[jhrmnn]

I think the intuition here is that an “intelligent” system would lean much more heavily on the heuristics than on the search, like humans. It’s hard to quantify in this case, but certainly in the case of chess, when engines were about as good as best human players, they were doing orders of magnitude larger search than humans. Which made them feel more like chess _engines_ than chess AI. AlphaZero certainly made a step in the right direction, but it’s still far from how humans play.

[killerstorm]

The comment above mentions "brute force" incorrectly. It's fundamentally impossible to brute force googol combinations...

[tsimionescu]

The key insight is this whole thread is that this Alpha Geometry only works because the search field is not a googol combinations. So, it doesn't really generalize to many other fields of math. We shouldn't expect an AlphaCategoryTheory or AlphaNumberTheory anytime soon.

[killerstorm]

"Brute force" = going through all combinations indiscriminately.

Using heuristics, beam search, etc, = "not brute force".

Calling something smart "brute force" is wrong.

> We shouldn't expect an AlphaCategoryTheory or AlphaNumberTheory anytime soon.

There's already a number of papers demonstrating use of LLMs for math in general:

"Autoformalization with Large Language Models" https://arxiv.org/abs/2205.12615

"Large language models can write informal proofs, translate them into formal ones, and achieve SoTA performance in proving competition-level maths problems!" https://twitter.com/AlbertQJiang/status/1584877475502301184

"Magnushammer: A Transformer-based Approach to Premise Selection" https://twitter.com/arankomatsuzaki/status/16336564683345879...

I find it rather sad that the reaction to amazing research is "but it's just..." and "it might not work on ...". Like, maybe appreciate it a bit and consider a possibility that something similar to it might work?

[tsimionescu]

AlphaGeometry works, in simplified terms, in two stages:

1. Heuristically add a candidate construction using an NN (a transformer)

2. Brute force search through all possible deductions from the current set of constructions using a symbolic solver

If you don't find a solution after step 2, repeat. There may be some backtracking involved to try a different set of constructions as well.

This approach only works because, in geometry, the set of possible deductions from a given construction is actually quite small.

Also, note that this approach overall is essentially an optimization, not amazing new capabilities. Replacing step 1 with a random construction still solves 10 problems on average in the given time, compared to 30 with the new approach. The existing algorithm, relying mostly on brute force search, is probably able to solve all of the geometry problems if given, say, 10 times as much time as the olympiad students (so not some absurd amount of time).

[hxypqr]

Key insight is the finiteness of reasoning parts in planar geometry that can be quickly solved by the SAT, which often does not exist in most first-order and second-order logics, such as number theory, algebra, or functional analysis

[nybsjytm]

The search space here is not so large. Read this paper, a key part of the present work: https://doi.org/10.1023/A:1006171315513