Dichotomy for Axiomatising Inclusion Dependencies on K-Databases

TL;DR

This paper investigates the logical foundations of inclusion dependencies in K-databases, establishing a dichotomy based on properties of the monoid K, and characterizing when standard axioms are sufficient for implication.

Contribution

It introduces a dichotomy for axiomatising inclusion dependencies over K-databases based on monoid properties, extending understanding of entailment in weighted database models.

Findings

Standard axioms are complete if K is weakly cancellative.

Additional axioms are needed if K is weakly absorptive.

Balance axiom generalizes distribution constraints.

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.