This exact same relationship was created as H(P,P), shown below.In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever...
For any program f that might determine if programs halt, a "pathological" program g, called with some input, can pass its own source and its input to f and then specifically do the opposite of what f predicts g will do. No f can exist that handles this case. https://en.wikipedia.org/wiki/Halting_problem
This is the overview of the method for proving that this analysis is correct:
(a) Verify that the execution trace of P by H is correct by comparing this execution trace to the ax86 source-code of P.
(b) Verify that this execution trace shows that P is stuck in infinitely nested simulation (a non-halting behavior).
This proof can only be understood only by those having sufficient technical competence in:
(a) software engineering (recognizing infinite recursion in C and x86 code)
(b) the x86 programming language
(c) the C programming language and
(d) the details of how C is translated into x86 by the Microsoft C compilers.
Code: Select all
#include <stdint.h>
#define u32 uint32_t
void P(u32 x)
{
if (H(x, x))
HERE: goto HERE;
return;
}
int main()
{
Output("Input_Halts = ", H((u32)P, (u32)P));
}
_P()
[00001352](01) 55 push ebp
[00001353](02) 8bec mov ebp,esp
[00001355](03) 8b4508 mov eax,[ebp+08]
[00001358](01) 50 push eax
[00001359](03) 8b4d08 mov ecx,[ebp+08]
[0000135c](01) 51 push ecx
[0000135d](05) e840feffff call 000011a2 // call H
[00001362](03) 83c408 add esp,+08
[00001365](02) 85c0 test eax,eax
[00001367](02) 7402 jz 0000136b
[00001369](02) ebfe jmp 00001369
[0000136b](01) 5d pop ebp
[0000136c](01) c3 ret
Size in bytes:(0027) [0000136c]
_main()
[00001372](01) 55 push ebp
[00001373](02) 8bec mov ebp,esp
[00001375](05) 6852130000 push 00001352 // push P
[0000137a](05) 6852130000 push 00001352 // push P
[0000137f](05) e81efeffff call 000011a2 // call H
[00001384](03) 83c408 add esp,+08
[00001387](01) 50 push eax
[00001388](05) 6823040000 push 00000423 // "Input_Halts = "
[0000138d](05) e8e0f0ffff call 00000472 // call Output
[00001392](03) 83c408 add esp,+08
[00001395](02) 33c0 xor eax,eax
[00001397](01) 5d pop ebp
[00001398](01) c3 ret
Size in bytes:(0039) [00001398]
machine stack stack machine assembly
address address data code language
======== ======== ======== ========= =============
...[00001372][0010229e][00000000] 55 push ebp
...[00001373][0010229e][00000000] 8bec mov ebp,esp
...[00001375][0010229a][00001352] 6852130000 push 00001352 // push P
...[0000137a][00102296][00001352] 6852130000 push 00001352 // push P
...[0000137f][00102292][00001384] e81efeffff call 000011a2 // call H
Begin Local Halt Decider Simulation Execution Trace Stored at:212352
...[00001352][0021233e][00212342] 55 push ebp // enter P
...[00001353][0021233e][00212342] 8bec mov ebp,esp
...[00001355][0021233e][00212342] 8b4508 mov eax,[ebp+08]
...[00001358][0021233a][00001352] 50 push eax // push P
...[00001359][0021233a][00001352] 8b4d08 mov ecx,[ebp+08]
...[0000135c][00212336][00001352] 51 push ecx // push P
...[0000135d][00212332][00001362] e840feffff call 000011a2 // call H
...[00001352][0025cd66][0025cd6a] 55 push ebp // enter P
...[00001353][0025cd66][0025cd6a] 8bec mov ebp,esp
...[00001355][0025cd66][0025cd6a] 8b4508 mov eax,[ebp+08]
...[00001358][0025cd62][00001352] 50 push eax // push P
...[00001359][0025cd62][00001352] 8b4d08 mov ecx,[ebp+08]
...[0000135c][0025cd5e][00001352] 51 push ecx // push P
...[0000135d][0025cd5a][00001362] e840feffff call 000011a2 // call H
Local Halt Decider: Infinite Recursion Detected Simulation Stopped
Code: Select all
...[00001384][0010229e][00000000] 83c408 add esp,+08
...[00001387][0010229a][00000000] 50 push eax
...[00001388][00102296][00000423] 6823040000 push 00000423 // "Input_Halts = "
---[0000138d][00102296][00000423] e8e0f0ffff call 00000472 // call Output
Input_Halts = 0
...[00001392][0010229e][00000000] 83c408 add esp,+08
...[00001395][0010229e][00000000] 33c0 xor eax,eax
...[00001397][001022a2][00100000] 5d pop ebp
...[00001398][001022a6][00000004] c3 ret
Number of Instructions Executed(15892) lines = 237 pages
https://www.researchgate.net/publicatio ... ulation_V5