Reconstructing cellular automata rules from observations at nonconsecutive times

Recent experiments by Springer and Kenyon have shown that a deep neural network can be trained to predict the action of t steps of Conway's Game of Life automaton given millions of examples of this action on random initial states. However, training was never completely successful for t>1, and even when successful, a reconstruction of the elementary rule (t=1) from t>1 data is not within the scope of what the neural network can deliver. We describe an alternative network-like method, based on constraint projections, where this is possible. From a single data item this method perfectly reconstructs not just the automaton rule but also the states in the time steps it did not see. For a unique reconstruction, the size of the initial state need only be large enough that it and the t-1 states it evolves into contain all possible automaton input patterns. We demonstrate the method on 1D binary cellular automata that take inputs from n adjacent cells. The unknown rules in our experiments are not restricted to simple rules derived from a few linear functions on the inputs (as in Game of Life), but include all 22n possible rules on n inputs. Our results extend to n=6, for which exhaustive rule-search is not feasible. By relaxing translational symmetry in space and also time, our method is attractive as a platform for the learning of binary data, since the discreteness of the variables does not pose the same challenge it does for gradient-based methods.