求解联合布尔网络中最小可观测性问题的多项式时间算法
A Polynomial-Time Algorithm for Solving the Minimal Observability Problem in Conjunctive Boolean Networks
AI总结：
本文提出了一种多项式时间算法，用于解决联合布尔网络中的最小可观测性问题，通过图论方法给出了可观测性的必要和充分条件，并通过研究随机联合布尔网络的性质验证了结果的实用性。
摘要：
许多生物、物理和工程中的复杂系统包含大量的状态变量，测量整个系统的完整状态通常是不可能的。通常会使用一组传感器来测量部分状态变量。如果这些测量能够重建整个系统的状态，则称该系统为可观测的。当系统不可观测时，一个重要且实用的问题是如何添加最少数量的传感器使系统变得可观测。这个最小可观测性问题在实践中和理论上都有价值，因为它能确定系统中最信息量大的节点。我们考虑了联合布尔网络（CBNs）这一重要特殊类别的最小可观测性问题。通过图论方法，我们为具有n个状态变量的CBN提供了可观测性的必要和充分条件，并给出了一个高效的O(n²)时间算法来解决最小可观测性问题。我们通过研究随机CBNs类别的性质来展示这些结果的实用性。
英文摘要：
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.
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