  Recently, two extensions of Wyner's common information\textemdash exact and
R\'enyi common informations\textemdash were introduced respectively by Kumar,
Li, and El Gamal (KLE), and the present authors. The class of common
information problems involves determining the minimum rate of the common input
to two independent processors needed to exactly or approximately generate a
target joint distribution. For the exact common information problem, exact
generation of the target distribution is required, while for Wyner's and
$\alpha$-R\'enyi common informations, the relative entropy and R\'enyi
divergence with order $\alpha$ were respectively used to quantify the
discrepancy between the synthesized and target distributions. The exact common
information is larger than or equal to Wyner's common information. However, it
was hitherto unknown whether the former is strictly larger than the latter for
some joint distributions. In this paper, we first establish the equivalence
between the exact and $\infty$-R\'enyi common informations, and then provide
single-letter upper and lower bounds for these two quantities. For doubly
symmetric binary sources, we show that the upper and lower bounds coincide,
which implies that for such sources, the exact and $\infty$-R\'enyi common
informations are completely characterized. Interestingly, we observe that for
such sources, these two common informations are strictly larger than Wyner's.
This answers an open problem posed by KLE. Furthermore, we extend Wyner's,
$\infty$-R\'enyi, and exact common informations to sources with countably
infinite or continuous alphabets, including Gaussian sources.
