Rough Set Logic

Origin. Pawlak (1982). Indiscernibility. Approximations. Incomplete information. Foundation of approximate reasoning.

Models. Approximation spaces. Lower and upper approximations. Boundary regions. Indiscernibility relations.

Formalism.

Approximation space: ⟨U, R⟩ where: U: universe of objects. R: equivalence relation (indiscernibility). [x]_R: equivalence class of x.

Lower approximation: R*(X) = {x ∈ U : [x]_R ⊆ X}. Certainly in X. All indiscernible objects in X.

Upper approximation: R*(X) = {x ∈ U : [x]_R ∩ X ≠ ∅}. Possibly in X. Some indiscernible object in X.

Boundary region: BN_R(X) = R*(X) - R*(X). Uncertain membership. Rough set: nonempty boundary. Crisp set: empty boundary.

Rough membership: μ_X(x) = |[x]_R ∩ X| / |[x]_R|. Degree of membership. 0 = certainly out. 1 = certainly in. Between = boundary.

Modal interpretation: □A ↔ lower approximation. ◇A ↔ upper approximation. S5 modality. Necessity = certainty.

Rough logic operators: Rough equality: X ≈ Y iff R*(X) = R*(Y) and R*(X) = R*(Y). Rough inclusion. Approximation operations.

Symbols.

Metatheory. Approximation spaces. Modal correspondence. Topological connection. Algebraic structure.

Applies to. Data mining. Incomplete information. Approximate reasoning. Knowledge discovery.

Limitations. Equivalence relations only (extended later). Attribute selection. Computational complexity. Discretization.