When a simulating halt decider rejects all inputs as non-halting
whenever it correctly detects that its correct and complete simulation
of its input would never reach the final state of this input then all
[these] inputs (including pathological inputs) are decided correctly.
*computation that halts* … the Turing machine will halt whenever it
enters a final state. (Linz:1990:234)
Linz, Peter 1990. An Introduction to Formal Languages and Automata.
Lexington/Toronto: D. C. Heath and Company. (317-320)
typedef void (*ptr)();
void P(ptr x)
if (H(x, x))
HERE: goto HERE;
Output("Input_Halts = ", H(P, P));
(01) 55 push ebp
(02) 8bec mov ebp,esp
(03) 8b4508 mov eax,[ebp+08]
(01) 50 push eax // push P
(03) 8b4d08 mov ecx,[ebp+08]
[0000135c](01) 51 push ecx // push P
[0000135d](05) e840feffff call 000011a2 // call H
(03) 83c408 add esp,+08
(02) 85c0 test eax,eax
(02) 7402 jz 0000136b
(02) ebfe jmp 00001369
[0000136b](01) 5d pop ebp
[0000136c](01) c3 ret
Size in bytes:(0027) [0000136c]
(1) It is an easily verified fact that when we assume that H is only an
x86 emulator that the correctly emulated P never reaches its "ret"
instruction it remains stuck in repeated cycles of emulation.
(2) It is an easily verified fact that if H has been adapted to
correctly detect (in a finite number of steps) that the correct and
complete x86 emulation of its input would never each its "ret"
instruction that H could abort its emulation and return 0 to report this.
(3) When the halt status criteria is defined as correctly determining
whether or not an x86 emulated input would ever reach its "ret"
instruction then it becomes an easily verified fact H(P,P) could
correctly reject its input as non-halting.
Correct deductive inference proves that all of these things are true
without any need what-so-ever to see either the source-code or the
execution trace of H.
The one thing that is not proved is whether or not an actual encoded
H(P,P) does indeed correctly determine that its input would never reach
its "ret" instruction as a pure function of its inputs. This aspect will
be confirmed by fully operational source-code.
Halting problem undecidability and infinitely nested simulation (V5)
Copyright 2022 Pete Olcott
"Talent hits a target no one else can hit;
Genius hits a target no one else can see."