The Undecidability of Conditional Affine Information Inequalities and Conditional Independence Implication with a Binary Constraint

TL;DR

This paper proves the undecidability of certain problems in information theory related to conditional inequalities and independence, using a reduction from the periodic tiling problem, indicating some problems are independent of standard set theory.

Contribution

It establishes the undecidability of conditional affine information inequalities and related problems, advancing understanding of their computational limits.

Findings

Undecidability of conditional affine information inequalities.

Undecidability of conditional independence implication with a binary variable.

Undecidability of intersection emptiness in entropic regions.

Abstract

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).