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A labeling of a graph is a function from the vertices of the graph to some finite set. In 1996, Albertson and Collins defined distinguishing labelings of undirected graphs. Their definition easily extends to directed graphs. Let $G$ be a directed graph associated to the $k$-block presentation of a Bernoulli scheme $X$. We determine the automorphism group of $G$, and thus the distinguishing labelings of $G$. A labeling of $G$ defines a finite factor of $X$. We define demarcating labelings anddoi:10.37236/6 fatcat:qrygkv6qjvdm5c7ogug75tind4