Abstractions of Finite-State Machines Optimal with Respect to Single Undetectable Output Faults

An observer, whose task is to monitor a large and complex system M̂ subject to malfunctions, may be interested in dealing with a simplified, abstracted model (M̂A of it, at the expense of some loss in fault-detection ability. Let M̂ be a finite- state machine whose inputs are modeled by stationary random variables. The abstraction A is effected by lumping M̂'s states, inputs, and outputs into classes, to obtain a smaller probabilistic machine M̂A. These ideas have been introduced in a previous paper, and the question of finding an optimal abstraction A* which minimizes the number of faults undetectable by the observer was posed. An algorithm for constructing the output component of the optimal abstraction A* is given in this paper. If there are no faults in the next-state map of M̂, this construction is sufficient to minimize the number of single faults in the output map that are undetectable by the observer because of the abstraction. Some experiments carried out using the algorithm provide general insight into the tradeoff between simplifying M̂ and making some faults in it undetectable. As a specific example, optimal output abstractions are found for a finite-state machine specification of the link level of the X.25 communication protocol.