Boundary optimization for rough sets

Abstract

Let n > m ≥ 2 be integers and let A = {A1, . . . , Am} be a partition of [n] = {1, . . . , n}. For X ⊆ [n], its A-boundary region A(X) is defined to be the union of those blocks Ai of A for which Ai ∩ X = ∅ and Ai ∩ ([n] \ X) = ∅. For three different probability distributions on the power set of [n], partitions A of [n] are determined such that the expected cardinality of the A-boundary region of a randomly chosen subset of [n] is minimal and maximal, respectively. The problem can be reduced to an optimization problem for integer partitions of n. In the most difficult case, the concave-convex shape of the corresponding weight function as well as several other inequalities are proved using an integral representation of the weight function. In one case, there is an interesting analogon to the AZ-identity. The study is motivated by the rough set theory.This number can be interpreted as the average of the average relative uncertainty for fixed cardinality of X.Now the question is, how small and how large the numbers µ j (A) can be if all partitions of [n] into m parts are allowed (throughout let j ∈ {1, 2, 3}).

Lattice Representations with Set Partitions Induced by Pairings

“…Granular Computing (briefly GrC) is an emerging paradigm which relies on the idea of partitioning a set of objects in some granules depending on some given criteria [29,30,38,39]. Many ideas and methods of GrC have been used in order to investigate discrete mathematical objects, such as matroids, set partitions and ordered structures [21,25,26,36,37].…”