A new look at the critical observability of finite state machines from an algebraic viewpoint

The traditional critical observability of FSMs refers to the possibility of detecting if the current state of an FSM belongs to a critical set for any input. In practical applications, the input is always of finite length, and we may only care about some critical states at a given time step or the inverse problem that how to find all possible inputs moving an FSM to a critical state of interest. Thus, we propose two new concepts about critical observability to cope with these two cases. Using the algebraic model of FSMs proposed in recent years, we propose several algebraic criteria for these types of critical observability. Moreover, the results are also applicable to large-scale networks of FSMs.