On entropic and almost multilinear representability of matroids

TL;DR

This paper proves that determining entropic and almost-multilinear representability of matroids is undecidable, impacting areas like network coding and secret sharing, by linking these problems to undecidable group theory problems.

Contribution

It establishes the undecidability of generalized matroid representations related to information theory and cryptography, answering longstanding open questions.

Findings

Deciding entropic representability is undecidable.

Deciding almost-multilinear representability is undecidable.

Implications for secret sharing and network coding are significant.

Abstract

This article studies two notions of generalized matroid representations motivated by algorithmic information theory and cryptographic secret sharing. The first (entropic representability) involves discrete random variables, while the second (almost-multilinear representability) deals with approximate subspace arrangements. In both cases, we prove that determining whether an input matroid has such a representation is undecidable. Consequently, the conditional independence implication problem is also undecidable, providing an independent answer to a question posed by Geiger and Pearl, recently resolved by Cheuk Ting Li. These problems are also closely related to characterizing achievable rates in network coding and constructing secret sharing schemes. For example, another corollary of our work is that deciding whether an access structure admits an ideal secret sharing scheme is undecidable. Our approach reduces undecidable problems from group theory to matroid representation problems. Specifically, we reduce the uniform word problem for finite groups to entropic representability and the word problem for sofic groups to almost-multilinear representability. A key part of this reduction involves modifying group presentations into forms where linear representations are generic in an appropriate sense when restricted to the generating set.