  In this work we study a non-linear generalization based on affine
transformations of probabilistic and quantum automata proposed recently by
D\'iaz-Caro and Yakary{\i}lmaz \cite{DCY16A} referred as affine automata.
First, we present efficient simulations of probabilistic and quantum automata
by means of affine automata which allows us to characterize the class of
exclusive stochastic languages. Then, we initiate a study on the succintness of
affine automata. In particular, we show that an infinite family of unary
regular languages can be recognized by 2-state affine automata but the state
numbers of quantum and probabilistic automata cannot be bounded. Finally, we
present the characterization of all (regular) unary languages recognized by
two-state affine automata.
