulate ⟨Ĥ⟩ ⟨Ĥ⟩ these steps would
    keep repeating:
    Ĥ copies its input ⟨Ĥ⟩ to ⟨Ĥ⟩ then embedded_H simulates ⟨Ĥ⟩ ⟨Ĥ⟩...

    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.

    It is the case that if embedded_H recognizes an infinitely repeating
    pattern in the behavior of its simulated input: ⟨Ĥ⟩ applied to ⟨Ĥ⟩ such
    that this correctly simulated input cannot possibly reach its own final
    state then this is complete proof that this simulated input never halts.

    Because a 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
    the behavior of the computation that it is contained within: Ĥ applied
    to ⟨Ĥ⟩ because this is not an actual input to embedded_H.




    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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