Verification of detectability in Probabilistic Finite Automata

In this paper we analyze state estimation in stochastic discrete event systems (SDES) that can be modeled as probabilistic finite automata (PFAs). For a given PFA, we obtain the necessary and sufficient conditions that guarantee exact state estimation, at least asymptotically, with increasing certainty as more information is acquired from observing the behavior of the given PFA, by defining the notion of AA-detectability, and providing necessary and sufficient conditions that can be used to verify it. The characterization and analysis of AA-detectability is transformed to a problem of classification between two (or more) PFAs, which capture the recurrent behavior of an underlying Markov process that is obtained by ignoring output behavior and focusing on state transitions in the given PFA. Our approach combines techniques used in classification between two (or more) PFAs with state estimation methods used in logical discrete event systems (DES). We prove that the proposed verification of AA-detectability is of polynomial complexity with respect to the size of the state space of the given PFA.