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This is actually an incredibly deep and difficult question to answer. I would expect no less from cperciva. :-)

As I mention in the talk, you get very bad tactical performance from taking a uniform random distribution and piping it into the emulator. The fuzzer is exponentially unlikely to hold the jump button for many successive frames without a break. In the fully general case, I think instead of maximum entropy, you want something more like Marcus Hutter's AIXI where you spend energy on inputs inversely proportional to their Kolmogorov complexity. Unfortunately, that's uncomputable, but it turns out that just switching to toggling bits with low probability does a lot better than pure randomness. The approach is analogous to swarm testing (https://users.cs.utah.edu/~regehr/papers/swarm12.pdf).

All of which is to say, the result that we show here is vastly less jumpy than our first tries. The reason it's still more jumpy than a human player is that our platform has no idea where it is in the game, or even that it's playing a game. So if a jump doesn't harm it in the exploration process, there's some chance the first input getting somewhere new will involve a jump, and that will then get locked in.

We do have the capability to do optimization on inputs (what conventional PBT calls "shrinking"), and indeed if you apply this to Mario you can get it to jump a lot less and complete levels a lot faster. That capability didn't exist yet when this video was recorded. We should totally do a another post on this topic!

3 comments

To reframe the problem with purely random "Tactics", some frequencies may be an inefficient way to get to desired states of the system. That is, it is useful to consider the correlation between random inputs across time.

This reminds me a lot of the problem space involving Perlin Noise. I'd hypothesize that by using a fractal noise pattern you can generate inputs with correlated values yet still remain random enough to get to any state. Have you tried using noise generators for random inputs?

> Unfortunately, that's uncomputable

Minor nitpick, but while Kolmogorov complexity as typically defined is uncomputable, I would argue that this result is only a theoretical curiosity and mostly irrelevant.

That is, the "uncomputable" Kolmogorov complexity computation presupposes that you have a Turing machine, i.e. a machine with literally infinite memory, which is not possible to construct in our universe. Or alternatively, it presupposes that "computable function" is one that can be computed by a machine with an infinite amount of storage, which amounts to the same thing as having a Turing machine.

You could probably define some version of Kolmogorov complexity that is parametrized by the memory size (e.g. of a linear bounded automaton or similar model that better represents a computer with finite resources), which should make it computable. 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).

While reading your comment, I kept having the vague feeling that your idea of "theoretically computable" (within the given memory bounds) will still leave some exponential time and/or space complexity, and so effectively will still be "practically not computable".

Your last paragraph then seemed to confirm that, i.e. there was no particular shortcut for this specific case that would make it any different from general Kolmogorov complexity.

In that sense, isn't your comment itself also "only a theoretical curiosity"? As we went to from "uncomputable", to "theoretically computable under the given constraints", to "practically uncomputable"?

While suffering from lack of rigor, I think a lot of times--probably even the majority of times outside purely cs-theoretical treatments--when we colloquially speak of "uncomputable", we are always talking about practical computers without an infinite touring tape, and so actually mean "practically uncomputable".

Because, yes, while everyone immediately understands that actual computers don't have infinite memory, at the same time everyone understands that "exponential time" is still "never" in practical terms.

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

Oh 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!

I 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...

Thank you. Comments like this are why I love HN.

What if you optimized inputs based on the strategy? E.g when you're in the air pressing down has no effect, or more measurably, the code paths between runs where you do or don't press down are very similar or completely converge. Similarly, jumping while running on a flat floor at full speed doesn't affect the game much, so prune it.

Maybe call it "strategy-informed tactics"