Generic halt deciding principle for inputs having the pathological
self-reference error

(a) Halt deciding simulator H only affects the behavior of input P in
one way and only under one condition. In all other cases a halt deciding
simulator H has no effect what-so-ever on the behavior of input P and
acts exactly like pure simulator S on input P.

(b) When an infinite computation is specified as input P to a simulating
halt decider H the halt decider H must stop the simulation of this input
P. When this same input is simulated by simulator S, its simulation
never stops.

(c) When a finite computation is specified as input P to a simulating
halt decider H the halt decider H simulates its input exactly as if H
was simulator S.

Because of the above simulating halt decider H can decide input P having
the pathological self-reference error on the basis of the proxy input P2
having the pathological self-reference error removed.

When input P invokes embedded simulating halt decider H on itself H
creates proxy input P2 replacing H with simulator S. The embedded halt
decider H examines the execution trace of proxy input P2. The embedded
halt decider correctly decides its input P on the basis of how it
decides input P2.

When we apply *the Generic halt deciding principle* to the embedded
simulating halt decider at state Ĥ.qx to its input ([Ĥ],[Ĥ]) we find
that the simulation of this input must be aborted.

Because the necessity to abort the simulation of an input is equated
with this input specifying an infinite computation the simulating halt
decider Ĥ.qx correctly transitions to its final Ĥ.qn state deciding not
halting on its input.

http://www.liarparadox.org/Halting_problem_undecidability_and_infinitely_nested_simulation.pdf

--
Copyright 2021 Pete Olcott

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

On 5/23/2021 2:08 PM, olcott wrote:
> Generic halt deciding principle for inputs having the pathological
> self-reference error
>
> (a) Halt deciding simulator H only affects the behavior of input P in
> one way and only under one condition. In all other cases a halt deciding
> simulator H has no effect what-so-ever on the behavior of input P and
> acts exactly like pure simulator S on input P.
>
> (b) When an infinite computation is specified as input P to a simulating
> halt decider H the halt decider H must stop the simulation of this input
> P. When this same input is simulated by simulator S, its simulation
> never stops.
>
> (c) When a finite computation is specified as input P to a simulating
> halt decider H the halt decider H simulates its input exactly as if H
> was simulator S.
>
> Because of the above simulating halt decider H can decide input P having
> the pathological self-reference error on the basis of the proxy input P2
> having the pathological self-reference error removed.
>
> When input P invokes embedded simulating halt decider H on itself H
> creates proxy input P2 replacing H with simulator S.

This is the same thing as H simply excluding its own execution from
halting analysis as was done with Halts((u32)H_Hat, (u32)H_Hat);

> The embedded halt
> decider H examines the execution trace of proxy input P2. The embedded
> halt decider correctly decides its input P on the basis of how it
> decides input P2.
>
> When we apply *the Generic halt deciding principle* to the embedded
> simulating halt decider at state Ĥ.qx to its input ([Ĥ],[Ĥ]) we find
> that the simulation of this input must be aborted.
>
> Because the necessity to abort the simulation of an input is equated
> with this input specifying an infinite computation the simulating halt
> decider Ĥ.qx correctly transitions to its final Ĥ.qn state deciding not
> halting on its input.
>
> http://www.liarparadox.org/Halting_problem_undecidability_and_infinitely_nested_simulation.pdf
>
>

--
Copyright 2021 Pete Olcott

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