Closure via functional dependence simplification

Knowledge discovery in social networks by using a logic-based treatment of implications

“…If oIa for all a ∈ A, then oIb for all b ∈ B. Now we briefly present the rules conforming the inference system (see [33] for further details). To begin with, SL F D considers reflexivity as axiom scheme [Ref] A ⊇ B A →B together with the rules of fragmentation, composition and simplification below:…”

“…This area originated in the late eighties with the seminal works of Ganter and Wille [19] and, since then, has proven to be a useful framework on which several applications have been developed; for instance, one can see papers ranging from ontology merging [9] and resolution of fuzzy or multi-adjoint relational equations [1,14], to applications to the Semantic Web by using the notion of concept similarity or rough sets [18], and from noise control in document classification [31] to the study of fuzzy databases, in areas such as functional dependencies [33], or even linguistics [16].…”

“…[33]). Let K = (G, M, I) a formal context and Γ a basis for K. For all A ⊆ M , the equality A + Γ = A holds.…”

“…[33]). Let A, B and C be sets of attributes, then the following equivalences hold:• If B ⊆ A then {∅ →A, B →C} ≡ {∅ →A ∪ C}.…”

“…(c) LCS = { abd, {ad} , abcd, {acd} } 8. The output of the previous sub-call is incorporated to the labeled closed sets, which is the final output of the algorithm: LCS = { ∅, {∅} , a, {a} , c, {c} , d, {d} , ac, {ac} , cd, {cd} , bd, {b} , abd, {ab, ad} , abcd, {bc, acd} }This method is strongly based on the SL F D Closure algorithm introduced in[33], and is based on Theorem 2 and the following proposition.…”