[Tainnor]

Solving differential equations symbolically[^1] is a classic pattern matching problem though. It takes a lot of practice recognising which equations can be solved with which method/heuristic, but in the end, the available heuristics are somewhat limited in number and so it's not unreasonable that an ML model would perform well here. Notice that you will still need to actually check your result in some way or the other, because ML models can generate false positives and you really don't want that in maths; so it's not that you're replacing rule-based systems, you're just augmenting them with other approaches (and of course, verifying the solution to a differential equation is basically a solved problem in symbolic analysis, since taking derivatives is a simple rule-based procedure[^2]).

I think theorem proving is much different. The space of possible heuristics you may apply to a proof is basically infinite. It can take a tremendous amount of creativity and intuition to come up with a complicated and novel proof. While I can see ML models being of use for simpler proofs or lemmas within bigger proofs (such as simple epsilon-delta proofs etc.), I have a hard time imagining that they will really be able to do real proofs anytime soon.

[^1]: I emphasise "symbolically" because it is my understanding that outside of simple situations and university lectures, most people don't bother with that, instead solving differential equations numerically.

[^2] There are some subtleties around the fact that, afaik, deciding whether two expressions are equal is undecidable in the general case because at some point you will have to compare e.g. coefficients and the equality of real numbers is undecidable. In practice, you will consider two numbers to be equal if their difference is below a certain threshold, which could in theory also yield false positives, but I find it unlikely that this would occur in practice in the situation of solving an equation.

[thomasahle]

Actually OpenAI just published an article about using Transformer for theorem proving: https://arxiv.org/abs/2009.03393

[riku_iki]

I think they can use transformer in the same way for proof path generation: do pattern matching for complex logical statements to find those where specific rule can be applied, similarly as it has be done in differential equations paper.

[Tainnor]

How does that help and/or address any of what I wrote?

You can generate possible transformations from one proposition to another one, yes. But the space of possible transformations is infinite, how does the system know which ones to try? Moreover, even if you try some local transformation, how do you know that you have made progress?

A system like that could probably perform well on typical undergraduate proofs: epsilon-delta, proof that such and such a thing is a norm, etc., but it's gonna have a hard time with more difficult problems.