On Verifying and Engineering the Well-gradedness of a Union-closed Family

Details On Verifying and Engineering the Well-gradedness of a Union-closed Family On Verifying and Engineering the Well-gradedness of a Union-closed Family

2007-04-23

arxiv.org/abs/0704.2919v3

J. Mathematical Psychology 53(1):34-39, 2009

Mathematics

Combinatorics

15 pages

Scientific paper

10.1016/j.jmp.2008.09.002

Current techniques for generating a knowledge space, such as QUERY, guarantees that the resulting structure is closed under union, but not that it satisfies wellgradedness, which is one of the defining conditions for a learning space. We give necessary and sufficient conditions on the base of a union-closed set family that ensures that the family is well-graded. We consider two cases, depending on whether or not the family contains the empty set. We also provide algorithms for efficiently testing these conditions, and for augmenting a set family in a minimal way to one that satisfies these conditions.

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