HALTING PROBLEM – given a syntactically correct program P and an input string of
characters X. (X may or may not be a valid input for P – it doesn't matter)
oQuestion – does P halt when given X?
oP may loop forever or halt
So, is there an algorithm for the Halting Problem?
By way of contradiction, we assume that there exists an algorithm, here called Q,
which examines (P, X) to report if P halts or does not halt on input X:
input: string X and program P (Note: P is just another
string of characters to Q.)
Q:
P halts when
given X
…Now, build another program, Q', by changing 2 statements of Q:
Lemma 1. If Q exists, then Q' exists
P halts on X if and only if Q prints "P halts on X";
P halts on X if and only if Q' loops forever;
P does not halt on X if and only if Q' halts on (P, X)
1
P X
HAL
T
P doesn’t
halt when
given X
P X
HAL
T
If P doesn’t
halt on X
then (no
op)
If P halts on
X then while
9 < 10 do
End (loops)
