Initial-and - Final-State Detectability of Nondeterministic Finite-State Automata

Detectability is a crucial notion regarding the state estimation problem of real-world systems. In this paper, we propose a new notion of detectability, namely initial-and-final-state detectability (IFD), which indicates the ability to determine the initial state, the current state, and subsequent states of a system. Similar to prior concepts of detectability, we categorize IFD into two variants known as weak initial-and-final-state detectability (WIFD) and strong initial-and-final-state detectability (SIFD). On the one hand, in the case of WIFD, we scrutinize its connection with two other detectability concepts: weak detectability and weak initial-state detectability. Similarly, in the context of SIFD, we explore its relationship with strong detectability and strong initial-state detectability. On the other hand, for a system modeled as a nondeterministic finite-state automaton, we present a method of constructing multiple I-labeled observers to verify its WIFD. We also demonstrate the feasibility of using a detector and a reversed verifier to verify SIFD for the system.