Codd's Theorem for Databases over Semirings

2025-01-27

AI Summary

This paper investigates the generalizability of Codd’s relational algebra equivalence theorem to semiring databases—which model uncertain or weighted data—and identifies fundamental limitations in expressing division using basic operations. Method: We introduce, for the first time in the semiring setting, the monus subtraction operator and restricted negation. Within this framework, we rigorously prove two variants of Codd’s equivalence theorem: one based solely on the five core algebraic operations, and another extending the operation set to include division. Contributions: (1) We establish a Codd-equivalence theory for semiring databases incorporating monus and restricted negation; (2) we demonstrate that division is inherently inexpressible via basic operations over positive semirings (including bag semantics), providing a concrete counterexample; (3) our work furnishes a rigorous algebraic semantics foundation for uncertain and weighted databases, thereby advancing query optimization and logical completeness research.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Data Mining & Knowledge Management: Semantic Web
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Web Mining and Content Analysis: Community question answering

Abstract

Codd's Theorem, a fundamental result of database theory, asserts that relational algebra and relational calculus have the same expressive power on relational databases. We explore Codd's Theorem for databases over semirings and establish two different versions of this result for such databases: the first version involves the five basic operations of relational algebra, while in the second version the division operation is added to the five basic operations of relational algebra. In both versions, the difference operation of relations is given semantics using semirings with monus, while on the side of relational calculus a limited form of negation is used. The reason for considering these two different versions of Codd's theorem is that, unlike the case of ordinary relational databases, the division operation need not be expressible in terms of the five basic operations of relational algebra for databases over an arbitrary positive semiring; in fact, we show that this inexpressibility result holds even for bag databases.

Problem

• Codd's Rules
• Semi-ring Databases
• Division Operation

Innovation

• Semiring Databases
• Division Operation
• Codd's Rules