  Nondeterministic weighted automata are finite automata with numerical weights
on transitions. They define quantitative languages L that assign to each word w
a real number L(w). The value of an infinite word w is computed as the maximal
value of all runs over w, and the value of a run as the maximum, limsup,
liminf, limit average, or discounted sum of the transition weights. We
introduce probabilistic weighted automata, in which the transitions are chosen
in a randomized (rather than nondeterministic) fashion. Under almost-sure
semantics (resp. positive semantics), the value of a word w is the largest real
v such that the runs over w have value at least v with probability 1 (resp.
positive probability).
  We study the classical questions of automata theory for probabilistic
weighted automata: emptiness and universality, expressiveness, and closure
under various operations on languages. For quantitative languages, emptiness
and universality are defined as whether the value of some (resp. every) word
exceeds a given threshold. We prove some of these questions to be decidable,
and others undecidable. Regarding expressive power, we show that probabilities
allow us to define a wide variety of new classes of quantitative languages,
except for discounted-sum automata, where probabilistic choice is no more
expressive than nondeterminism. Finally, we give an almost complete picture of
the closure of various classes of probabilistic weighted automata for the
following pointwise operations on quantitative languages: max, min, sum, and
numerical complement.
