> thus we can solve the halting problem in finite time by simply iterating over all possible configurations, that is a somewhat uninteresting observation It is

> thus we can solve the halting problem in finite time by simply iterating over all possible configurations, that is a somewhat uninteresting observation It is, but that's not the observation I was making. You only mentioned one way of solving the problem, but that's not the only way. > That is, it turns out that the only way to verify whether an arbitrary finite-state Turing machine will halt on a specific input is to check all possible states What do you mean by all possible states? If you mean literally all possible states, that's not true. I mean, yes, you could iterate over all possible states to solve that problem, but that's probably the least efficient way to solve it. There are already-known algorithms which always solve the Halting problem for machines with finite state, and they don't need to iterate over all possible states. They do, however, need to iterate over all state transitions that the machine actually goes through (multiple times, even). However, these algorithms that I'm mentioning (i.e. cycle detection algorithms) are also quite dumb. They don't exploit any knowledge about the state transitions in order to analyze whether the machines halt or not, they just simulate the machine step by step (this is due to the definition of the cycle detection problem itself, which does not allow inspecting the program). In principle, and even in practice, it's possible to make those algorithms significantly more efficient, at least for many of the machines (i.e. programs) that we care about. I suspect it is not possible to make such a (fully automatic) algorithm significantly more efficient for all possible programs (even the nonsensical ones), although I don't think such a proof exists (if it does, I would like to see it). The closest I've been pointed to is a paper possibly implying that such an algorithm would have to be EXPTIME-complete, although even the person that pointed me to that paper had some difficulty interpreting it -- and even if that were true, that says nothing about its real-world efficiency. > That is, it turns out that the only way to verify whether an arbitrary finite-state Turing machine will halt on a specific input is to check all possible states Can you point me to a source that proves this claim? Not only I'm doubting it, but even if you are right, I'd be really interested in reading such a proof. > For an example, an arbitrary program for a computer with as much memory as the infamous "640KB is enough for anyone" quote would require at least 640,000^255 (~10^1480) computational steps to check if it halts. Again, I'm wondering why you are claiming that the only possible algorithms which can check whether arbitrary finite-state programs halt have to iterate over all possible states (or even all the actual state transitions).