Undecidability of Network Coding, Conditional Information Inequalities, and Conditional Independence Implication

TL;DR

This paper proves that key problems in network coding, information inequalities, and conditional independence are undecidable, resolving long-standing open questions in information theory and network analysis.

Contribution

It establishes the undecidability of these problems using novel constructions inspired by database dependencies and group automorphisms.

Findings

Decidability of network coding is impossible in general.

Conditional information inequalities are undecidable.

Implication of conditional independence among variables is undecidable.

Abstract

We resolve three long-standing open problems, namely the (algorithmic) decidability of network coding, the decidability of conditional information inequalities, and the decidability of conditional independence implication among random variables, by showing that these problems are undecidable. The proof utilizes a construction inspired by Herrmann's arguments on embedded multivalued database dependencies, a network studied by Dougherty, Freiling and Zeger, together with a novel construction to represent group automorphisms on top of the network.