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We study a representation of an antimatroid by Horn rules, motivated by itsrecent application to computer-aided educational systems. We associate any set$\mathcal{R}$ of Horn rules with the unique maximal antimatroid$\mathcal{A}(\mathcal{R})$ that is contained in the union-closed family$\mathcal{K}(\mathcal{R})$ naturally determined by ${\cal R}$. We addressalgorithmic and Boolean function theoretic aspects on the association ${\cal R}\mapsto \mathcal{A}(\mathcal{R})$, where ${\cal R}$ is viewed as the input. Wepresent linear time algorithms to solve the membership problem and theinference problem for ${\cal A}({\cal R})$. We also provide efficientalgorithms for generating all members and all implicates of ${\cal A}({\calR})$. We show that this representation is essentially equivalent to theKorte-Lov\'{a}sz representation of antimatroids by rooted sets. Based on theequivalence, we provide a quadratic time algorithm to construct theuniquely-determined minimal representation. % These results have potentialapplications to computer-aided educational systems, where an antimatroid isused as a model of the space of possible knowledge states of learners, and isconstructed by giving Horn queries to a human expert.

A representation of antimatroids by Horn rules and its application to educational systems