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Categorical Calculus and Algebra for Multi-Model Data

Jiaheng Lu

TL;DR

This paper proposes two formal query languages: categorical calculus and categorical algebra, by extending relational calculus and relational algebra respectively, and demonstrates the equivalence between these two languages of queries.

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.

Categorical Calculus and Algebra for Multi-Model Data

TL;DR

This paper proposes two formal query languages: categorical calculus and categorical algebra, by extending relational calculus and relational algebra respectively, and demonstrates the equivalence between these two languages of queries.

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.
Paper Structure (13 sections, 6 theorems, 41 equations, 3 figures, 2 tables)

This paper contains 13 sections, 6 theorems, 41 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Categorical Calculus
  4. Categorical Algebra
  5. Set Operators
  6. Unary Set Operators
  7. Binary Set Operators
  8. Ternary operators with graph data
  9. Category operators
  10. Sets and Functions to Category
  11. Category to Set
  12. Algebraic Transformation Rules
  13. Conclusion and future work

Key Result

Theorem 8

The categorical calculus and categorical algebra are equivalent.

Figures (3)

  • Figure 1: This toy example shows three types of data including relation, XML and graph. They have a unified categorical representation.
  • Figure 2: An illustrative example for student course category
  • Figure 3: Categorical schemata for trees and graphs

Theorems & Definitions (14)

  • Definition 1
  • Example 2
  • Example 3
  • Example 4
  • Example 5
  • Example 6
  • Example 7
  • Theorem 8
  • Lemma 9
  • Lemma 10
  • ...and 4 more