The opacity of stochastic discrete event system is studied for stochastic system models. Firstly, the notions of stochastic infinite-step opacity and stochastic K-step opacity are formalized. By constructing a validator based on a two-way observer, a necessary and sufficient condition is obtained to verify the stochastic infinite-step opacity and stochastic K-step opacity of systems by using the validator of two-way observer. Finally, the verification algorithm for stochastic infinite-step opacity and stochastic K-step opacity based on two-way observer is proposed.

Aiming at the problem that the existing fault diagnosis method has too high requirements for systems, the weak diagnosability of fuzzy discrete event system (FDES) is studied, and a weak fuzzy diagnosability method is proposed, which extends the weak diagnosability method of classical discrete event system (DES) to the fuzzy system. Firstly, the notion of weak fuzzy diagnosability of FDES is formalized; In order to verify the weak fuzzy diagnosability of fuzzy systems, a verifier automaton is constructed, and the necessary and sufficient conditions for the weak fuzzy diagnosability of FDES are obtained, in which the weak fuzzy fault diagnosis of fuzzy systems is realized. This method is suitable for weak fault diagnosis of both FDES and classical DES.

Most existing studies of decentralized stochastic discrete-event systems (SDESs) usually focused on faults caused by a single event, while in many practical applications, faults were usually resulted from the successive occurrence of a series of events. To address the problem of detecting such faults, in this paper, a pattern safe diagnosis method is proposed for decentralized SDESs. First, the notion of pattern safe codiagnosability of decentralized stochastic systems is formalized. Then, by constructing a pattern safe codiagnoser, a sufficient and necessary condition of pattern safe codiagnosability is presented based on the pattern safe codiagnoser for decentralized SDESs. As a result, the pattern safe diagnosis for decentralized SDESs can be performed.

In recent years, the opacity of discrete event systems has achieved remarkable application results in information security mechanisms, attracting increasing attention from the researchers of the world. With the aim to address the security issue of leakage paths in the system, this paper proposes a method to enhance the $k$-step opacity of the system through active control based on the framework of stochastic discrete event systems, ensuring that all secret states in the system are not disclosed. First, the $k$-step active opacity of stochastic discrete event systems is formalized. Second, by constructing a $k$-step active opacity verifier, a controllable event string with the lowest leakage probability is found for each secret state in the system, allowing the system to enter the secret state through the safest path by enforcing controllable events. In this way, the $k$-step opacity can be enhanced, and additionally a necessary and sufficient condition for the $k$-step active opacity of stochastic discrete event systems is obtained, which can be used to verify the $k$-step active opacity of stochastic discrete event systems.

In many practical applications, faults are frequently caused by the occurrence of specific events in succession (i.e., pattern faults) rather than a single failure event. The safe diagnosis method based on pattern faults can diagnose the event string that triggers faults. However, the static observation of the original system may not be able to fully capture system faults for complex systems due to the pre-defined and fixed set of observable and unobservable events, leading to the omission of the fault diagnosis. The purpose of this research is to address the limitations of traditional static observation methods. For this reason, a safe diagnosis method based on the dynamic observation of pattern faults in stochastic discrete-event systems (SDESs) is proposed. First, the concepts of S-type and T-type pattern safe diagnosability for SDESs under dynamic observations are formally introduced. Then,a pattern-safe diagnoser and a forbidden language recognizer are constructed to diagnose the pattern faults. Finally, the necessary and sufficient conditions for pattern-safe diagnosability of SDESs under dynamic observations are presented, and an example is provided to illustrate the results.

In recent years, the research on opacity of discrete event systems has attracted widespread attention due to the successful application of opacity in information security of cyber-physical systems. In this research, the enforcement issue of the current state opacity is investigated under the framework of fuzzy discrete event systems modeled by fuzzy automata. The notion of active opacity of fuzzy systems is formalized. For a fuzzy system that does not possess current state opacity, an active control strategy is introduced and then an approach to active opacity is proposed. By constructing a controllability verifier of fuzzy secret states, a controllable event sequence is identified for each secret state, which reinforces the opacity of the fuzzy system by allowing the system to enter a secret state undetected by intruders. Based on the controllability verifier, the necessary and sufficient condition for the active opacity of fuzzy systems is presented, and a corresponding verification algorithm is proposed.