On the complexity of acyclic modules in automata networks

TL;DR

This paper studies acyclic modules in automata networks, providing bounds on attractors and complexity results for cycle detection, advancing understanding of their computational properties.

Contribution

It extends the formalism of modules to acyclic cases and analyzes their limit behavior and complexity in automata networks.

Findings

Upper bound on the number of attractors in acyclic modules

Complexity results for cycle detection depending on inputs and cycle size

A formal description of acyclic modules' limit behavior via output functions

Abstract

Modules were introduced as an extension of Boolean automata networks. They have inputs which are used in the computation said modules perform, and can be used to wire modules with each other. In the present paper we extend this new formalism and study the specific case of acyclic modules. These modules prove to be well described in their limit behavior by functions called output functions. We provide other results that offer an upper bound on the number of attractors in an acyclic module when wired recursively into an automata network, alongside a diversity of complexity results around the difficulty of deciding the existence of cycles depending on the number of inputs and the size of said cycle.