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computers / comp.ai.philosophy / Concise refutation of halting problem proofs V56 [ key essence ]

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* Concise refutation of halting problem proofs V56 [ key essence ]olcott
`- Re: Concise refutation of halting problem proofs V56 [ key essence ]olcott

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Subject: Concise refutation of halting problem proofs V56 [ key essence ]
From: olcott
Newsgroups: comp.theory, comp.ai.philosophy, sci.logic, sci.math
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Date: Sat, 29 Jan 2022 23:51 UTC
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Subject: Concise refutation of halting problem proofs V56 [ key essence ]
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Correctly answering this single question refutes the Linz proof:
Which state does Ĥ applied to ⟨Ĥ⟩ transition to correctly ?


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 there are no finite number of steps of the pure simulation of ⟨Ĥ⟩ applied to ⟨Ĥ⟩ by embedded_H such that this simulated input meets the Linz definition of halting:

computation that halts … the Turing machine will halt whenever it enters a final state. (Linz:1990:234)

Therefore it is correct to say that the input to embedded_H specifies a sequence of configurations that never halts.

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


Subject: Re: Concise refutation of halting problem proofs V56 [ key essence ]
From: olcott
Newsgroups: comp.theory, comp.ai.philosophy, sci.logic, sci.math
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Date: Sun, 30 Jan 2022 01:08 UTC
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Subject: Re: Concise refutation of halting problem proofs V56 [ key essence ]
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On 1/29/2022 5:51 PM, olcott wrote:
Correctly answering this single question refutes the Linz proof:
Which state does Ĥ applied to ⟨Ĥ⟩ transition to correctly ?


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.⊢* ∞
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

Because there are no finite number of steps of the pure simulation of ⟨Ĥ⟩ applied to ⟨Ĥ⟩ by embedded_H such that this simulated input meets the Linz definition of halting:

computation that halts … the Turing machine will halt whenever it enters a final state. (Linz:1990:234)

// Halting defined
Mathematically this is known as existential quantification:

∃n ∈ N number of simulated steps such that:
embedded_H simulates ⟨Ĥ⟩ applied to ⟨Ĥ⟩ ⊢* ⟨Ĥ⟩.qy or
embedded_H simulates ⟨Ĥ⟩ applied to ⟨Ĥ⟩ ⊢* ⟨Ĥ⟩.qn


Therefore it is correct to say that the input to embedded_H specifies a sequence of configurations that never halts.



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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