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

TL;DR

This paper presents a novel method combining rule properties and neural network classification to efficiently generate de Bruijn sequences from non-Markovian rules with memory.

Contribution

It introduces a new methodology that reduces the search space and employs neural networks to identify rules producing de Bruijn sequences.

Findings

Method effectively generates de Bruijn sequences for large memory lengths

Significant reduction in rule search space achieved

Neural network classifier accurately identifies suitable rules

Abstract

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.