  We consider partially observable Markov decision processes (POMDPs) with
limit-average payoff, where a reward value in the interval [0,1] is associated
to every transition, and the payoff of an infinite path is the long-run average
of the rewards. We consider two types of path constraints: (i) quantitative
constraint defines the set of paths where the payoff is at least a given
threshold {\lambda} in (0, 1]; and (ii) qualitative constraint which is a
special case of quantitative constraint with {\lambda} = 1. We consider the
computation of the almost-sure winning set, where the controller needs to
ensure that the path constraint is satisfied with probability 1. Our main
results for qualitative path constraint are as follows: (i) the problem of
deciding the existence of a finite-memory controller is EXPTIME-complete; and
(ii) the problem of deciding the existence of an infinite-memory controller is
undecidable. For quantitative path constraint we show that the problem of
deciding the existence of a finite-memory controller is undecidable.
