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

We are proposing that when an input P has an embedded simulating halt
decider H we can replace H with Simulator S defining proxy input P2.

When the original simulating halt decider H decides halting on the proxy
input P2 this decision can be correctly used as a proxy decision for the
original input.

The following analyzes why we can replace the invocations of halt
decider H embedded in input P with a simulator S defining input S2 as
the proxy halt deciding basis for deciding P.

The following must be analyzed at the architectural level of detail
ignoring how an embedded simulating halt decider is recognized and
replaced. These issues become moot at the next level of analysis.

(a) Simulating halt decider H and simulator S are equivalent
computations for all inputs P that halt.

(b) Simulating halt decider H and simulator S are equivalent
computations for all inputs P that do not halt up to the point where H
stops simulating P.

(c) When we replace embedded halt decider H in input P with embedded
simulator S defining proxy input P2 and the simulation of P2 halts
because of (a) we can know that H would decide halting on P.

(d) When we replace embedded halt decider H in input P with an embedded
simulator S defining proxy input P2 and the simulation of P2 does not
halt because of (b) we can know that H would decide not halting on P.

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

Constantly updated:
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

*Halting problem undecidability and infinitely nested simulation*

When input P to simulating halt decider H has the pathological
self-reference error (PSRE) simulating halt decider H decides this input
P on the basis of proxy input P2 that has the pathological
self-reference error removed.

void H_Hat(u32 P)
{ u32 Input_Halts = Halts(P, P);
if (Input_Halts)
HERE: goto HERE;
}

The pathological self-reference error arises in the halting theorem from
the fact that both return values: {0, 1} from Halts() to H_Hat() are
incorrect when H_Hat() is invoked with its own machine address as input.

The above input is defined to immediately halt if the halt decider
decides that it will never halt and to run forever when the halt decider
decides that it will halt.

In computability theory and computational complexity theory, a decision
problem is a problem that can be posed as a yes-no question of the input
values. (Wikipedia: Decision problem)

Every decision problem: Question/Input lacking a correct yes/no return
value is incorrect.

A yes or no question lacking a correct yes or no answer is an incorrect
question. When an incorrect question is translated into a correct
equivalent question then the original question can be answered by its
equivalent proxy question.

When define H_Hat2() by replacing the call to Halts() with a call to
Simulate()

u32 Simulate(u32 P, u32 I)
{ ((void(*)(int))P)(I);
return 1;
}

H_Hat2() never halts on its own machine address as input.
This key fact is leveraged to correctly decide halting:
Halts((u32)H_Hat, (u32)H_Hat);

The concept of proxy inputs to a simulating halt decider having inputs
with the pathological self-reference error (PSRE) is a teaching device.
This teaching device is used to prove that another equivalent process
does correctly decide the halting status of these original inputs.

The first step of beginning to understand proxy inputs is understanding
this hypothetical situation: If there is a decidable equivalent proxy
input P2 to undecidable decision problem Question/Input:P pair then
deciding this proxy input P2 is equivalent to deciding the original
input P.

The following will analyze the generic notion of proxy inputs P2 to
undecidable halting problem instances thus showing that H_Hat2() is such
a proxy input for input H_Hat().

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

We are proposing that when an input P has an embedded simulating halt
decider H we can replace H with Simulator S defining proxy input P2.

When the original simulating halt decider H decides halting on the proxy
input P2 this decision can be correctly used as a proxy decision for the
original input.

The following analyzes why we can replace the invocations of halt
decider H embedded in input P with a simulator S defining input S2 as
the proxy halt deciding basis for deciding P.

The following must be analyzed at the architectural level of detail
ignoring how an embedded simulating halt decider is recognized and
replaced. These issues become moot at the next level of analysis.

(a) Simulating halt decider H and simulator S are equivalent
computations for all inputs P that halt.

(b) Simulating halt decider H and simulator S are equivalent
computations for all inputs P that do not halt up to the point where H
stops simulating P.

(c) When we replace embedded halt decider H in input P with embedded
simulator S defining proxy input P2 and the simulation of P2 halts
because of (a) we can know that H would decide halting on P.

(d) When we replace embedded halt decider H in input P with an embedded
simulator S defining proxy input P2 and the simulation of P2 does not
halt because of (b) we can know that H would decide not halting on P.

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

--
Copyright 2021 Pete Olcott

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