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

Abstract

Consider separate encoding of correlated sources <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">X^{n}=(X_{l}, \cdots ,X_{n}), Y^{n} = (Y_{l}, \cdots ,Y_{n})</tex> for the decoder to reliably reproduce a function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\{F(X_{i}, Y_{i})\}^{n}_{i=1}</tex> . 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p(x,y)</tex> is positive for all <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(x,y)</tex> .