Finite Input and Given Time Critical Observabilities of Finite State Machines Based on the Semi-tensor Product of Matrices

The notion of safety in safety-critical applications can be represented by the concept of critical observability. The traditional concept of critical observability refers to possibility of detecting if the current state of a nondeterministic finite state machine (NFSM) belongs to a so-called critical set for any input and any initial state. In practice, the input is always of finite length and we may only care about the state of an FSM at a given time step or the inverse problem that how to find all possible inputs moving the FSM to a critical state of interest. Thus, we propose two new concepts about critical observability, critical observability with finite input and critical observability at a given time step, to cope with these two cases. From the view of algebraic model resulting from the theory of semi-tensor product (STP), we propose corresponding algebraic criteria and algorithms. They are based on matrix operation, which means that they naturally avoid the design of critical observers and can be solved efficiently by computers.