Majority Boolean networks classifying density: structural characterization and complexity

TL;DR

This paper characterizes the structural properties of majority Boolean networks that solve the Density Classification Task, identifying forbidden patterns and their computational complexity.

Contribution

It provides a structural characterization of networks solving DCT and proves the complexity of detecting forbidden patterns is NP and PSPACE complete.

Findings

Characterization of networks solving DCT via forbidden patterns

Detection of forbidden patterns is NP-complete

Detection of forbidden patterns is PSPACE-complete

Abstract

Given a set of entities each holding a Boolean state, the Density Classification Task (DCT) asks them to converge to the most represented state. Given a directed graph of entities where each node synchronously updates to the local majority among its in-neighbors, we characterize in terms of three forbidden patterns when it solves DCT, and show that discovering these patterns is complete for NP and PSPACE.