What exactly is the halting problem?

Solution 1:

EDIT (much later than original answer): MarkCC of Good Math, Bad Math recently wrote up an excellent discussion of the Halting problem with concrete examples.

The halting problem is basically a formal way of asking if you can tell whether or not an arbitrary program will eventually halt.

In other words, can you write a program called a halting oracle, HaltingOracle(program, input), which returns true if program(input) would eventually halt, and which returns false if it wouldn’t?

The answer is: no, you can’t.

Following up on questions about whether the input to the Halting problem is relevant or a red herring: Yes, the input is important. Also, there seems to be some confusion in that I see "infinite" being used where "arbitrary" is more correct.

Practical example: Imagine that you are working in a QA position and you are to write a halting checker program (aka an oracle) that will confirm that for any arbitrary program written by the development team (D) and any arbitrary input provided by the end-user (I), program D will eventually halt when given input I.

Cue manager voice: "Ho ho, those goofy users, let's make sure that no matter what garbage they type, our server tasks will never end up in an endless loop. Make it so, code monkey!"

This seems like a great idea, right? You don't want your server to hang, right?

What the halting problem is telling you is that you are being handed an unsolvable task. Instead, in this particular case, you need to plan for tasks that run past a threshold time and be ready to cancel them.

Mark uses code instead of input to illustrate the problem:

def Deciever(i):
  oracle = i[0]
  in = i[1]
  if oracle(Deceiver, i):
    while True:
      continue
  else:
    return i

In my discussion in the comments, I went the route of malicious input manipulation to force an unsolvable problem. Mark's example is far more elegant, using the halting oracle to defeat itself:

So, the input to Deceiver is actually a list of two elements: the first one is a proposed halting oracle. The second is another input. What the halting killer does is ask the Oracle: “Do you think I’ll halt for input i?”. If the oracle says, “Yes, you’ll halt”, then the program goes into an infinite loop. If the oracle says “No, you won’t halt”, then it halts. So no matter what the oracle says, it’s wrong.

Said another way, without cheating, reformatting inputs, countable / uncountable infinities or anything other distractions, Mark has written a piece of code that can defeat any halting oracle program. You cannot write an oracle that answers the question of whether Deceiver ever halts.

Original answer:

From the great Wikipedia:

In computability theory, the halting problem is a decision problem which can be stated as follows: given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input.

Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. We say that the halting problem is undecidable over Turing machines. Copeland (2004) attributes the actual term halting problem to Martin Davis.

One of the critical points is that you have no control over either the program or the input. You are handed those and it's up to you to answer the question.

Note also that Turing machines are the basis for effective models of computability. Said another way, everything that you do in modern computer languages can be mapped back to these archetypical Turing machines. As a result, the halting problem is undecidable in any useful modern language.

Solution 2:

To solve the halting problem, you'd have to develop an algorithm that could determine whether any arbitrary program halts for any arbitrary input, not just the relatively simple cases in your examples.