Full characterisation of attractors of two intersected asynchronous Boolean automata cycles

TL;DR

This paper characterizes the transient and long-term dynamics of intersecting asynchronous Boolean automata cycles, advancing understanding of their computational capabilities through a new formalism inspired by algorithms.

Contribution

It provides a novel characterization of the dynamics of double-cycle automata networks, introducing an efficient formalism for describing their update sequences.

Findings

Characterization of transient and asymptotic behaviors of double-cycles

Introduction of an algorithm-inspired formalism for update sequences

Enhanced understanding of automata network complexity

Abstract

The understanding of Boolean automata networks dynamics takes an important place in various domains of computer science such as computability, complexity and discrete dynamical systems. In this paper, we make a step further in this understanding by focusing on their cycles, whose necessity in networks is known as the brick of their complexity. We present new results that provide a characterisation of the transient and asymptotic dynamics, i.e. of the computational abilities, of asynchronous Boolean automata networks composed of two cycles that intersect at one automaton, the so-called double-cycles. To do so, we introduce an efficient formalism inspired by algorithms to define long sequences of updates, that allows a better description of their dynamics than previous works in this area.