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

TL;DR

This paper advances understanding of the finite controllability conjecture for bounded derivation depth rule sets by showing universal models cannot contain large tournaments without implying a loop query, narrowing potential counterexamples.

Contribution

It demonstrates that universal models generated by bdd rule sets cannot contain arbitrarily large tournaments without entailing a loop query, providing new insights into the conjecture.

Findings

Universal models cannot contain arbitrarily large tournaments.

Large tournaments imply the existence of a loop query.

Results narrow the space of potential counterexamples.

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.