A decomposition theorem for probabilistic transition systems

Abstract

In this paper we prove that every finite Maikov chain can be decomposed 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 [Paz71] concerning the decomposability of probabilistic systems. Our result is based on the surprisingly-original observation that in probabilistic transition systems, “randomness” and “memory” can be separated in such a way that allows the non-random part to be treated using common deterministic automata-theoretic techniques. The same separation technique can be applied as well to other kinds of non-determinism.