Decidable and expressive classes of probabilistic automata

k-Hierarchical probabilistic automata (HPA) are probabilistic automata whose states are partitioned into k+1 levels such that for any state and input symbol, at most one transition goes to a state at the same level, and others go to higher level states. We show that 1-HPA, with acceptance threshold 1/2 (in the finite and infinite word cases) can recognize non-regular languages, and the non-emptiness and non-universality problems for 1-HPA with threshold 1/2 are decidable in EXPTIME and are PSPACE-hard. We present a new sufficient condition when 1-HPA recognize regular languages. We show that the emptiness problem is undecidable for 2-HPAs.