Topological Relational Theory: A Simplicial-Complex View of Functional Dependencies, Lossless Decomposition, and Acyclicity

TL;DR

This paper introduces a topological approach to relational schema design by encoding functional dependencies as simplicial complexes, enabling homological analysis to diagnose cyclic dependencies and improve schema decomposition.

Contribution

It develops a novel topological framework using simplicial complexes and homology to analyze and diagnose cyclic dependencies in relational schemas, extending classical database theory.

Findings

Homological invariants diagnose cyclic dependency structures.

Lossless decomposition characterized by topological intersection conditions.

Betti numbers serve as lightweight diagnostics for dependency cycles.

Abstract

We develop a topological lens on relational schema design by encoding functional dependencies (FDs) as simplices of an abstract simplicial complex. This dependency complex exposes multi-attribute interactions and enables homological invariants (Betti numbers) to diagnose cyclic dependency structure. We define Simplicial Normal Form (SNF) as homological acyclicity of the dependency complex in positive dimensions, i.e., vanishing reduced homology for all $n \ge 1$. SNF is intentionally weaker than contractibility and does not identify homology with homotopy. For decompositions, we give a topological reformulation of the classical binary lossless-join criterion: assuming dependency preservation, a decomposition is lossless exactly when the intersection attributes form a key for at least one component. Topologically, this yields a strong deformation retraction that trivializes the relevant Mayer--Vietoris boundary map. For multiway decompositions, we show how the nerve of a cover by induced subcomplexes provides a computable certificate: a 1-cycle in the nerve (detected by $H_1$) obstructs join-tree structure and aligns with cyclic join behavior in acyclic-scheme theory. Finally, we discuss an algorithmic consequence: Betti numbers of the dependency complex (or of a decomposition nerve) can be computed from boundary matrices and used as a lightweight schema diagnostic to localize "unexplained" dependency cycles, complementing standard FD-chase tests.