The joy of implications, aka pure Horn functions: mainly a survey

TL;DR

This paper surveys the mathematical theory of pure Horn functions, highlighting key results from the last thirty years and introducing some new insights, with applications in databases, data mining, and formal concept analysis.

Contribution

It provides a comprehensive overview of pure Horn functions, including recent results and a few novel contributions, along with open problems for future research.

Findings

Discussion of implicational bases and their properties

Introduction of new results on optimum bases and generating closed sets

Presentation of open problems to guide future research

Abstract

Apart from a brief look at applications (Relational Databases, Formal Concept Analysis, data mining) this article is devoted to the mathematical t h e o r y of implications (=pure Horn formulas). It is mainly a survey of results obtained in the last thirty years, but features a few novelties as well. Some keywords: The Duquenne-Guiges (implicational) base, the canonical direct base, prime implicates, the consensus method, implications and meet irreducible closed sets, optimum bases for certain lattices, ordered direct bases, generating all closed sets, general (i.e. impure) Horn functions. We pose four open problems to stimulate further research.