How about axioms of logic as legal moves, and asking it to go from a set of axioms to open mathematical problems?
Or chemical procedures and components as moves, and asking it to tackle a disease with a known molecular structure?
It is not as straightforward as I make it sound, but these are complete information problems.
>Or chemical procedures and components as moves, and asking it to tackle a disease with a known molecular structure?
>It is not as straightforward as I make it sound, but these are complete information problems.
However they are not adversarial games. Self-play only works for adversarial games. Also each mathematical problem is different and needs to be solved only once, and likewise for molecular structures. Additionally, they only need to be solved once, so there is no "gradient of skill" that we know to climb.
You can extend this idea to full first-order intuitionistic logic and probably also to higher-order logics, as well as many different modal logics. There are also formulations of classical logic as a single player game, but that doesn't seem to be very useful here.
While chess if very complicated to play, the state can be represented by (at worst) an 8x8x6 boolean array (board, one of 6 possible pieces), for go a 17x17x2 boolean array. There is nothing similar for logic and and deep learning breaks down (in my experience) once you don't have that nice regular input space.
I'm not sure how you could train it against itself. The branching factor would be infinite as well so I don't know how you'd constrain the legal moves. For example, maybe rewriting (a * 2) to (a + a) or (a * 4 - a * 2) and so on for every theorem you know would be useful in a proof, plus you can invent your own theorems as intermediate steps.