I’ve seen quite a few critiques of the practicality of time complexity analysis but I’ve never seen an article describe what we should replace it with. Time complexity analysis, as given, has given some structure to the infinite zoo of problems even if imperfect and composes relatively nicely and is relatively simple. It’s also tractable, has mathematical structure, and is often practical as a heuristic (the kinds of algorithms people typically encounter or come up are typically reasonable under this kind of analysis). In order to be a proper substitute, something would have to satisfy all these qualities. Without such a substitute, we are back to analyzing problems and their complexity without any sort of framework which seems even worse than just remembering that sometimes the big O is not practical.
The only candidate that would be more practical that I know of is cache complexity for cache oblivious algorithms but that’s much more complicated.
This author seems to really mis-understand complexity theory and in what ways it matters. The choice of exponential = bad and polynomial = good may have been quasi-arbitrary initially, but it since found a much more solid footing.
Exponential complexity is what you automatically get when your algorithm is "generate a possible solution, verify if it's correct, repeat if it's not". Polynomial complexity is what you get when the problem itself has some kind of structure that you can exploit to either directly construct a solution, or to generate a very small set (relative to the problem size) of possible solutions that have to be checked by hand.
Now, it is true that for all practical problem-solving purposes, "P = NP but only with algorithms that require n^10^80 or more time (with this as a proven lower bound for even the typical case, no hope of improving)" would be exactly equivalent to "P != NP". However, it would still be a mathematically extremely interesting result, as it would tell you that any problem where it is easy to verify if a solution is correct carries some hidden structure that makes it "easy" to generate the solutions. And it could still open the door to finding new sub-classes of actually easy instances of these problems that are not known today, even if the worse case remains just as intractable.
Now, note the refutation of ETH with the new algorithm found by AI is not really that impressive, it doesn't seem to have that deep of an implication. Sub-exponential running times are not that fundamentally different from exponential ones - they basically just tell you that the problem has just enough structure to eliminate a certain class of possible solutions outright, instead of spending the time to actually verify them. If the algorithm found had been polynomial, even of a huge degree, this would have been a completely different discussion. As it is, we'll just replace the ETH with a "nearly-E"TH and nothing much changes.
> Exponential complexity is what you automatically get when your algorithm is "generate a possible solution, verify if it's correct, repeat if it's not".
I might be misunderstanding you, but this surely depends on the size of the candidate set in the problem. For a problem like "determine whether a given list of integers of length `n` contains a pair that sum to 100" the naive brute-force algorithm is polynomial. If you replace "pair" by "subset" then the naive brute-force algorithm becomes exponential.
Moreover, it was already known that some NP-complete problems admit sub-exponential algorithms. For example, planar graph 3-colorability can be solved in 2^O(sqrt(n)) time. This is not only sub-exponential, but vastly better than the running time of the new 3-SAT algorithm.
Still, the result is very interesting, since many problems can be solved asymptotically faster once subproblems that can be represented as 3-SAT can be solved faster. For example, we now have now have a deterministic interger factoring algorithm with run time N^o(1), where N is the integer being factored. Heuristic algorithms already achieved this bound, but the new claim is a deterministic worst-case guarantee. The result is a fairly straightforward application of the new 3-SAT solver.
There is literally not one aspect of the universe that is provably finite. Not in any direction, not at any scale, not in any way you can think to look at it.
Maybe I misunderstand you, but I don’t think I agree with that.
E.g. the observable universe is finite. Its observable age is finite. The number of known elementary particle types is finite. The number of particles within the observable universe is finite (according to the current understanding). Etc.
There are many examples.
Do we disagree on this, or am I perhaps misunderstanding your position here?
Observable universe is just the part of the universe from where light had time to hit the Earth already. It is growing pretty much at the speed of light, not because of physical changes to the space-time, but because the light from farther away finally arrived. So while it's definitely finite, I wouldn't say it disproves the claim that nothing about the universe is known to be finite.
Because of inflation, there is a distance at which the objects not already in our light cone are moving away faster than the speed of light. So the amount of stuff that we will be able to observe is limited.
Key word is provably. We can assume the universe is finite because we don't have evidence against it, but no one knows if it's possible to prove or reject that for good.
Infinity is an abstraction. The universe cannot be "infinite" in the mathematical sense, as the mathematical infinity is an abstraction. The only meaningful question is "is mathematical infinity required to produce good models of the universe/physics?" and the answer so far is "no". We use infinity because it is convenient, but it is not really needed.
There is no reason to think that "infinity" will be essential to describing the universe any more than any other possible abstraction that we have not so far constructed.
I have the exact opposite experience. There nothing I can think of that is infinite in nature. Not only is it impossible to prove infinity empirically ("in the universe"; the only thing that actually matters) but there are exactly zero actual things with an infinite nature of any kind that I can think of. It's a useful imaginary construct that makes fancy math work, that's all.
For example, the special theory of relativity doesn't work if space-time has any "lowest scale" (i.e. if the distance between any two events can be expressed as n * min_dist, where n is an integer and min_dist is a constant that has the same value for all observers), it you can't infinitely sub-divide it, since then lengths couldn't contract the way that the Lorrentz transforms predicts. Lots of math in QM and QFT also requires space and time to be continuous for other reasons as well.
Also, none of our current laws put any kind of limit on the life span of the Universe as a whole, so space-time is also infinitely increasing, even if heat death or big crunch models are correct.
Of course, none of this is provable per se - it is fundamentally impossible to run an experiment in finite time that proves whether something is finite or infinite. It could be that SR and QFT are wrong in their continuous space assumptions, but only in ways that are too subtle to measure. Still, as it stands, we don't have any theory of how the world works which allows for a discrete space-time, even as an approximation - we need space-time to be continuous for our current theories to work at all.
On the other hand, every single observation anyone has ever made was a finite value of something.
Even when things are potentially infinite, there seems to be a conspiracy of the universe to make it finite. The size of space may be infinite, but due to a finite speed of light and the finite age of the universe we can only ever observe a finite slice. When attempting to zoom in to an infinite degree onto supposedly point-like particles, quantum mechanics makes everything fuzzy and finite again. Black holes appear to have infinite curvature at the center, but it's unobservable due to the event horizon.
This principle seems so natural that we even apply it in advance: In quantum field theory, there are infinities in the equations when done naively, which we deem so unnatural that we apply funny tricks which make mathematicians cringe, but they miraculously work out so well that we get theory and experiments to agree to an astonishing degree.
And yet the vast majority of physicists treat infinity in the equations as a sign of them being incomplete, and dismiss the possibility of the universe being infinite in any way
I would like to point out that there is also no proof of God not existing either
Who needs calculus anyways? I'd rather focus learning linear algebra especially after realizing a derivative is just a linear transformation and integration problems share a lot with determinants (areas).
Maybe so many more people would engage with mathemathics when they realize this.
It would be great to prove that there is or isn't a non-galactic polynomial-time algorithm for NP-complete problems. But we can't even solve the easier problem of proving there is or isn't a galactic polynomial-time algorithm, so what hope is there for the harder problem?
If you could prove that any polynomial-time algorithm for NP problems had to be galactic, you would be the most celebrated computer scientist since Turing. It's just no one has any idea how to do that.
Yep, ranting against galactic polynomial algorithms reminds me of the people who oppose the use of infinity. Like, yes, it's not physically realistic construction, but it makes things simple enough that we can do analytic solutions to problems and simplify terribly complicated things.
Polynomials are closed under multiplication, so you can do a poly algorithms poly number of times and still have a poly algorithm. That's very convenient.
The article makes the point that not all polynomial algorithms are created equal. If you combine ten O(n^2) algorithms, you can end up with an O(n^1024) algorithm, which isn’t practical anymore, despite being polynomial. That’s inconvenient.
It's just irrelevant for big-O complexity what happens for instances below a constant threshold, so the algorithm uses the simplest approach and moves on. If you have an exponential-time algorithm that's faster than brute force for small instances, you can use that and get a faster overall runtime at the same big-O complexity. If you can lower the switchover threshold, ditto. It's a bit weird to dismiss the subexponential algorithm just because it's leaving an obvious optimization on the table... it can only get faster from here.
I guess in engineering what you care about is: how fast is it on given input size X relevant for the practical problem. Or: how much input size can we do in time bound Y.
This reminds me of the (perhaps unrelated meme) hello I would like (graph of zeta function) number of apples. Statements dreamt up by the utterly deranged.
Indeed I have met professors who claim since something is O(1) it doesn't matter if you run it 4 times. No need to optimize. I think it's just part of the wider divide between theory and application. For instance big O also requires a register machine to make sense, which most courses don't define. Somehow addition and multiplication are O(1) in some places and O(n) or O(n^2) somewhere else. Also quite common is the fact that sets are slower than lists for lookup for small n, despite being O(1).
It's just a tool and we shouldn't miss the trees for the forest. I also understand this comment has nothing to do with the ETH or the article, but just some thoughts I have after reading the first chunk on complexity
Did you ever pause to consider that theoretical computer science might not be done for the sake of midwit code monkeys whose deepest contact with complexity theory was when they learned to reverse a linked list in O(n) for their SWE interview?
I'm not sure that retreating to the ivory tower is a winning rhetorical strategy.
Not sure why the hostility anyways. If appreciating TCS results really is a higher brow activity, then GP's indignation at computational models shouldn't upset you, quite the opposite. It's definitionally the normal experience.
And while I do find the blogpost in the OP trite, that is mostly because it's from someone who should absolutely be more intimate with the subject than such "midwitted code monkeys" (aka. normal people) you refer to, yet pretends to feel duped by OAI's results. Quite the different case, and I think is much more worthy of ridicule as a result. They're the high brow who should be appreciating, understanding, yet fail to.
I think the common expression is to miss the forest (big picture) for the trees (details). But maybe you're playing on words and saying we shouldn't let the big picture (big O notation/analysis) obscure our attention to the details (an algorithm's actual efficiency and performance).
I don't think the author's criticism of "Big-O Cabal" is very nuanced, but to the second half of the essay it is funny/concerning/sobering to find that the OpenAI proof is doing brute-force search for all instances below some super-astronomic threshold. It kind of buries the lede if the proof did not mention this clearly and the author has to comb through it to find out (I asked ChatGPT the same on the pdf file, but it came up with some other threshold number).
I don't think it matters? It's completely standard for "galactic" algorithms and not relevant to big-O runtime.
The complaint that these algorithms aren't "realistic" is ostensibly valid, but you could just as well aim that cannon at 95%+ of math research too. Complexity theory is interesting and fun. Isn't that enough?
> It's completely standard for "galactic" algorithms and not relevant to big-O runtime.
This is true, but if doing brute-force search for all instances below some super-astronomic threshold is part of the algorithm, you typically want to understand
- where this constant comes from,
- how much this constant can be decreased (or why it can't)
- etc.
So, proving the correctness of an algorithm which is doing brute-force search for all instances below some super-astronomic threshold is typically just the first step towards getting an understanding of the complexity-theoretic phenomenon.
So: There is indeed nothing wrong with such aspects of galactic algorithms, but exactly understanding and improving the super-astronomic thresholds involved is the next step.
I hate to use absolutely but it absolutely does matter from an education perspective. To put things in perspective to a layperson explain to them this particular galactic theorem requires full search below some titanic |x|. That should prompt some excellent lay questions that a serious computer scientist ought to reckon with.
It is not to say the approach has been wrong (as the author strongly believes in his reframing) but that this clarity does matter for purposes of, ahem, mathematical understanding.
Author is a math professor who somehow wants you to believe that he didn't know what a computational model is until OpenAI's manuscript dump a few days ago. In this post, he now extends that pretending game to galactic algorithms.
Very gonzo, but maybe I prefer to learn about these things with a little less circlejerk.
The only candidate that would be more practical that I know of is cache complexity for cache oblivious algorithms but that’s much more complicated.
Exponential complexity is what you automatically get when your algorithm is "generate a possible solution, verify if it's correct, repeat if it's not". Polynomial complexity is what you get when the problem itself has some kind of structure that you can exploit to either directly construct a solution, or to generate a very small set (relative to the problem size) of possible solutions that have to be checked by hand.
Now, it is true that for all practical problem-solving purposes, "P = NP but only with algorithms that require n^10^80 or more time (with this as a proven lower bound for even the typical case, no hope of improving)" would be exactly equivalent to "P != NP". However, it would still be a mathematically extremely interesting result, as it would tell you that any problem where it is easy to verify if a solution is correct carries some hidden structure that makes it "easy" to generate the solutions. And it could still open the door to finding new sub-classes of actually easy instances of these problems that are not known today, even if the worse case remains just as intractable.
Now, note the refutation of ETH with the new algorithm found by AI is not really that impressive, it doesn't seem to have that deep of an implication. Sub-exponential running times are not that fundamentally different from exponential ones - they basically just tell you that the problem has just enough structure to eliminate a certain class of possible solutions outright, instead of spending the time to actually verify them. If the algorithm found had been polynomial, even of a huge degree, this would have been a completely different discussion. As it is, we'll just replace the ETH with a "nearly-E"TH and nothing much changes.
I might be misunderstanding you, but this surely depends on the size of the candidate set in the problem. For a problem like "determine whether a given list of integers of length `n` contains a pair that sum to 100" the naive brute-force algorithm is polynomial. If you replace "pair" by "subset" then the naive brute-force algorithm becomes exponential.
Still, the result is very interesting, since many problems can be solved asymptotically faster once subproblems that can be represented as 3-SAT can be solved faster. For example, we now have now have a deterministic interger factoring algorithm with run time N^o(1), where N is the integer being factored. Heuristic algorithms already achieved this bound, but the new claim is a deterministic worst-case guarantee. The result is a fairly straightforward application of the new 3-SAT solver.
E.g. the observable universe is finite. Its observable age is finite. The number of known elementary particle types is finite. The number of particles within the observable universe is finite (according to the current understanding). Etc.
There are many examples.
Do we disagree on this, or am I perhaps misunderstanding your position here?
The observable universe is growing much faster than that, because space itself is expanding.
There is no reason to think that "infinity" will be essential to describing the universe any more than any other possible abstraction that we have not so far constructed.
For example, the special theory of relativity doesn't work if space-time has any "lowest scale" (i.e. if the distance between any two events can be expressed as n * min_dist, where n is an integer and min_dist is a constant that has the same value for all observers), it you can't infinitely sub-divide it, since then lengths couldn't contract the way that the Lorrentz transforms predicts. Lots of math in QM and QFT also requires space and time to be continuous for other reasons as well.
Also, none of our current laws put any kind of limit on the life span of the Universe as a whole, so space-time is also infinitely increasing, even if heat death or big crunch models are correct.
Of course, none of this is provable per se - it is fundamentally impossible to run an experiment in finite time that proves whether something is finite or infinite. It could be that SR and QFT are wrong in their continuous space assumptions, but only in ways that are too subtle to measure. Still, as it stands, we don't have any theory of how the world works which allows for a discrete space-time, even as an approximation - we need space-time to be continuous for our current theories to work at all.
Or provaly infinite.
Even when things are potentially infinite, there seems to be a conspiracy of the universe to make it finite. The size of space may be infinite, but due to a finite speed of light and the finite age of the universe we can only ever observe a finite slice. When attempting to zoom in to an infinite degree onto supposedly point-like particles, quantum mechanics makes everything fuzzy and finite again. Black holes appear to have infinite curvature at the center, but it's unobservable due to the event horizon.
This principle seems so natural that we even apply it in advance: In quantum field theory, there are infinities in the equations when done naively, which we deem so unnatural that we apply funny tricks which make mathematicians cringe, but they miraculously work out so well that we get theory and experiments to agree to an astonishing degree.
I would like to point out that there is also no proof of God not existing either
Maybe so many more people would engage with mathemathics when they realize this.
It would be great to prove that there is or isn't a non-galactic polynomial-time algorithm for NP-complete problems. But we can't even solve the easier problem of proving there is or isn't a galactic polynomial-time algorithm, so what hope is there for the harder problem?
If you could prove that any polynomial-time algorithm for NP problems had to be galactic, you would be the most celebrated computer scientist since Turing. It's just no one has any idea how to do that.
Polynomials are closed under multiplication, so you can do a poly algorithms poly number of times and still have a poly algorithm. That's very convenient.
Surely it cannot be necessary for the complexity result?!
Indeed I have met professors who claim since something is O(1) it doesn't matter if you run it 4 times. No need to optimize. I think it's just part of the wider divide between theory and application. For instance big O also requires a register machine to make sense, which most courses don't define. Somehow addition and multiplication are O(1) in some places and O(n) or O(n^2) somewhere else. Also quite common is the fact that sets are slower than lists for lookup for small n, despite being O(1).
It's just a tool and we shouldn't miss the trees for the forest. I also understand this comment has nothing to do with the ETH or the article, but just some thoughts I have after reading the first chunk on complexity
Not sure why the hostility anyways. If appreciating TCS results really is a higher brow activity, then GP's indignation at computational models shouldn't upset you, quite the opposite. It's definitionally the normal experience.
And while I do find the blogpost in the OP trite, that is mostly because it's from someone who should absolutely be more intimate with the subject than such "midwitted code monkeys" (aka. normal people) you refer to, yet pretends to feel duped by OAI's results. Quite the different case, and I think is much more worthy of ridicule as a result. They're the high brow who should be appreciating, understanding, yet fail to.
I think the common expression is to miss the forest (big picture) for the trees (details). But maybe you're playing on words and saying we shouldn't let the big picture (big O notation/analysis) obscure our attention to the details (an algorithm's actual efficiency and performance).
The complaint that these algorithms aren't "realistic" is ostensibly valid, but you could just as well aim that cannon at 95%+ of math research too. Complexity theory is interesting and fun. Isn't that enough?
This is true, but if doing brute-force search for all instances below some super-astronomic threshold is part of the algorithm, you typically want to understand
- where this constant comes from,
- how much this constant can be decreased (or why it can't)
- etc.
So, proving the correctness of an algorithm which is doing brute-force search for all instances below some super-astronomic threshold is typically just the first step towards getting an understanding of the complexity-theoretic phenomenon.
So: There is indeed nothing wrong with such aspects of galactic algorithms, but exactly understanding and improving the super-astronomic thresholds involved is the next step.
It is not to say the approach has been wrong (as the author strongly believes in his reframing) but that this clarity does matter for purposes of, ahem, mathematical understanding.
Very gonzo, but maybe I prefer to learn about these things with a little less circlejerk.