• Concise refutation of halting problem proofs V44 [honest dialogue]

    From olcott@21:1/5 to All on Sun Jan 2 22:06:20 2022
    XPost: comp.theory, sci.logic, sci.math

    For simplicity we will refer to the copy of Linz H at Ĥ.qx embedded_H.

    Simplified syntax adapted from bottom of page 319:
    Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
    Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

    Instead of defining what embedded_H actually does we are defining the
    set of behaviors such that every embedded_H having these behaviors is necessarily correct.

    It can be easily seen that the simulated input to embedded_H never
    reaches the final state of this input no matter how many steps are
    simulated.

    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)

    That a simulated input cannot possibly reach it final state conclusively
    proves that this input never halts.

    This proves that every simulating halt decider at Ĥ.qx that somehow
    computes the mapping from its input ⟨Ĥ⟩ ⟨Ĥ⟩ to Ĥ.qn it is necessarily
    correct.



    Peter Linz HP Proof
    https://www.liarparadox.org/Peter_Linz_HP_315-320.pdf

    --
    Copyright 2021 Pete Olcott

    Talent hits a target no one else can hit;
    Genius hits a target no one else can see.
    Arthur Schopenhauer

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)