Coding-Logic Correspondence: Turning Information and Communication Networks into Logical Formulae via Hypergraph Heyting Algebra

TL;DR

This paper introduces a novel hypergraph-based logical framework for modeling and optimizing communication networks, enabling direct computation of optimal coding schemes through a logical algebraic approach.

Contribution

It establishes a new correspondence between information networks and logical formulae using hypergraph Heyting algebra, facilitating direct calculation of optimal coding schemes.

Findings

Hypergraph Heyting algebra models information and communication networks.

Optimal coding schemes are computed via hypergraph entropy.

A logical formula representation simplifies network coding optimization.

Abstract

We propose using confusion hypergraphs (hyperconfusions) as a model of information. In contrast to the conventional approach using random variables, we can now perform conjunction, disjunction and implication of information, forming a Heyting algebra. Using the connection between Heyting algebra and intuitionistic logic, we can express the requirements of a communication network (e.g., network coding, index coding, Slepian-Wolf coding) as a logical formula, allowing us to use the hypergraph Heyting algebra to directly compute the optimal coding scheme. The optimal communication cost is simply given by the entropy of the hypergraph (within a logarithmic gap). This gives a surprising correspondence between coding settings and logical formulae, similar to the Curry-Howard correspondence between proofs and computer programs.