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No Cliques Allowed: The Next Step Towards BDD/FC Conjecture

Lucas Larroque, Piotr Ostropolski-Nalewaja, Michaël Thomazo

TL;DR

It is demonstrated that universal models generated by BDD rule sets cannot contain arbitrarily large tournaments (arbitrarily directed cliques) without entailing a loop query, E(x,x).

Abstract

This paper addresses one of the fundamental open questions in the realm of existential rules: the conjecture on the finite controllability of bounded derivation depth rule sets (bdd $\Rightarrow$ fc). We take a step toward a positive resolution of this conjecture by demonstrating that universal models generated by bdd rule sets cannot contain arbitrarily large tournaments (arbitrarily directed cliques) without entailing a loop query, $\exists{x} E(x, x)$. This simple yet elegant result narrows the space of potential counterexamples to the (bdd $\Rightarrow$ fc) conjecture.

No Cliques Allowed: The Next Step Towards BDD/FC Conjecture

TL;DR

It is demonstrated that universal models generated by BDD rule sets cannot contain arbitrarily large tournaments (arbitrarily directed cliques) without entailing a loop query, E(x,x).

Abstract

This paper addresses one of the fundamental open questions in the realm of existential rules: the conjecture on the finite controllability of bounded derivation depth rule sets (bdd fc). We take a step toward a positive resolution of this conjecture by demonstrating that universal models generated by bdd rule sets cannot contain arbitrarily large tournaments (arbitrarily directed cliques) without entailing a loop query, . This simple yet elegant result narrows the space of potential counterexamples to the (bdd fc) conjecture.
Paper Structure (29 sections, 30 theorems, 30 equations)

This paper contains 29 sections, 30 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Decidability of OBQA
  3. OBQA in the finite
  4. Finite Controllability
  5. Contributions
  6. Organization of the paper
  7. Preliminaries
  8. First-Order Logic
  9. Existential rules
  10. Homomorphisms
  11. Queries
  12. The Chase
  13. UCQ-rewritability
  14. Other Useful Notions
  15. Graph Theoretical Notions
  16. ...and 14 more sections

Key Result

Theorem 1

For every $\mathtt{bdd}$ rule set $\mathcal{R}$ and every instance $\mathcal{I}$ we have:

Theorems & Definitions (69)

  • Example 1
  • Theorem 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Lemma 5
  • proof
  • Definition \ref{def:ucq-rewritability}: rephrased
  • Proposition 6
  • proof
  • ...and 59 more