[Draft] High-order estimation-based properties and high-order observers for labeled finite-state automata

In this paper, we consider labeled finite-state automata (LFSAs), extend some state estimation-based properties from a single agent to a finite ordered set of agents. We also extend the notion of observer to \emph{high-order observer} using our \emph{concurrent composition}. As a result, a general framework for characterizing high-order estimation-based properties is built, in which each agent infers its preceding agent's estimation via all agents in front. The high-order observer plays the role of a basic tool to verify such properties. In more detail, in our general framework, the system's structure is publicly known to all agents $A_1,\dots,A_n$; each agent $A_i$ has its own observable event set $E_i$, and additionally knows all its preceding agents' observable events but can only observe its own observable events. The intuitive meaning of our high-order observer is to characterize what agent $A_n$ knows about what $A_{n-1}$ knows about \dots what $A_2$ knows about $A_1$'s state estimate of the system. This general framework can be regarded as an automata representation of dynamic epistemic logic. Compared with the classical representation of dynamic epistemic logic based on fragments of logic, our representation has advantages in property verification and flexibly changing agents to enforce properties. As case studies, this general framework applies to basic properties such as current-state opacity, strong current-state opacity, regular-language-based opacity, critical observability, high-order opacity, etc. Special cases for which verification can be done more efficiently are also discussed.