Optimum basis of finite convex geometry

TL;DR

This paper demonstrates that the F-basis is optimal for convex geometries, especially in key parts of the basis, and explores conditions under which further optimization is possible, with implications for computational tractability.

Contribution

It establishes the optimality of the F-basis in convex geometries and identifies conditions for further basis optimization, extending previous results and highlighting open problems.

Findings

F-basis is optimal in convex geometries

Optimization of non-binary implications under certain properties

Tractable basis for convex geometries of order convex subsets

Abstract

Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right sides (conclusions) of binary implications and left sides (premises) of non-binary ones. The right sides of non-binary implications can also be optimized, when the convex geometry either satisfies the Carousel property, or does not have $D$-cycles. The latter generalizes a result of P.L.~Hammer and A.~Kogan for acyclic Horn Boolean functions. Convex geometries of order convex subsets in a poset also have tractable optimum basis. The problem of tractability of optimum basis in convex geometries in general remains to be open. [1] K. Adaricheva and J.B.Nation, On implicational bases of closure systems with unique critical sets, arxiv:1205.2881