A Polynomial-Time Algorithm for Solving the Minimal Observability Problem in Conjunctive Boolean Networks
This work solves a key theoretical and practical problem for a specific class of Boolean networks, offering an efficient algorithm for sensor placement in systems modeled by CBNs.
The authors provide a necessary and sufficient condition for observability in conjunctive Boolean networks and an O(n^2)-time algorithm to solve the minimal observability problem, demonstrated on random CBNs.
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.