On entropic and almost multilinear representability of matroids

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.