Towards the Complexity of Recognizing Pseudo-intents

• Authors: Barış Sertkaya
• Venue: ICCS '09 Proceedings of the 17th International Conference on Conceptual Structures: Conceptual Structures: Leveraging Semantic Technologies
• Year: 2009

Pseudo-intents play a key rôle in Formal Concept Analysis. They are the premises of the implications in the Duquenne-Guigues Base, which is a minimum cardinality base for the set of implications that hold in a formal context. It has been shown that checking whether a set is a pseudo-intent is in conp . However, it is still open whether this problem is conp -hard, or it is solvable in polynomial time. In the current work we prove a first lower bound for this problem by showing that it is at least as hard as transversal hypergraph , which is the problem of identifying the minimal transversals of a given hypergraph. This is a prominent open problem in hypergraph theory that is conjectured to form a complexity class properly contained between p and conp . Our result explains why the attempts to find a polynomial algorithm for recognizing pseudo-intents have failed until now. We also formulate a decision problem, namely first pseudo-intent , and show that if this problem is not polynomial, then, unless p = np , pseudo-intents cannot be enumerated with polynomial delay in a specified lexicographic order.