A Representation of Antimatroids by Horn Rules and Its Application to Educational Systems

Abstract

We study representations of an antimatroid by Horn rules, motivated by its recent application to computer-aided educational systems. Since an antimatroid is a special union-closed family, it is represented by Horn rules in a natural way. This representation, however, is inconvenient since not every set of Horn rules corresponds to an antimatroid. In this paper, we introduce a useful representation of antimatroids by Horn rules, which associates every set R of Horn rules with an antimatroid A(R). Our representation is computationally implementable. We show that the following basic problems can be solved in linear time, as in the case of the natural representation: Membership problem: given a set R of Horn rules and a set X, is X a member of A(R)? Inference problem: given a set R of Horn rules and a Horn rule (A; q), does (A; q) accept A(R)? Our representation is essentially equivalent to the `circuit' representation of antimatroids by Korte and Lovasz. We establish their relationship, and give a polynomial time algorithm to construct the uniquely-determined minimal circuit representation from a given set of Horn rules. We explain that our results have potential applications to computer-aided educational systems, where an antimatroid is used as a model of the space of possible knowledge states of learners, and is constructed by giving Horn queries to a human expert.