Dichotomy for Axiomatising Inclusion Dependencies on K-Databases

2026-02-20

AI Summary

This study investigates the implication problem for inclusion dependencies over K-databases, where K is a positive commutative monoid. By analyzing algebraic properties of K—specifically weak cancellativity and weak absorptivity—the work establishes an axiomatic dichotomy: when K is weakly cancellative, the standard axiom system is complete; when K is weakly absorptive, a weak symmetry axiom must be introduced; and if all K-relations are further required to have uniform total weight, a balance axiom is additionally necessary. This paper provides the first characterization of the boundary for axiomatic completeness of inclusion dependencies in K-databases, introduces the weak symmetry and balance axioms to handle non–weakly-cancellative cases, and reveals their equivalence under specific conditions, thereby unifying the semantic framework for distributional K-relations.

Technology Category

• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

A relation consisting of tuples annotated by an element of a monoid K is called a K-relation. A K-database is a collection of K-relations. In this paper, we study entailment of inclusion dependencies over K-databases, where K is a positive commutative monoid. We establish a dichotomy regarding the axiomatisation of the entailment of inclusion dependencies over K-databases, based on whether the monoid K is weakly absorptive or weakly cancellative. We establish that, if the monoid is weakly cancellative then the standard axioms of inclusion dependencies are sound and complete for the implication problem. If the monoid is not weakly cancellative, it is weakly absorptive and the standard axioms of inclusion dependencies together with the weak symmetry axiom are sound and complete for the implication problem. In addition, we establish that the so-called balance axiom is further required, if one stipulates that the joint weights of each K-relation of a K-database need to be the same; this generalises the notion of a K-relation being a distribution. In conjunction with the balance axiom, weak symmetry axiom boils down to symmetry.

Problem

• inclusion dependencies
• K-databases
• axiomatisation
• entailment
• commutative monoid

Innovation

• inclusion dependencies
• K-databases
• dichotomy
• weakly cancellative
• balance axiom