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

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, ∃ E (x,x). This simple yet elegant result narrows the space of potential counterexamples to the (bdd⇒fc) conjecture.