Non interactive simulation of correlated distributions is decidable

A basic problem in information theory is the following: Let P = (X;Y) be an arbitrary distribution where the marginals X and Y are (potentially) correlated. Let Alice and Bob be two players where Alice gets samples fxigi1 and Bob gets samples fyigi1 and for all i, (xi; yi) P. What joint distributions Q can be simulated by Alice and Bob without any interaction? Classical works in information theory by Gacs-Körner and Wyner answer this question when at least one of P or Q is the distribution Eq (Eq is defined as uniform over the points (0; 0) and (1; 1)). However, other than this special case, the answer to this question is understood in very few cases. Recently, Ghazi, Kamath and Sudan showed that this problem is decidable for Q supported on f0; 1gf0; 1g. We extend their result to Q supported on any finite alphabet. Moreover, we show that If Q can be simulated, our algorithm also provides a (non-interactive) simulation protocol. We rely on recent results in Gaussian geometry (by the authors) as well as a new smoothing argument inspired by the method of boosting from learning theory and potential function arguments from complexity theory and additive combinatorics.