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

• Published in: IEEE transactions on automatic control Vol. 64; no. 7; pp. 2727 - 2736
• Main Authors: Weiss, Eyal, Margaliot, Michael
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.07.2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Biological system modeling; Boolean algebra; Boolean functions; Boolean networks (BNs); Complex systems; Complexity; computational complexity; Graph theory; Irrigation; logical systems; Observability; Observability (systems); Observers; Polynomials; random graphs; Sensor systems; Sensors; social networks; State variable; systems biology
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2018.2882154

Many complex systems in biology, physics, and engineering include a large number of state variables (SVs), and measuring the full state of the system is often impossible. Typically, a set of sensors is used to measure a part of the SVs. 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 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 (BNs), called conjunctive BNs (CBNs). Using a graph-theoretic approach, we provide a necessary and sufficient condition for observability of a CBN with <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula> SVs and an efficient algorithm for solving the minimal observability problem. The algorithm time complexity is linear in the length of the description of the CBN and in particular it is <inline-formula><tex-math notation="LaTeX">O(n^2)</tex-math></inline-formula>. We demonstrate the usefulness of these results by studying the properties of a class of randomly generated CBNs.