The picture
Each row is one generation of the automaton, read off the described machine's tape the moment it finishes. The tick beside it means it matches the automaton computed directly.
Press Draw.
What is running
Three machines, stacked:
- The network: one transformer layer, two attention heads, a 64-wide residual stream. Each forward pass reads three tokens of its own transcript and appends the next.
- A universal Turing machine (34 states, 23 symbols, 559 rules), which is what the network learned, from random machines only.
- A cellular-automaton machine (8 states, 18 rules), written on the universal machine's tape as data:
Rule 110 is itself universal (Cook, 2004), so the last picture is a universal machine all the way down.
The automaton machine sweeps each row once, left to right, remembering the old values of the two cells behind it. It writes each new cell one position to the right of where it belongs; because a row also widens by one cell on each side, every row still starts in the same place. Its rules:
The network is evaluated once for each different window it meets (a few thousand), then reused; a window of three tokens is a finite function, and that is the point. Nothing here is precomputed. The checks beside the picture count any step it never saw in training and any disagreement with the universal machine's exact rule.