A dichotomy of functionsF(X, Y)of correlated sources(X, Y)

Consider separate encoding of correlated sources
X^{n}=(X_{l}, \cdots ,X_{n}), Y^{n} = (Y_{l}, \cdots ,Y_{n})
for the decoder to reliably reproduce a function
\{F(X_{i}, Y_{i})\}^{n}_{i=1}
. We establish the necessary and sufficient condition for the set of all achievable rates to coincide with the Slepian-Wolf region whenever the probability density
p(x,y)
is positive for all
(x,y)
.