Coatom Enumeration in Hypergraph Horn Functions: Rank-Three Representations of Horn Model Posets

Abstract

For a finite hypergraph H, the complements of the models of its associated definite Horn CNF are exactly the stopping sets of H; hence the complements of its coatoms are the inclusion-minimal nonempty stopping sets. We study their output-sensitive enumeration from the hypergraph incidence representation. Our main result is a representation of arbitrary Horn model posets whose incidence size is linear in the incidence length of the normalized Horn input. Given a Horn CNF $Γ$, we construct a hypergraph $C(Γ)$ of rank at most three whose proper-model poset is inclusion-order isomorphic to the model poset of $Γ$; equivalently, each source model has a unique extension to a proper target model. Thus maximal models of $Γ$ correspond bijectively to target coatoms. Combining this representation with the maximal-model construction of Kavvadias, Sideri, and Stavropoulos shows that coatom enumeration is not in OutputP unless P=NP, even when all hyperedges have size two or three. Incidence splitting reduces maximum element frequency to three while preserving the stopping-set poset, and a local replacement of two-element hyperedges yields the same lower bound for three-uniform hypergraphs of maximum element frequency at most three. These thresholds are conditionally sharp for arbitrary-order enumeration: rank at most two and maximum element frequency at most two both admit output-linear total-time generation; in the frequency-two case, a polynomial-delay, polynomial-space algorithm is also available. In contrast, coatom extension is NP-complete already for three-uniform hypergraphs of exact element frequency two.