Amazing paper, I re-read it in more detail today. It feels very rich, like almost a new field of study —- congratulations to the authors.

I’m ninjaing in here to ask a q — you point out in the checkerboard initial discussion that the 5(!) circuit game of life implementation shows bottom left to top right bias — very intriguing.

However, when you show larger versions of the circuit, and in all future demonstrations, the animations are top left to bottom right. Is this because you trained a different circuit, and it had a different bias, or because you forgot and rotated them differently, or some other reason? Either way, I’d recommend you at least mention it in the later sections (or rotate the graphs if that aligns with the science) since you rightly called it out in the first instance.

Author here. Thank you! You're seeing that correctly. The directional bias is the result of some initial symmetry breaking and likely random-seed dependent. The version that constructs the checkerboard from the top-right down was trained asynchronously, and the one from the bottom-left up was trained synchronously. The resulting circuits are different.