Rule-based Generation of de Bruijn Sequences: Memory and Learning

We investigate binary sequences generated by non-Markovian rules with memory length $\mu$, similar to those adopted in Elementary Cellular Automata. This generation procedure is equivalente to a shift register and certain rules produce sequences with maximal periods, known as de Bruijn sequences. We introduce a novel methodology for generating de Bruijn sequences that combines: (i) a set of derived properties that significantly reduce the space of feasible generating rules, and (ii) a neural network-based classifier that identifies which rules produce de Bruijn sequences. Experiments for large values of $\mu$ demonstrate the approach's effectiveness and computational efficiency.