  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. We study expressiveness and closure
questions about these quantitative languages.
  We first show that the set of words with value greater than a threshold can
be non-$\omega$-regular for deterministic limit-average and discounted-sum
automata, while this set is always $\omega$-regular when the threshold is
isolated (i.e., some neighborhood around the threshold contains no word). In
the latter case, we prove that the $\omega$-regular language is robust against
small perturbations of the transition weights.
  We next consider automata with transition weights 0 or 1 and show that they
are as expressive as general weighted automata in the limit-average case, but
not in the discounted-sum case.
  Third, for quantitative languages $L_1$ and $L_2$, we consider the operations
$\max(L_1,L_2)$, $\min(L_1,L_2)$, and $1-L_1$, which generalize the boolean
operations on languages, as well as the sum $L_1 + L_2$. We establish the
closure properties of all classes of quantitative languages with respect to
these four operations.
