Equivalence of Fair Diagnosability and Stochastic Diagnosability of Discrete Event Systems

The failure diagnosis problem for a wide variety of systems has been studied using Discrete Event System (DES) models. The DES framework declares a fault as non-diagnosable if there is a cycle through failure states which cannot be distinguished from a similar cycle through normal states. Thorsley et al. showed that mere presence of such a cycle through faulty states to declare the failure non-diagnosable, does not hold for many system e.g., once having continuous dynamics. Thorsley et al. proposed a new DES paradigm where the classical DES model was augmented with probabilities of transitions. In the stochastic framework, failure is considered diagnosed when it is found that probability of the system traversing though failure states is higher than a threshold. Latter, Biswas et al. have proposed another DES paradigm to handle similar systems, where fairness was augmented to the classical DES model. The claim was, the abstraction employed in obtaining DES models from many systems e.g., those having continuous dynamics often obliterates the fairness property. The diagnosability condition in this case checks if there exists equivalent Strongly Connected Components (SCCs) involving failure states and normal states. The present paper establishes formal equivalence of stochastic DES and Fair DES frameworks and the diagnosability conditions.