Math. 55
The Halting Problem
March 13, 1999 5:26 pm
Prof. W. Kahan Page 1
On pp. 181-2 of our textbook,
Discrete Mathematics etc
., 4th ed., by Kenneth Rosen, the
treatment of the Halting Problem is not so clear as I would like. The following treatment may be
easier to understand. Some strings of characters will be processed as potential programs much as
compilers do it; when a string P is presented to a compiler it produces a compiled program that
the computer can run, or else an error-message to the effect that string P is not valid source-
program text. Other strings will be treated as inputs to programs; when a running compiled
program reads such a string I it produces an output ( or error message ) after running for a finite
time, or else runs forever ( or until stopped by some external agency ).
Can a program H(P, I) be written to accept any two strings and then, in a finite time, tell whether
the program compiled from string P ultimately halts ( perhaps with an error-message ) or runs
forever after reading input string I ? Alan Turing showed why no such program H can exist.
This conclusion saddens us because we often wish to know whether a program P(I) will ever
halt; and we can’t be satisfied by just running it until it halts or until we lose patience and stop it
perhaps prematurely. What we desire is some kind of computation by way of proof that assures us
in advance that P will not run forever after reading I in. But Turing showed why, unless P is
designed with such a proof in mind, no routine way exists to decide whether P(I) halts.
Suppose, for the sake of argument, that such a program H(P, I) did exist; we shall try to infer a
contradiction that will prove no such H exists. The text of H would look something like this:
string function H(string P, string I):
begin
Text of a program that reads string P and checks its validity as a program,
then reads string I and computes whether P(I) would halt.
If (
P is a valid program and P(I) would never halt
),
return(
“Program text runs forever with Input text”
)
else return(
“Program is invalid or Input text lets it halt”
) ;
end program H .
From H a program K(P) could be constructed by copying the italicized text from H after the
substitution of “
P
” for “
I
” and adding a few other changes. The text of K would look like this:
string function K(string P):
begin
Text of a program that reads string P and checks its validity as a program,
then reads string P and computes whether P(P) would halt.
If (
P is a valid program and P(P) would never halt
),
return(
“ Program K(K) halts.”
)
else while true do { run forever } ;
end program K .
Let’s run K(K) , submitting the text of K as input to K . What would happen? If H(K, K)
would have put out “Program is invalid or Input text lets it halt” then K(K) would run forever;
but if H(K, K) would have put out “Program text runs forever with Input text” then K(K)
would return “ Program K(K) halts.” Either way, program H(K, K) would have put out the
wrong message, so no such H(P, I) can predict correctly for
every
program P whether it halts.
