Observability Properties of Colored Graphs
This work addresses observability issues for agents in systems modeled as colored graphs, such as hidden Markov models, but it is incremental as it builds on existing research.
The paper tackles the problem of observability in colored graphs, where an agent tries to determine its current or past positions based on observed colors, by unifying prior notions into a common framework and showing that a graph modification problem is NP-Complete.
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).