General Horn false-positive damage bound

Establish that, for every integer s≥2, there exists a finite constant C_s such that, in every definite Horn theory whose rule bodies contain at most s atoms, a single false-positive answer at an atom a outside a closed truth set M⋆ has uniformly random-order expected damage at most C_s c_a^{1−1/s}, where c_a is the number of newly true atoms generated by closing M⋆∪{a}, including a.

Background

The paper analyzes how a wrong oracle answer propagates through greedy constraint pruning. For definite Horn theories, a false positive at an atom can activate additional rule conclusions, and the resulting damage depends on both the size of the affected closure and the number of premises in each rule.

The authors prove an exact asymptotic law for a shared-premise Horn hub: with at most s premises per rule, the expected damage grows as k^{1−1/s}. The conjecture proposes that this exponent, expressed in terms of the cone size c_a, is a universal upper bound for all definite Horn theories with rule bodies of size at most s. The hub construction shows that the exponent cannot be improved.