Don't dismiss search, it might be brute force but it goes beyond human level in Go and silver at IMO. Search is also what powers evolution which created us, also by a lot of brute forcing, and is at the core of scientific method (re)search.
What makes solving IMO problems hard is usually the limits of human memory, pattern-matching, and search, not creativity. After all, these are problems that are already solved, and it is expected that many people can solve the problems in about 1 hour's time.
That makes it, in principle, similar or even easier than a champsionship-level chess move, which often take more than 1 hour for a professional human (with more training than an IMO high school student) to solve.
Another interesting concern is that when posing a problem to humans, it's fine to pose an "easy" brute-forceable problem, but humans, being slow brute-searchers, need to find more clever solutions. But if you give such a problem to a computer, it can trivialize it. So to test a computer, you need to pose non- easily-brute-forceable problems, which are harder for the computer than the others, but equally difficult for the humans as the other problems are.
Why do you say these are problems that are already solved? Sure, they're often variations on existing themes, but the same is true for chess positions and, honestly, almost everything else in any field of human endeavor.
Agreed that the absolute upper tier of chess players have trained longer and harder than most or all IMO contestants. Though I do wonder which (top-tier chess or the IMO) draws on a larger talent pool. To my understanding, a significant fraction of all high school students on Earth take some form of qualifying exam which can channel them into an IMO training program.
And as far as the being amenable to brute force (relative difficulty for humans vs. computers): it seems that chess was comparatively easier for computers, IMO problems are comparatively easier for humans, and the game of Go is somewhere in between.
These problems are literally already solved? Of course, the IMO problem designers make sure the problems have solutions before the use them. That's very different than math research, where it's not known in advance what the answer is, or even that there is good answer.
I'm saying they weren't solved until the problem composer (created and) solved them. They're not, in general, problems for which solutions have been lying around. So "these are problems that are already solved" isn't introducing anything interesting or useful into the discussion. The post I was replying to was trying to draw a contrast with chess moves, presumably on the grounds that (after the opening) each position in a chess game is novel, but IMO problems are equally novel.
It's true that IMO problems are vetted as being solvable, but that still doesn't really shed any information on how the difficulty of an IMO problem compares to the difficulty of chess play.
The main difficulty to scaling Alpha Proof is finding theorems to train it with. AlphaGo didn't have that problem because it could generate it's own data.
And I understand the upper time limit for each question was 4.5 hours. So it solved one question almost immediately, two well over the allotted time (60 hrs), and two not at all. No medal for you, Grasshopper.
Contestants get 4.5 hours for each of the two days of competition. They have to solve three problems in that time, so on average you can spend 1.5 hours per problem (if you're aiming to finish all three).
That said, the gap from "can't do it at all" to "can do it in 60 hours" is probably quite a bit larger than the gap from 60 hours to 1.5 hours.
Timing something that can be ran faster by throwing better hardware at it honestly feels conceptually irrelevant, as long as the complexity is actually tractable.
Search is great, search works, but there was not a tonne to learn from the AlphaGeometry paper unless you were specifically interested in solving geometry problems.
I would argue that no actually searchable solution space is really infinite (if only because infinite turing machines can't exist). Finite solution spaces can get more than large enough to be intractable.
What about ℕ? Seems pretty infinite to me, unless with "actually" you mean finite in time and space, which would make your argument a tautology. Or am I missing something?
Almost every "number" "in" N doesn't actually exist. In the search for numbers that exist, we will most likely only ever find a finite set of numbers before the Universe or humanity dies.
Searches happen in finite time an space and, more importantly, systems performing those searches have practical finite limits on parameters that determine size of the space within which that search can take place (such as available memory).
Even within fairly modest finite limits, you can produce a solution space that cannot be significantly searched with the available finite matter and time available in the observable universe.
Thus, the problem with using search isn't that solution spaces can be infinite, but that finite solution spaces can be unimaginably large.
For some problems validation is expensive. Like the particle collider or space telescope, or testing the COVID vaccine. It's actually validation that is the bottleneck in search not ideation.
You mean that by improving search we can solve any problem? What if solution field is infinite, even if we make search algo 10x100 more performant, solution field will still be infinite, no?
Don't dismiss search, it might be brute force but it goes beyond human level in Go and silver at IMO. Search is also what powers evolution which created us, also by a lot of brute forcing, and is at the core of scientific method (re)search.