Emptiness Under Isolation and the Value Problem for Hierarchical Probabilistic Automata

k-Hierarchical probabilistic automata (k-HPA) are probabilistic automata whose states are stratified into k + 1 levels such that from any state, on any input symbol, at most one successor belongs to the same level, while the remaining belong to higher levels. Our main result shows that the emptiness and universality problems are decidable for k-HPAs with isolated cut-points; recall that a cut-point x is isolated if the acceptance probability of every word is bounded away from x. Our algorithm for establishing this result relies on computing an approximation of the value of an HPA; the value of a probabilistic automaton is the supremum of the acceptance probabilities of all words. Computing the exact value of a probabilistic automaton is an equally important problem and we show that the problem is co-R.E.-complete for k-HPAs, for k ≥ 2 (as opposed to Π2-complete for general probabilistic automata). On the other hand, we also show that for 1-HPAs the value can be computed in exponential time.