General Undecidable Rule (V4) General Public Domain

[wij]

Copyright General Public Domain by WIJ 2021
===============================

Since the conventional HP only mentions a specific halting problem, which is
often believed to be an invalid proof and limited in use.
See https://groups.google.com/g/comp.theory/c/RO9Z9eCabeE/m/Ka8-xS2rdEEJ

I hereby proudly announce the General Undecidable Rule (2021 WIJ):
+-----------------------------------------------------------------------------------+
| No TM U can decide the property of a TM P if that property can be defied by TM P. |
+-----------------------------------------------------------------------------------+

//---------------------------
// Example 1: U is a function
//
// [Syn] bool U(void (*f)())
// U decides whether f prints character 'Y' or not.
//
// [Ret] true: f prints 'Y'
// false: f does not print 'Y'
//
inline bool U(void (*f)()) {
// Rewrite the supposed function U here.
// The rewrite can be in very different but functionally equivalent ways, isomorphic or not..
}

void P() { // P always prints a character not predicted by U
if(U(P)) {
printf("b");
} else {
printf("Y");
}
}

//------------------------------
// Example 2: U is an executable
/*
[Syn] U <prog>
U Decides whether the program <prog> will return or not,

[Exit Status] non-zero: <prog> will return (Or, TM stops at final accept or reject state)
zero: otherwise (<prog> will not return)
*/

/*
Program: P.c

Build: gcc -o P P.c
*/
#include <stdlib.h>

void P() { // P always behaves not predicted by U
int r=system("U P");

if(r) {
for(;;) {}; // infinite loop
}
};

int main() {
P();
};

==============================
The general construct of P (proof of General Undecidable Rule) is intuitive
and above all, REPRODUCIBLE, VERIFIABLE.

// [Ret] true: f has the (dynamic)property Q
// false: otherwise
//
typedef void (*Func)();
inline bool U(Func f) {
// Rewrite U here.
// The rewrite can be in very different but functionally equivalent ways, isomorphic or not..
}

void P() {
if(U(P)) {
// do whatever makes Q wrong
} else {
// do whatever Q defines
}
};

+-------------+
| Acknowledge |
+-------------+
I would like to acknowledge Olcott tirelessly refuted various wrong
conventional HP proofs over these years for me. So I need not to do the
same work again, though not necessary.

In Example 2, a halting decider U can not return correctly. TM in such a state is
referred to as "undecidable" by conventional HP proof. The notion "undecidable" conveys is
crucial but I do not know who to attribute.

+-------------+
| Post Script |
+-------------+
GUR is an improvised work to shorten discussions in Google forum comp.theory. I find it
particularly useful to refute various kind of telepathic programs and pathological logic,
if compared with using the conventional HP proofs (no blame here, the conventional HP proofs
were made earlier than what we now know as 'computer').

I think I had brought down the faulty equality of limit and Cantor's theory(partly) and the
mathematics based on them, e.g. 0.999...=1. But I keep this to myself, I am not content with
the destructive result, but the constructive side which I am working hard to make, although
I am not a mathematician.

[wij]

Copyright correction: Public Domain