On implicational bases of closure systems with unique critical sets

TL;DR

This paper explores the structure of closure systems with unique critical sets, introduces new bases like the K-basis and E-basis, and discusses algorithms for their recognition and optimization, with implications for Horn Boolean functions.

Contribution

It introduces the K-basis and E-basis for closure systems, provides polynomial algorithms for their recognition, and analyzes their optimality and subclasses with unique critical sets.

Findings

Every optimum basis is also right-side optimum.

Polynomial algorithms exist for recognizing D-relations and certain bases.

Optimization of some basis parts remains NP-complete.

Abstract

We show that every optimum basis of a finite closure system, in D.Maier's sense, is also right-side optimum, which is a parameter of a minimum CNF representation of a Horn Boolean function. New parameters for the size of the binary part are also established. We introduce a K-basis of a general closure system, which is a refinement of the canonical basis of Duquenne and Guigues, and discuss a polynomial algorithm to obtain it. We study closure systems with the unique criticals and some of its subclasses, where the K-basis is unique. A further refinement in the form of the E-basis is possible for closure systems without D-cycles. There is a polynomial algorithm to recognize the D-relation from a K-basis. Thus, closure systems without D-cycles can be effectively recognized. While E-basis achieves an optimum in one of its parts, the optimization of the others is an NP-complete problem.