Concise refutation of halting problem proofs V22 [ precisely defined sets ]

[olcott]

#include <stdint.h>
#include <stdio.h>
typedef int (*ptr)();

int H(ptr x, ptr y)
{
x(y); // direct execution of P(P)
return 1;
}

// Minimal essence of Linz(1990) Ĥ
// and Strachey(1965) P
int P(ptr x)
{
H(x, x);
return 1; // Give P a last instruction at the "c" level
}

int main(void)
{
H(P, P);
}

Computation that halts
a computation is said to halt whenever it enters a final state.
(Linz:1990:234)

PSR set: Combinations of H/P having pathological self-reference
Every H of H(P,P) invoked from main() where P(P) calls this same H(P,P)
and H simulates or executes its input and aborts or does not abort its
input P never reaches its last instruction.

PSR subset: Because we know that the input to H(P,P) never halts for the
whole PSR set and a subset of these H/P combinations aborts the
execution or simulation of its input then we know that for this entire
PSR subset the input to H(P,P) never halts and H(P,P) halts.

When int main(void) { P(P); } is invoked on H/P elements of the above
PSR subset, then we have cases where the input to H(P,P) never halts and
P(P) halts. The fact that the input to H(P,P) never halts is not
contradicted by the fact that P(P) halts.

Decidable_PSR subset: The subset of the PSR subset where H returns 0 on
the basis that H correctly detects that P specifies infinite recursion
defines the decidable domain of function H.

Halt decider (Olcott 2021)
Function H maps elements of its domain D to {0,1}.
Domain D is comprised of elements that specify a sequence of
configurations.
H maps elements E of D to {0,1} on the basis of whether or not E reaches
its final state.

The above H could detect that its simulated P is calling H(P,P) with the
same parameters that it was called with, thus specifying infinite
recursion.


Halting problem undecidability and infinitely nested simulation V2

https://www.researchgate.net/publication/356105750_Halting_problem_undecidability_and_infinitely_nested_simulation_V2)


--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer