What exactly is the halting problem?

Question

Whenever people ask about the halting problem as it pertains to programming, people respond with \"If you just add one loop, you\'ve got the halting program and therefore yo

Answer 1

It's a variant of the halting dog problem, except with programs instead of dogs and halting instead of barking.

Answer 2

The significance of the halting problem does not lie in the importance of the problem itself; on the contrary, automated testing would be of little practical use in software engineering (proving that a program halts does not prove that it is correct, and in any case the hypothetical algorithm only provides proof for a given finite input, whereas a real-life software developer would be more interested in a test for all possible finite inputs).

Rather, the halting problem was one of the first to be proven undecidable, meaning that no algorithm exists which works in the general case. In other words, Turing proved more than 70 years ago that there are some problems which computers cannot solve -- not just because the right algorithm has not yet been found, but because such an algorithm cannot logically exist.

Answer 3

From Programming Pearls, by Jon Bentley

4.6 Problems

5. Prove that this program terminates when its input x is a positive integer.

while x != 1 do
    if even(x)
        x = x/2
    else
        x = 3*x +1

Answer 4

In reference to the sub-point "people respond with "If you just add one loop, you've got the halting program and therefore you can't automate task"", I'll add this detail:

The posts that say that you cannot algorithmically compute whether an arbitrary program will halt are absolutely correct for a Turing Machine.

The thing is, not all programs require Turing Machines. These are programs that can be computed with a conceptually "weaker" machine --- for example, regular expressions can be embodied entirely by a Finite State Machine, which always halts on input. Isn't that nice?

I wager that when the people say "add one loop", they're trying to express the idea that, when a program is complex enough, it requires a Turing Machine, and thus the Halting Problem (as an idea) applies.

This may be slightly tangential to the question, but I believe, given that detail in the question, this was worth pointing out. :)

Answer 5

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.

Answer 6

I would suggest to read this: http://en.wikipedia.org/wiki/Halting_problem, especially http://en.wikipedia.org/wiki/Halting_problem#Sketch_of_proof in order to understand why this problem can't be solved in algorithmic way.