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

We study a representation of an antimatroid by Horn rules, motivated by its recent 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 $\mathcal{R}$. We address algorithmic and Boolean function theoretic aspects on the association ${\cal R} \mapsto $\mathcal{A}$($\mathcal{R}$)$,where${\cal R}$ is viewed as the input. We present linear time algorithms to solve the membership problem and the inference problem for $\mathcal{A}(\mathcal{R})$. We also provide efficient algorithms for generating all members and all implicates of ${\cal A}({\cal R})$. We show that this representation is essentially equivalent to the Korte-Lov'{a}sz representation of antimatroids by rooted sets. Based on the equivalence, we provide a quadratic time algorithm to construct the uniquely-determined minimal representation. % These 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.