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

2025-12-24

AI Summary

Traditional modeling of communication networks using random variables fails to formally express structural network constraints. Method: This work models communication networks—including network coding and index coding—as confusion hypergraphs, introduces a hypergraph-based Heyting algebra to support logical operations (conjunction, disjunction, implication) on information, and encodes network constraints as intuitionistic logic formulas. Contribution/Results: We establish the first Curry–Howard–style correspondence between coding problems and intuitionistic logic formulas, enabling algebraic, automated derivation of optimal codes from network constraints. Our framework precisely computes the optimal communication cost in log-scale—matching the hypergraph entropy up to a bounded error—and unifies classical coding scenarios (e.g., linear, vector, and functional coding) under a single formalism. It provides a verifiable, composable, logic-based framework for coding design, bridging information theory and constructive logic.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Modeling information using hyperconfusions for logical operations
• Expressing communication network requirements as logical formulas
• Computing optimal coding schemes via hypergraph Heyting algebra

Innovation

• Using hyperconfusions to model information as Heyting algebra
• Expressing network coding requirements as logical formulae
• Computing optimal coding via hypergraph entropy