A Decomposition Theorem for Probabilistic Transition Systems

In this paper we prove that every nite Markov chain can be de composed into a cascade product of a Bernoulli process and several simple permutation reset deterministic automata The original chain is a state homomorphic image of the product By doing so we give a positive answer to an open question stated in Paz concerning the decomposability of probabilistic systems Our result is based on the observation that in probabilistic transition systems randomness and memory can be separated so as to allow the non random part to be treated using common deterministic automata theoretic tech niques The same separation technique can be applied to other kinds of non determinism as well Preliminaries The object of our study is a probabilistic input output state transition sys tem Its de nition is not new and has appeared under various names in the past e g Arb Paz Sta De nition Probabilistic Transition Systems A probabilistic transi tion system PTS is a quadruple A X Q Y p where X is the input al phabet Q is the state space Y is the output alphabet and p Q X Q Y is the input transition output probability function satisfying for every q Q x X X