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I would somewhat agree with your comment, but I think you're missing some important points: > everyone understands that "exponential time" is still "never" in
by someplaceguy 2y ago
I would somewhat agree with your comment, but I think you're missing some important points:
> everyone understands that "exponential time" is still "never" in practical terms.
It's important to note that this is not necessarily true:
1. "exponential time" is somewhat ambiguous. An algorithm might be exponential time yet have a very low exponential base (e.g. 1.0001), so in practical terms it might be practically computable for reasonable sizes.
2. Even if the exponential base is high, it still doesn't say anything about whether an algorithm can be used practically or not. The algorithm might still be very efficient for reasonable problem sizes even though it has a high exponential base (important question: exponential in terms of what, exactly?).
3. Exponential time algorithms might only be so in the worst case but might not necessarily be exponential in the average case, or even the vast majority of interesting cases. As an example, it might be easy (or at least doable) to solve the halting problem for normal computer programs. Humans do this all the time with real-world programs when performing formal verification (as these programming languages, logics and tools force you to prove that loops and recursive functions always terminate, even when assuming a model equivalent to a Turing machine), and AFAIK there's no proof that computers can't efficiently do the same for the vast majority of real-world programs (cryptographic algorithms being the usual exception).
4. Whenever someone mentions that the halting problem and Kolmogorov complexity are uncomputable, the discussion ends there. But notice that when I pointed out that it's in fact computable, the discussion turned into one about how efficient the computation might be (which I argue, is how all such discussions should be).
5. As a side note, every single time I argued this point in the past, usually in the context of the halting problem, someone always argued that such an algorithm would necessarily have a time complexity of 2^N, where N=nr. of bits of the machine. This is not true. It would only be true for the simplest and most naive solution to the halting problem, which is inevitably what that person has in mind. In fact, there's already a family of algorithms that solve the halting problem with less complexity for almost all programs: "Floyd's tortoise and hare" and similar ones (see [1]). Note that these algorithms don't even inspect the program, they just run it step by step. This leads me to think that there are undiscovered algorithms that are far more efficient by virtue of exploiting knowledge about the program being analyzed.
[1] https://en.wikipedia.org/wiki/Cycle_detection https://en.wikipedia.org/wiki/Cycle_detection
- anyfoo 2y agoOh but that was specifically what I was trying to get at, i.e. point #4 of this reply of yours: Are there any shortcuts that make this reasonably computable or not? In your original reply, you started with (transcribing) "Kolmogorov complexity is not technically uncomputable in the practical case of not having an infinite tape", which gave me hope that we would start talking about how there are some reasonable shortcuts in this particular case (as you mention in this answer now). But then you ended with (literally) "That said, in practice it would probably take an unreasonable amount of time to perform this computation (but that is orthogonal to whether it's computable or not)", which squashed my hopes, seemed to just have replaced "theoretically uncomputable" with "practically uncomputable", and made me wonder how it changed anything that OP already wrote in practical terms, namely: "Unfortunately, that's uncomputable" But now it seems we're back to (potentially) discussing how in this particular use case there might be tractable ways to (limited, but useful) computability, which is good again!
- someplaceguy 2y agoI should note that I'm not a computer scientist and I'm currently a bit sleep-deprived, but to continue discussing what you're interested in: I tend to think that an efficient computation of some finite-state version of Kolmogorov complexity would necessarily require an efficient computation of a finite-state version of the halting problem, but I'm not entirely sure of this. Naively, it seems that this Kolmogorov calculation would require enumerating all programs (in increasing program size) and then running each of them until they either 1) enter an infinite loop, 2) produce the input string and halt, or 3) start producing a different string. However, I'm not sure this would be the most efficient algorithm. For example, it might be easy to inspect each program and discard almost all that would "obviously" not produce the input string before we even try to run them. Or better yet, never even enumerate such programs that can be proven not to produce the input string. In other words, cut the search space significantly. Perhaps there might even be a shortcut to directly construct the smallest program that produces a given string, or at least, a family of small candidate programs that would be a very small subset of all possible programs and yet would be guaranteed to contain the solution. As you might have noticed, unfortunately I don't know if there are such extraordinarily efficient shortcuts, I'm only speculating that they might exist. That said, I still suspect that this might be intractable due to having to account for the worst case, i.e. undecipherable random-looking programs. In the case of the halting problem in the context of formal verification, I'm more optimistic since we usually don't need to care about such random-looking programs, only human-constructed ones (usually), which might be far easier to analyze algorithmically. I don't know if that makes sense...