  Interval Markov chains extend classical Markov chains with the possibility to
describe transition probabilities using intervals, rather than exact values.
While the standard formulation of interval Markov chains features closed
intervals, previous work has considered also open interval Markov chains, in
which the intervals can also be open or half-open. In this paper we focus on
qualitative reachability problems for open interval Markov chains, which
consider whether the optimal (maximum or minimum) probability with which a
certain set of states can be reached is equal to 0 or 1. We present
polynomial-time algorithms for these problems for both of the standard
semantics of interval Markov chains. Our methods do not rely on the closure of
open intervals, in contrast to previous approaches for open interval Markov
chains, and can characterise situations in which probability 0 or 1 can be
attained not exactly but arbitrarily closely.
