A novel matrix approach to observability analysis of finite automata

In this paper, the observability of finite automata (acronym is FA) that contain both deterministic finite automata and non-deterministic finite automata is investigated under the framework of the semi-tensor product of matrices. For both initial state and current state cases, two different observability definitions with or without input information are considered, respectively. First, we show that how the observability problem of initial state of FA can be transformed to the construction problem of an initial state-outputs matrix that presents the relationship between initial state and outputs. Second, a current state-outputs matrix to verify the observability problem of current state is given. When two matrices are obtained, four theorems to verify the observability of initial state and current state are presented, respectively. In particular, compared with the existing approach, the proposed approach not only provides a unified verification for the two types of observability of both initial state and current state but also reduces the computational complexity considerably. An illustrative example is presented to show the theoretical results.