Weighted automata are nondeterministic automata with numerical weights on
transitions. They can define quantitative languages~$L$ that assign to each
word~$w$ a real number~$L(w)$. In the case of infinite words, the value of a
run is naturally computed as the maximum, limsup, liminf, limit-average, or
discounted-sum of the transition weights. The value of a word $w$ is the
supremum of the values of the runs over $w$. We study expressiveness and
closure questions about these quantitative languages.

We study observation-based strategies for partially-observable Markov
decision processes (POMDPs) with omega-regular objectives. An observation-based
strategy relies on partial information about the history of a play, namely, on
the past sequence of observations. We consider the qualitative analysis
problem: given a POMDP with an omega-regular objective, whether there is an
observation-based strategy to achieve the objective with probability~1
(almost-sure winning), or with positive probability (positive winning). Our
main results are twofold.