Learning Spaces, and How to Build Them

Doignon, Jean-Paul

doi:10.1007/978-3-319-07248-7_1

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…This construction of an antimatroid from a union-closed family was implicit in [11]. Its importance was recently recognized by [5,14] in the literature of Knowledge Space Theory (KST) [8]. We show that this antimatroid coincides with the above join-closed subset of shortest galleries between the two flags.…”

Section: Introductionmentioning

confidence: 89%

Two flags in a semimodular lattice generate an antimatroid

Hayashi¹,

Hirai²

2022

Preprint

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A basic property in a modular lattice is that any two flags generate a distributive sublattice. It is shown , Herscovic 1998 that two flags in a semimodular lattice no longer generate such a good sublattice, whereas shortest galleries connecting them form a relatively good join-sublattice. In this note, we sharpen this investigation to establish an analogue of the two-flag generation theorem for a semimodular lattice. We consider the notion of a modular convex subset, which is a subset closed under the join and meet only for modular pairs, and show that the modular convex hull of two flags in a semimodular lattice of rank n is isomorphic to a union-closed family on [n]. This family uniquely determines an antimatroid, which coincides with the join-sublattice of shortest galleries of the two flags.

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Section: Introductionmentioning

confidence: 89%

Two flags in a semimodular lattice generate an antimatroid

Hayashi¹,

Hirai²

2022

Preprint

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“…We thank Jean-Paul Doignon for helpful comments and references [14,16,27,35] of KST, and the referees for helpful comments and references [2,3,9,25] of implicational systems. We also thank Marcel Wild for remarks.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…We present a simple revision of the QUERY algorithm to output an antimatroid. Our revision is obtained by replacing K(P) with A(P), and is understood as a concrete realization of Doignon's adjusted QUERY algorithm [14].…”

Section: Generating All Nontrivial Implicates Of A(r)mentioning

confidence: 99%

“…Therefore the resulting A(P) might be a reasonable outer approximation of the true antimatroid L. Also sufficiently many queries uncover L as L = A(P); see [14,Proposition 13]. Doignon asked what about the revised QUERY is applied to the expert following a union-closed family L. Also, in this case, L ⊆ K(P) holds throughout iterations.…”

mentioning

confidence: 99%

“…Therefore the resulting A(P) might be a reasonable outer approximation of the true antimatroid L. Also sufficiently many queries uncover L as L = A(P); see [14,Proposition 13].…”

mentioning

confidence: 99%

See 2 more Smart Citations

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

Yoshikawa

Hirai

Makino

2017

Journal of Mathematical Psychology

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We study a representation of an antimatroid by Horn rules, motivated by its recent application to computer-aided educational systems. We associate any set R of Horn rules with the unique maximal antimatroid A(R) that is contained in the union-closed family K(R) naturally determined by R. We address algorithmic and Boolean function theoretic aspects on the association R → A(R), where R is viewed as the input. We present linear time algorithms to solve the membership problem and the inference problem for A(R). We also provide efficient algorithms for generating all members and all implicates of A(R). We show that this representation is essentially equivalent to the Korte-Lovász 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.

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Investigating Markov Model Accuracy in Representing Student Programming Behaviours

Kandjimi,

Suleman

2024

Communications in Computer and Information Science

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Learning Spaces, and How to Build Them

Cited by 10 publications

References 15 publications

Two flags in a semimodular lattice generate an antimatroid

Two flags in a semimodular lattice generate an antimatroid

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

Investigating Markov Model Accuracy in Representing Student Programming Behaviours

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