  We investigate logics and equivalence relations that capture the qualitative
behavior of Markov Decision Processes (MDPs). We present Qualitative Randomized
CTL (QRCTL): formulas of this logic can express the fact that certain temporal
properties hold over all paths, or with probability 0 or 1, but they do not
distinguish among intermediate probability values. We present a symbolic,
polynomial time model-checking algorithm for QRCTL on MDPs.
  The logic QRCTL induces an equivalence relation over states of an MDP that we
call qualitative equivalence: informally, two states are qualitatively
equivalent if the sets of formulas that hold with probability 0 or 1 at the two
states are the same. We show that for finite alternating MDPs, where
nondeterministic and probabilistic choices occur in different states,
qualitative equivalence coincides with alternating bisimulation, and can thus
be computed via efficient partition-refinement algorithms. On the other hand,
in non-alternating MDPs the equivalence relations cannot be computed via
partition-refinement algorithms, but rather, they require non-local
computation. Finally, we consider QRCTL*, that extends QRCTL with nested
temporal operators in the same manner in which CTL* extends CTL. We show that
QRCTL and QRCTL* induce the same qualitative equivalence on alternating MDPs,
while on non-alternating MDPs, the equivalence arising from QRCTL* can be
strictly finer. We also provide a full characterization of the relation between
qualitative equivalence, bisimulation, and alternating bisimulation, according
to whether the MDPs are finite, and to whether their transition relations are
finitely-branching.
