  We establish the undecidability of conditional affine information
inequalities, the undecidability of the conditional independence implication
problem with a constraint that one random variable is binary, and the
undecidability of the problem of deciding whether the intersection of the
entropic region and a given affine subspace is empty. This is a step towards
the conjecture on the undecidability of conditional independence implication.
The undecidability is proved via a reduction from the periodic tiling problem
(a variant of the domino problem). Hence, one can construct examples of the
aforementioned problems that are independent of ZFC (assuming ZFC is
consistent).
