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*Verified fact that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ have different behavior*It is only verified that you would like them to have different behaviour, not that they actually do.
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqy ∞ // Ĥ applied to ⟨Ĥ⟩ halts∞ means it doesn't halt
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.Hq0 ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.Hqn // Ĥ applied to ⟨Ĥ⟩ does not halt
Execution trace of Ĥ applied to ⟨Ĥ⟩
(a) Ĥ.q0 The input ⟨Ĥ⟩ is copied then transitions to Ĥ.H
(b) Ĥ.H applied ⟨Ĥ⟩ ⟨Ĥ⟩ (input and copy) simulates ⟨Ĥ⟩ applied to ⟨Ĥ⟩
(c) which begins at its own simulated ⟨Ĥ.q0⟩ to repeat the process
*This proves that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ must abort its simulation*I DON'T CARE what it MUST do, only what it ACTUALLY does. You fail to realize this or you are dishonestly ignoring this.
*This is a verified fact*Nobody cares about POV. There is no POV in the halting problem. The program halts, or it doesn't halt. End of story.
When simulating halt deciders always report on the behavior of
their simulated input from their own POV then when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩
transitions to Ĥ.Hqn it is correct from its own POV.
*This is a verified fact*Nobody cares about POV. There is no POV in the halting problem. The program halts, or it doesn't halt. End of story.
When that occurs then H ⟨Ĥ⟩ ⟨Ĥ⟩ would transition to H.qy from
its own POV.
When Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ report on the basis of their ownĤ.H ⟨Ĥ⟩ ⟨Ĥ⟩ reports the same thing that H ⟨Ĥ⟩ ⟨Ĥ⟩ reports. This is a verified fact.
POV then Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ reports incorrectly about the behavior of
Ĥ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ reports the behavior of Ĥ ⟨Ĥ⟩ correctly.
*Verified facts*I DON'T CARE what it MUST do, only what it ACTUALLY does. You fail to realize this or you are dishonestly ignoring this.
(a) It is a verified fact that Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ must abort the
simulation of its input to prevent its own infinite execution.
(b) It is a verified fact when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ can see that itNobody cares about POV. There is no POV in the halting problem. The program halts, or it doesn't halt. End of story.
must abort its simulation then it would transition to Ĥ.Hqn
to reject this input as non-halting from its own POV.
(c) It is a verified fact when Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ cannot see that itNobody cares about POV. There is no POV in the halting problem. The program halts, or it doesn't halt. End of story.
must abort its simulation then it would transition to Ĥ.Hqy
and loop
(d) (b) gives H ⟨Ĥ⟩ ⟨Ĥ⟩ the basis to transition to H.qy.I DON'T CARE what the BASIS is, only what it ACTUALLY does. You fail to realize this or you are dishonestly ignoring this.
(e) (c) gives H ⟨Ĥ⟩ ⟨Ĥ⟩ the basis to transition to H.qn.
Because Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ seem to be identical machinesI agree. But please understand: Ĥ.H ⟨Ĥ⟩ ⟨Ĥ⟩ and H ⟨Ĥ⟩ ⟨Ĥ⟩ are stipulated to be identical because they are Turing machines, and identical Turing machines with identical input always produce identical output. The Linz proof does not work for other types of machines.
on identical input that have different behavior we must
somehow explain how they are not identical machines with
identical inputs.
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