The joy of implications, aka pure Horn formulas: Mainly a survey

Wild, Marcel

doi:10.1016/j.tcs.2016.03.018

Cited by 32 publications

(54 citation statements)

References 45 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here we first summarize the basic relationship between Horn rules, entailments, and closure operators. This is essentially one given by Dowling [16,Section 2]; see [9,36] from the view of implicational systems. We then discuss entailments for antimatroids (or convex geometries).…”

Section: Closure Operator and Entailmentmentioning

confidence: 98%

“…As explained in several papers of implicational systems, a set R of Horn rules is naturally identified with a (pure) Horn Boolean CNF ϕ, and K(R) is identified with the set of true points of ϕ; see [9,Section 5] and [36,Section 3.4]. Various Boolean function theoretic concepts and algorithms are easily adapted to our setting.…”

Section: Preliminariesmentioning

confidence: 99%

“…For completeness, we give a direct proof in Appendix. It should be noted that the circuit concept is extended to general closure spaces/union-closed families [25], and an analogous characterization holds [36,Section 3.3].…”

Section: Korte-lovász Representationmentioning

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

View full text Add to dashboard Cite

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.

show abstract

Section: Closure Operator and Entailmentmentioning

confidence: 98%

Section: Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

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

Yoshikawa

Hirai

Makino

2017

Journal of Mathematical Psychology

View full text Add to dashboard Cite

show abstract

“…The study of implicational bases of closure systems, and convex geometries in particular is quite active, see K. Adaricheva and J.B. Nation [1] and M. Wild [8].…”

Section: Definitionsmentioning

confidence: 99%

Description of closure operators in convex geometries of segments on the line

Adaricheva

Gjonbalaj

2019

Algebra Univers.

View full text Add to dashboard Cite

Convex geometry is a closure space (G, φ) with the anti-exchange property. A classical result of Edelman and Jamison (1985) claims that every finite convex geometry is a join of several linear sub-geometries, and the smallest number of such sub-geometries necessary for representation is called the convex dimension. In our work we find necessary and sufficient conditions on a closure operator φ of convex geometry (G, φ) so that its convex dimension equals 2, equivalently, they are represented by segments on a line. These conditions can be checked in polynomial time in two parameters: the size of the base set |G| and the size of the implicational basis of (G, φ).2010 Mathematics Subject Classification. 05A05, 06A15, 06B99, 52B55.

show abstract

“…For a detailed study about several properties related with the size of these implicational systems, we refer to [21,2]. Finally, there are several algorithms in the literature to calculate from a formal context both the concept lattice and an implicational system.…”

Section: Preliminariesmentioning

confidence: 99%