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Dichotomy for Axiomatising Inclusion Dependencies on K-Databases

Miika Hannula, Teymur Ismikhanov, Jonni Virtema

TL;DR

A dichotomy regarding the axiomatisation of the entailment of inclusion dependencies over K-databases is established, based on whether the monoid K is weakly absorptive or weakly cancellative, and the so-called balance axiom is established, if one stipulates that the joint weights of each K-relation of a K-database need to be the same.

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.

Dichotomy for Axiomatising Inclusion Dependencies on K-Databases

TL;DR

A dichotomy regarding the axiomatisation of the entailment of inclusion dependencies over K-databases is established, based on whether the monoid K is weakly absorptive or weakly cancellative, and the so-called balance axiom is established, if one stipulates that the joint weights of each K-relation of a K-database need to be the same.

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.
Paper Structure (13 sections, 27 theorems, 40 equations, 2 figures, 6 tables)

This paper contains 13 sections, 27 theorems, 40 equations, 2 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inclusion Dependencies
  4. $\oplus$-Chase
  5. Weakly Cancellative Monoids
  6. Weakly Absorptive Monoids
  7. Countably Absorptive Monoids
  8. Not Countably Absorptive Monoids
  9. Main Theorem: Weakly Absorptive Monoids
  10. Conclusion
  11. Proof of Lemma \ref{['lem:terminates']}
  12. Proof of Lemma \ref{['lem:wcembed']}
  13. Proofs for Lemmas \ref{['lem:wa']} and \ref{['lem:supppreserved']}

Key Result

Proposition 3

Let $\mathbb{K}$ be a positive monoid, and $\mathfrak{D}$ and $\mathfrak{D}'$ be $\mathbb{K}$-databases over a shared schema. Then $\mathrm{Supp}(\mathfrak{D}\oplus \mathfrak{D}')=\mathrm{Supp}(\mathfrak{D})\cup \mathrm{Supp}(\mathfrak{D}')$.

Figures (2)

  • Figure 1: Relationships between monoid properties.
  • Figure 2: Chains generated in WA&$\neg$CA monoids. Top: an unbounded $b$-tail (when natural order is antisymmetric over multiplicities of $b$). Bottom: an eventually periodic $b$-tail, with $(m+\ell)b=mb$ (when natural order is not antisymmetric over multiplicities of $b$).

Theorems & Definitions (56)

  • Example 1: Budgets and expenses
  • Example 2
  • Proposition 3
  • Definition 4
  • Proposition 5
  • Proposition 6
  • proof
  • Proposition 7
  • proof
  • Example 8
  • ...and 46 more