Algorithm Design and Analysis 4/3/2008
Lecture 31: Halting problem and reductions
Instructor: Mike Kowalczyk
1 Halting problem
Consider the function programHalts in Halt.java. Is it possible to write a correct implementation?
After several attempts, it appears to be impossible, but can we prove it? Assuming that we have
a correct implementation of programHalts, then we could also write the method thisProgramHalts,
and then write our main program to do the opposite of whatever our program supposedly was going
to do. Intuitively, such a function would allow us to predict the future outcome of our program as
it is still running! Let’s reason this more carefully:
If the program Halt.java halts, then the if statment evaluates to true, and then the program
infinite loops, so Halt.java doesn’t halt. But if the program Halt.java doesn’t halt, then the else
statment will execute, and the program will halt. Thus, we have a contradiction and our assumption
that the method programHalts could be written is false. So we conclude that there is no algorithm
that can always determine whether or not any given program halts on any given input.
A decision problem with this property that no algorithm solves it in general is called an unde-
cidable problem. So the halting problem is an example. Calculating the Kolmogorov complexity of
a given string is another example of an undecidable problem.
Does nondeterminism help? If we used a nondeterministic machine, could we solve an undecid-
able problem? A deterministic machine (such as an NMU laptop) can simulate a nondeterministic
machine, it just takes lots of time. Therefore, even a nondeterministic machine cannot solve the
halting problem.
Note that it is possible to solve the halting problem for specific cases, but not in general. An
interesting question to ask yourself is: “Is it possible to solve the halting problem if we only restrict
inputs to problems other than the same program itself?”, which throws a slight monkey wrench in
our proof technique above.
2 Reductions
A reduction, informally, is a way solving a problem using another. Example: getting to Chicago
from Marquette reduces to the problem of finding a map. It also reduces to loading up Google
maps, which in turn reduces to the problem of finding a computer with an Internet connection and
a browser. We will use reductions to relate the complexity of different problems.
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