Fuzzy Description Logics - A Survey

Mathematical Fuzzy Logics [51,60] have a long tradition with roots going back to the many-valued logics of Łukasiewicz, Gödel, and Kleene [57, 68, 73] and the Fuzzy Set Theory of Zadeh [111]. Their purpose is to model vagueness or imprecision in the real world, by introducing new degrees of truth as additional shades of gray between the Boolean true and false. For example, one can express the distinction between a person x having a high fever or a low fever as the degree of truth of the logical statement Fever(x). One of the central properties of fuzzy logics is truth functionality—the truth degree of a complex logical formula is uniquely determined by the truth degrees of its subformulas. This is a fundamental difference to other quantitative logics like probabilistic or possibilistic logics [56,83]. The semantics of fuzzy logics are thus given by functions interpreting the logical constructors conjunction, implication, and negation. For example, the truth degree of the conjunction (Fever ∧ Headache)(x) can be computed as a function of the degrees of Fever(x) and Headache(x). The functions proposed by Zadeh [111] are a popular choice, because they lead to good computational properties. A different approach uses operators called triangular norms (t-norms) and their associated residua to interpret conjunction and implication [60,69]. More recently, Description Logics (DLs) were developed as fragments of firstorder logic, with a focus on their computational properties [3]. They use concepts (unary predicates, such as Fever and Headache) and roles (binary predicates like hasSymptom) to describe knowledge about the world. For example, the description logic axiom