  Stochastic languages are the languages recognized by probabilistic finite
automata (PFAs) with cutpoint over the field of real numbers. More general
computational models over the same field such as generalized finite automata
(GFAs) and quantum finite automata (QFAs) define the same class. In 1963, Rabin
proved the set of stochastic languages to be uncountable presenting a single
2-state PFA over the binary alphabet recognizing uncountably many languages
depending on the cutpoint. In this paper, we show the same result for unary
stochastic languages. Namely, we exhibit a 2-state unary GFA, a 2-state unary
QFA, and a family of 3-state unary PFAs recognizing uncountably many languages;
all these numbers of states are optimal. After this, we completely characterize
the class of languages recognized by 1-state GFAs, which is the only nontrivial
class of languages recognized by 1-state automata. Finally, we consider the
variations of PFAs, QFAs, and GFAs based on the notion of inclusive/exclusive
cutpoint, and present some results on their expressive power.
