A combinatorial problem about binary necklaces and attractors of Boolean automata networks

TL;DR

This paper explores a combinatorial problem involving binary necklaces with forbidden factors and relates it to the dynamics of Boolean automata networks, showing how cycle intersections reduce attractor counts exponentially.

Contribution

It introduces a new combinatorial problem involving constrained binary necklaces and connects it to the behavior of Boolean automata networks, providing formal support for attractor reduction.

Findings

Adding cycle intersections causes exponential decrease in attractors

Constrained necklaces relate to automata network behaviors

Formal proof supports attractor reduction hypothesis

Abstract

It is known that there are no more Lyndon words of length n than there are periodic necklaces of same length. This paper considers a similar problem where, additionally, the necklaces must be without some forbidden factors. This problem relates to a different context, concerned with the behaviours of particular discrete dynamical systems, namely, Boolean automata networks. A formal argument supporting the following idea is provided: addition of cycle intersections in network structures causes exponential reduction of the networks' number of attractors.