Average Size of Implicational Bases

TL;DR

This paper demonstrates that the average size of implicational bases, specifically the base of proper premises, is quasi-polynomial, using probabilistic analysis of minimal transversals in random hypergraphs.

Contribution

It introduces an average-case analysis showing that implicational bases are typically much smaller than worst-case exponential bounds.

Findings

Average size of proper premises base is quasi-polynomial.

Utilizes results on minimal transversals in random hypergraphs.

Provides probabilistic bounds on implicational base sizes.

Abstract

Implicational bases are objects of interest in formal concept analysis and its applications. Unfortunately, even the smallest base, the Duquenne-Guigues base, has an exponential size in the worst case. In this paper, we use results on the average number of minimal transversals in random hypergraphs to show that the base of proper premises is, on average, of quasi-polynomial size.