Fundamental characterization for state detectability of logical dynamical systems

The problem of using input and output sequences to determine the current state of logical dynamical systems has applications in many fields such as systems biology, sequential logic circuits, finite-strategic dynamical noncooperative games, and etc., and has been widely investigated during the recent years. There is an algorithm for verifying whether all sufficiently long input sequences determine the corresponding current states that runs in time quadratic polynomial of the number of states, inputs and outputs in the literature. In this paper, under the framework of the semi-tensor product of matrices, we propose a quadratic polynomial time algorithm for verifying a weaker and more widely used state detectability stating that whether there is an input sequence that determines the current state. This algorithm is much more effective than the existing algorithms for verifying the weaker state detectability in the literature, and is also applied to the fault diagnosis problem of oxidative stress response and sequential logic circuits.