You can always always tradeoff time vs space. It's actually a fundamental part of complexity theory called Blums speedup theorem by the guy who invented complexity theory no less.
As an example you give me an nlogn solution and i take a portion of the inner loop, put it in a lookup table (who cares if the lookup table is 2^n in size, that's not what we're measuring here since we're specifically measuring time not space) and suddenly i have something better than nlogn since that lookup table is O(1).
That's not to say there's some merit to all of this. It's just that people shouldn't be as astonished by these results as they are. It's complexity theory. There's literally no such thing as a hard wall where something is nlogn in time and can't get lower. It can always get lower. But you need both the time and space complexities together to reason about this.
Like, there exist problems that have a lower bound of n time where algorithms can get arbitrarily close to O(n) complexity but there is no program that actually reaches it (there is one that runs in time O(n^{1.1}), one that runs in O(n^{1.01}), etc).
But that doesn’t mean that there aren’t hard walls, it just means there are hard walls where every program’s runtime is > that wall, rather than >= that wall. But maybe I’m misunderstanding, or there are some corollaries I’m missing?
As an aside, as someone who studied more on the pure math side, Blums speed up theorem is rather intuitive. It looks like there might be a relatively elegant proof using topology. Blums speed up theorem is basically just a statement that the set of programs that implement a particular computable function doesn’t satisfy the greatest lower bound property. This is also the case for, for example, the set of rational numbers: there exist subsets of the rational numbers with a lower bound but not a greatest lower bound (like the set of all positive rational numbers q where q^2 >= 2). I’m hand waving, but I think in that same vein you can show that the set of programs under the topology of ordering by runtime is completely disconnected.
Not always, isn't it? The theorem says that there exists a function that can be sped up, but not that every function can be. A trivial counter example would be the identity function on a TM. It's simply a HALT/ACCEPT. There's no way to reduce that.
The space/time trade-off is also not infinitely flexible, since it takes time to fill space. And lookup tables have to be constructed at run-time, or they'll be of finite size, and hence only a constant speed-up. Perhaps you were thinking of that theorem, the linear as wikipedia calls it? https://en.wikipedia.org/wiki/Linear_speedup_theorem
Isn't this an ironic example to use, since one of the only lower bounds we have actually found is nlogn for comparison-based sorting?
And anyways, this point seems too strong: the fact that TCS people believe NP != P is another example that we believe a lower bound exists, no? (And relatedly, the large portion of TCS that is dedicated to trying to figure out how to lower-bound anything, while admittedly not so successful yet, shows that researchers believe lower bounds exist. I mean, any complexity class non-equality hypothesis is an example of this, no?)
I don't know whether the log of the size of the table needed has a linear relationship to log(n) in this case.
If the things you have to memoize are super-linear, suppose you need to store some value related to all subsets of an input set of size n.
There’s no way you can do this in O(1), even assuming infinitely large memory.
But your right in that we don’t have a huge LUT as it’s not worth it.
Small one in the ALU though, for partial products to add can be worth the additional wire length.
You don't need to go to these extremes regardless though. For small speedups take an nlogn algorithm and just move a very small part of the inner portion out in a similar tradeoff. Eg. "Every nth input is now part of a lookup." If you look at the openAIs <nlogn results there was a series of small improvements of this type (we take this part out, precomputing it, and likewise with this part and that small portion, aha! <nlogn!). I finally loaded the link above and it's 262million lines of such tricks in fact.
A 2^64bits lookup table only has O(1) lookup if you ignore the impossibility of creating a machine that has 2 exabytes of storage where all those bytes have the same access time. Or you'd have to redefine O(1) to be “at most the worst case”. But strictly speaking from the mathematical definition it’s O(1). Which is why no one in practical engineering takes most complexity results seriously as the entire field maps to practice poorly precisely for these reasons (all our machines use fixed integer math and have limited tape storage)
The whole point of complexity theory is that there are no bounds on input size.
That's just incorrect. Are you assuming the table starts pre-filled, and has O(1) lookup? Nobody is using a model like that for this problem.
2^n space complexity doesn't automatically mean you get to fill that space with 2^n chosen computed values for free.
There's models with "oracle"s for various sub-problems and such (that's essentially what you're describing as the ability you're baking in to your model of computation, it can look up the answer to any multiplication for free), but that's not an interesting model for this problem.
If you do that you're just asking "hey, how long does it take to multiply two numbers if multiplying two numbers is free". Boring question, and not the one anyone else is asking.
If you don't like that convention, you are free to use your own, but it is going to confuse people who use the standard convention. And the convention makes a lot of sense when you stop and think about the problem that people are trying to solve.
I guess you're saying that this holds in "realistic" computational models? Sorting in the decision tree model is famously \Omega(n log n)-time.
number multiplication is basically convolution and closely tied with FFTs, so I wouldn't personally be surprised even if a lot of the "FFT's work" could somehow be stuffed into exact arithmetic operations, and the remaining costs just track the packing and unpacking from that representation. It's just an exploit of the book-keeping.
That's ignoring the other complaints about this being a galactic algorithm.
Similarly, yes, it's a galactic algorithm, but these things are still viewed as meaningful. The fact that it is even possible at all with a unit cost RAM asymptotically is an incredible result that nobody expected.
One can of course make a puzzle out of anything, but it behooves asking what situation is being modelled by that problem.
And yes, this criticism might apply more broadly to TCS rather than just this specific artifact from OpenAI. It's reward hacking regardless of whether humans do it or LLMs.
Also the 10^-13 isn't an optimized number. Current results are up about 10 orders of magnitude (10^-3) and show little sign of stopping. it's seems entirely plausible that we might get to nsqrt(log(n)) (with a likely still astronomical constant)
But isn't the assumption that arithmetic is "atomic" standard in TCS? Costs do scale with the bit length, presumably they aren't assuming that addition is "constant time" with respect to # of bits (which would defeat the whole point of the exercise).
Maybe I'm missing something, but none of the things mentioned in that blogpost seem particularly "surprising" to me, most of it focuses on the fact that computational models used in TCS don't align with the real-world model of computers, but that seems like something you learn in day 1 of algorithm analysis? If you want something closer to real-world you'd use word ram model.
One can do some interesting things in this model, such as the van Emde Boas data structure for O(log log n) integer set queries, and tricks where O(log n) size subproblems are converted to lookup tables.
Whether it's an oversight in just this work, or TCS more broadly, I do not know. But it's a fundamental point nonetheless.
Did they start with OpenAI's Lean proof and continue work on that?
Does the Lean proof grow significantly, or are they also trying to keep the Lean proof short-ish?
Engineers don't have to know why something worked in particular in order to use it, they just do.
Science later tries to figure that out.
Both are valuable, but the idea that science must be "first" is, at best, unsupported by the evidence.
https://nitter.kareem.one/RohanArun/status/21093360158149592...
(Replace top-level with healthy instances or x.com as needed)