Categorical Calculus and Algebra for Multi-Model Data

2026-03-08

Electronic Proceedings in Theoretical Computer Science

AI Summary

This work addresses the challenges of query expression and optimization arising from the coexistence of heterogeneous data models—such as relational, hierarchical, and graph models—in multi-model databases. It introduces, for the first time, a systematic application of category-theoretic methods to this problem, proposing two formal query languages: categorical calculus and categorical algebra, which extend relational calculus and relational algebra, respectively. The study establishes the semantic equivalence of these two languages, develops a set of algebraic transformation rules enabling query optimization, and rigorously characterizes the expressive power and computational complexity of this unified query framework. Collectively, these contributions provide a theoretically sound foundation that simultaneously supports high expressiveness and effective query optimization across diverse data models.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Intelligent Query Processing
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Multi-model databases are designed to store, manage, and query data in various models, such as relational, hierarchical, and graph data, simultaneously. In this paper, we provide a theoretical basis for querying categorical databases. We propose two formal query languages: categorical calculus and categorical algebra, by extending relational calculus and relational algebra respectively. We demonstrate the equivalence between these two languages of queries. We propose a series of transformation rules of categorical algebra to facilitate query optimization. Finally, we analyze the expressive power and computation complexity for the proposed query languages.

Problem

• multi-model databases
• categorical calculus
• categorical algebra
• query languages
• data models

Innovation

• categorical calculus
• categorical algebra
• multi-model databases
• query optimization
• expressive power