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

Eyal Weiss, Michael Margaliot

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


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 n 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 O(n 2). We demonstrate the usefulness of these results by studying the properties of a class of randomly generated CBNs.

Original languageEnglish
Article number8540082
Pages (from-to)2727-2736
Number of pages10
JournalIEEE Transactions on Automatic Control
Issue number7
StatePublished - Jul 2019
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 1963-2012 IEEE.


Manuscript received December 6, 2017; revised December 19, 2017 and May 16, 2018; accepted August 24, 2018. Date of publication November 19, 2018; date of current version June 26, 2019. This work was supported by the Israel Science Foundation under ISF Grant 410/15. Recommended by Associate Editor M. Alamir. (Corresponding author: Michael Margaliot.) The authors are with the School of Electrical Engineering-Systems, Tel-Aviv University, Tel Aviv 69978, Israel (e-mail:,[email protected]; [email protected]).

FundersFunder number
Iowa Science Foundation410/15
Israel Science Foundation


    • Boolean networks (BNs)
    • computational complexity
    • logical systems
    • observability
    • random graphs
    • social networks
    • systems biology


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