Direct and Binary Direct Bases for One-set Updates of a Closure System

TL;DR

This paper introduces binary-direct bases for closure systems, providing an optimal algorithmic approach for updating bases when sets are added or removed, improving efficiency in managing closure systems.

Contribution

It defines the binary-direct basis concept and develops algorithms for optimal basis updates in closure systems after set modifications.

Findings

Shortest binary-direct basis exists and is known as the D-basis.

Algorithms for basis updates are effective for singleton set additions or removals.

The approach maintains the basis type after updates, ensuring consistency.

Abstract

We introduce a concept of a binary-direct implicational basis and show that the shortest binary-direct basis exists and it is known as the $D$-basis introduced in Adaricheva, Nation, Rand [Disc.Appl.Math. 2013]. Using this concept we approach the algorithmic solution to the Singleton Horn Extension problem, as well as the one set removal problem, when the closure system is given by the canonical direct or binary-direct basis. In this problem, a new closed set is added to or removed from the closure system forcing the re-write of a given basis. Our goal is to obtain the same type of implicational basis for the new closure system as was given for original closure system and to make the basis update an optimal process.