GPT-5.6 Pro solves 30-year complexity theory problem

The problem concerns finite closure systems—mathematical structures consisting of families of subsets that contain a universal set U and remain closed under intersection. These systems admit two elementary specifications. An implicational specification uses rules of the form A → b (if A is a subset of X, then b must be in X), and describes all subsets X satisfying every rule. An intersection specification lists subsets M1, ..., Mt and consists of all possible intersections of subfamilies from this list. The core question: do an implicational specification and an intersection specification define exactly the same closure system?

This simple-to-state question has resisted resolution across multiple fields. In 1995, Roni Khardon showed that efficiently translating between Horn formulas and their characteristic models is equivalent to deciding whether a proposed list of characteristic models is complete, but the exact computational complexity remained open. By ISAAC 2025, this problem and its variants—including the enumeration of irreducible closed sets from acyclic implications—were still described as "widely open." Related open questions appeared in Formal Concept Analysis (pseudo-intents and the Duquenne–Guigues basis) and in database theory (whether two functional dependency specifications define the same dependencies).

The new result proves that the equivalence test is coNP-complete. The hardness holds even under restrictive assumptions: when the implications are acyclic, when each premise contains at most three elements, and even when every listed subset is provably correct and cannot be removed without changing the resulting closure system. In this constrained setting, the only hard remaining step is deciding whether a required set is missing from the implicational specification. Because coNP-completeness implies the problem is at least as hard as any problem in the complexity class coNP, and because P equals NP remains unproven (and is believed to be false), this result rules out polynomial-time algorithms for the general case. The implications cascade through three research communities: through standard correspondences, the theorem makes the Characteristic Models Identification problem coNP-complete, the FD–Relation Equivalence problem coNP-complete, and rules out output-polynomial algorithms for generating all pseudo-intents, the complete Duquenne–Guigues basis, and minimum functional-dependency covers.

The proof was generated by OpenAI's GPT-5.6 Pro through ChatGPT and checked by the author, who published the work on July 18, 2026. The author takes full responsibility for the mathematical claims and the final manuscript, and has made the proof publicly available.

The problem at the center of this result has roots stretching back to the mid-1990s, when Khardon connected Horn formula translation to the completeness of characteristic models but left the complexity unresolved. Over three decades, the question persisted across multiple research communities, appearing under different names: in Formal Concept Analysis as the completeness of pseudo-intents and the Duquenne–Guigues basis, and in database theory as functional dependency equivalence and Armstrong relation completeness. Despite its clear statement—does one specification match the other?—the problem resisted solution for thirty years, even for restricted cases like acyclic convex geometries.

The breakthrough proves the equivalence test is coNP-complete, meaning it sits at the boundary of computational hardness. Even in the restricted case where premises contain at most three elements and all listed subsets are correct, the problem remains hard. This hardness result immediately implies that no polynomial-time algorithm can solve the general case unless P equals NP, a universally believed but unproven conjecture in computer science. The theorem therefore closes off an entire class of algorithmic approaches: one cannot hope to generate complete canonical lists in time proportional to the input size plus output size, even in the most constrained geometric setting.

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