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

Bilge Senturk, Faruk Alpay

TL;DR

A topological lens on relational schema design is developed by encoding functional dependencies (FDs) as simplices of an abstract simplicial complex that exposes multi-attribute interactions and enables homological invariants (Betti numbers) to diagnose cyclic dependency structure.

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.

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

TL;DR

A topological lens on relational schema design is developed by encoding functional dependencies (FDs) as simplices of an abstract simplicial complex that exposes multi-attribute interactions and enables homological invariants (Betti numbers) to diagnose cyclic dependency structure.

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 . 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 ) 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.
Paper Structure (20 sections, 3 theorems, 19 equations, 2 figures)

This paper contains 20 sections, 3 theorems, 19 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries
  4. Relational Schemas and Lossless Join
  5. Simplicial Complexes and Homology
  6. Dependency Complex and Simplicial Normal Form
  7. Canonical Covers
  8. Dependency Complex
  9. Simplicial Normal Form (SNF)
  10. A Case Where SNF is Strictly Different from Directed-Cycle Checks
  11. Lossless Join Through a Topological Lens
  12. Induced Subcomplexes
  13. Binary Lossless Join = Key Intersection = Retraction
  14. Multiway Decompositions and the Nerve Obstruction
  15. Algorithmic Consequences
  16. ...and 5 more sections

Key Result

Proposition 4.1

Let $R(U,F)$ be a schema with canonical cover $F_c$ and dependency complex $K_F$. Consider a dependency-preserving binary decomposition into $U_1,U_2$ and induced subcomplexes $K_1,K_2,K_{12}$ as above. If then there exists a simplicial map (induced by repeated FD application in the closure computation) that collapses $K_1$ onto $K_{12}$ in the following sense: the inclusion $K_{12}\hookrightarro

Figures (2)

  • Figure 1: A three-way decomposition whose nerve has a 1-cycle: all pairwise overlaps exist, but the triple overlap is empty, yielding a cyclic (non-tree) nerve.
  • Figure 2: A join tree for the query $Q=\Join_{i=1}^6 R_i$ with edge labels denoting separators (shared attributes). Acyclicity (tree-shaped structure satisfying the running-intersection property) supports semijoin-based reduction and an optimized evaluation order along the tree.

Theorems & Definitions (17)

  • Definition 3.1: Dependency complex
  • Remark 3.1
  • Example 3.1
  • Definition 3.2: Simplicial Normal Form (SNF)
  • Remark 3.2: What SNF does and does not mean
  • Example 3.2: Directed cycle present, but $H_1(K_F)=0$
  • Proposition 4.1: Topological reformulation of the binary lossless-join test
  • proof
  • Theorem 4.2: Homological Lossless Join Theorem (binary case)
  • proof
  • ...and 7 more