# A Polynomial-Time Algorithm for Solving the Minimal Observability Problem in Conjunctive Boolean Networks

## TL;DR

This paper presents a polynomial-time algorithm for determining the minimal set of sensors needed to make conjunctive Boolean networks observable, enhancing system monitoring in biology, physics, and engineering.

## Contribution

It provides a necessary and sufficient graph-theoretic condition and an efficient algorithm for the minimal observability problem in conjunctive Boolean networks.

## Key findings

- Efficient $O(n^2)$ algorithm for minimal observability
- Characterization of observability conditions for CBNs
- Application to random CBNs analysis

## Abstract

Many complex systems in biology, physics, and engineering include a large number of state-variables, and measuring the full state of the system is often impossible. Typically, a set of sensors is used to measure part of the state-variables. A system is called observable if these measurements allow to reconstruct the entire state of the system. When the system is not observable, an important and practical problem is how to add a \emph{minimal} number of sensors so that the system becomes observable. This minimal observability problem is practically useful and theoretically interesting, as it pinpoints the most informative nodes in the system. We consider the minimal observability problem for an important special class of Boolean networks, called conjunctive Boolean networks (CBNs). Using a graph-theoretic approach, we provide a necessary and sufficient condition for observability of a CBN with $n$ state-variables, and an efficient~$O(n^2)$-time algorithm for solving the minimal observability problem. We demonstrate the usefulness of these results by studying the properties of a class of random CBNs.