A colored graph is a directed graph in which nodes or edges have been assigned colors that are not necessarily unique. Observability problems in such graphs consider whether an agent observing the colors of edges or nodes traversed on a path in the graph can determine which node they are at currently or which nodes were visited earlier in the traversal. Previous research efforts have identified several different notions of observability as well as the associated properties of graphs for which those observability properties hold. This paper unifies the prior work into a common framework with several new results about relationships between those notions and associated graph properties. The new framework provides an intuitive way to reason about the attainable accuracy as a function of lag and time spent observing, and identifies simple modifications to improve the observability of a given graph. We show that one form of the graph modification problem is in NP-Complete. The intuition of the new framework is borne out with numerical experiments. This work has implications for problems that can be described in terms of an agent traversing a colored graph, including the reconstruction of hidden states in a hidden Markov model (HMM).

中文翻译：

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彩色图的可观察性属性

彩色图是一种有向图，其中节点或边已被分配不一定唯一的颜色。此类图中的可观察性问题考虑了观察在图中路径上遍历的边或节点的颜色的代理是否可以确定它们当前位于哪个节点或在遍历中较早访问了哪些节点。先前的研究工作已经确定了几种不同的可观察性概念以及这些可观察性属性所适用的图的相关属性。本文将先前的工作统一到一个通用框架中，其中包含关于这些概念和相关图属性之间关系的几个新结果。新框架提供了一种直观的方式来推断可达到的准确度作为滞后和观察时间的函数，并确定简单的修改以提高给定图的可观察性。我们表明图修改问题的一种形式是在 NP-Complete 中。新框架的直觉通过数值实验得到证实。这项工作对可以用代理遍历彩色图来描述的问题具有影响，包括隐藏马尔可夫模型 (HMM) 中隐藏状态的重建。