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    <p>There are lots of good answers already, but I haven't seen anyone address the fact that, in a sort of selective blending of theory and practicality, the Halting Problem really is solvable.</p> <p>So first of all, the Halting Problem is basically the task of writing a program which takes any arbitrary second program and determines whether the secondary program will halt on an arbitrary input. So you say "Yes this program will halt on this input" or "No it won't". And in fact, it is unsolvable in the general case (other people seem to have provided proofs of this already) on a Turing Machine. The real problem is that you can kind of find out whether something is going to halt by running it (just wait until it halts), but you can't really find out whether something is going to NOT halt by running it (you'll just keep waiting forever).</p> <p>This is a problem on a Turing Machine which, by definition, has an infinite amount of memory and thus infinitely many states. However, our computers have only a finite amount of memory. There are only so many bits on the computer. So if you could somehow keep track of all of the previous states (bit configurations) you've seen while running the program, you can guarantee that your checker will never go into an infinite loop. If the secondary program eventually halts, you say "Yes, this program will halt on this input". If you see the same bit configuration twice before it halts, you know "No it won't". Probably not of great technical importance, but it's good to know that a lot of times the really "hard" problems we face are harder in theory than in practice.</p>
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    1. POWhat exactly is the halting problem?
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    1. USAmbuoroko
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    1. COOh dear. You need to have a think about how many possible states a computer with 4 GB of RAM can get into!
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      1. USDaniel Earwicker
    2. COYou can consider computers DFAs, and then yes the halting problem is solvable. However, I wouldn't call this practical by any means. If you consider the hard-drive as part of the state machine, you've got more states than you could ever hope to enumerate.
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      1. USKevin Montrose
    3. COSure... it's still not practically solvable. But it's theoretically solvable. And who doesn't enjoy a little useless theory?
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