Completeness of Canonical Closure Representations Is coNP-Complete

Abstract

A finite closure system on a finite set $U$ is a family of subsets that contains $U$ and is closed under intersections. It can be specified in two elementary ways: an implicational specification lists rules $A \to b$ and consists of all $X \subseteq U$ satisfying every rule, while an intersection specification lists subsets $M_1, \ldots, M_t$ and consists of all intersections of subfamilies of that list. We ask whether one specification of each kind defines the same family. This question has remained open in several guises for about thirty years: Khardon (1995) showed that translating between Horn formulas and their characteristic models is equivalent to deciding completeness of a proposed list of characteristic models, leaving the exact complexity open; at ISAAC 2025, enumerating irreducible closed sets from implications was still described as "widely open," even for acyclic convex geometries; closely related questions concerned pseudo-intents and the Duquenne-Guigues basis in Formal Concept Analysis, and functional dependencies and Armstrong relations in databases. We prove that the equivalence test is coNP-complete. Hardness holds for acyclic implications with premises of size at most three, even when every listed subset is correct and none can be removed without changing the closure system generated by the list; the hard part is deciding whether a required set is missing. Unless $P = NP$, the complete canonical lists cannot be generated in time polynomial in the input plus the total output size, even for acyclic convex geometries. Through standard correspondences, the theorem makes Characteristic Models Identification and FD-Relation Equivalence coNP-complete and rules out output-polynomial algorithms for Horn characteristic models, all pseudo-intents of a given formal context (equivalently, its Duquenne-Guigues basis), and premises of minimum functional-dependency covers.