XPost: comp.theory, sci.logic, sci.math

The halting theorem counter-examples present infinitely nested
simulation (non-halting) behavior to every simulating halt decider.

-----1-----------2--3----------
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞

If the simulation of the 2nd ⟨Ĥ⟩ applied to
the 3rd ⟨Ĥ⟩ at Ĥ.qx reaches its final state.


-----1-----------2--3----------
Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn

If the simulation of the 2nd ⟨Ĥ⟩ applied to
the 3rd ⟨Ĥ⟩ at Ĥ.qx never reaches its final state.


https://www.researchgate.net/publication/351947980_Halting_problem_undecidability_and_infinitely_nested_simulation


--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein

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

XPost: comp.theory, sci.logic, sci.math

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 simulating halt decider is embedded at Ĥ.qx. It has been annotated so
that it only shows Ĥ applied to ⟨Ĥ⟩, converting the variables to constants.

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
if the simulated input to Ĥ.qx ⟨Ĥ⟩ applied to ⟨Ĥ⟩ reaches its final state. (Halts)

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
if the simulated input to Ĥ.qx ⟨Ĥ⟩ applied to ⟨Ĥ⟩ never reaches its final state. (Does not halt)

https://www.researchgate.net/publication/351947980_Halting_problem_undecidability_and_infinitely_nested_simulation




--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein

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