About Subsumption in Fuzzy EL

Classical Description Logics (DLs) [2] cannot properly deal with the endemic imprecision of biomedical knowledge. For example, the current version of the SNOMEDCT ontology defines a “Perinatal Cyanotic Attack” as a cardiovascular disorder occurring in the perinatal period and manifested through cyanosis. This definition depends on two vague notions, namely the perinatal period—the period of time around birth—and cyanosis—a bluish discoloration of the skin. While it is possible to say that one year after birth is not perinatal, and a few hours from birth is, there is no precise threshold on the end of the perinatal period. However, it makes sense to say that every child is less in its perinatal period as time goes by. A similar consideration can be made for skin turning from red to blue in cases of cyanosis. The use of several degrees of truth has been proposed for dealing with these gradual changes, as well as other kinds of imprecisions. Mathematical Fuzzy Logic [12] generalizes classical logic by allowing real numbers from the interval [0, 1] to act as truth degrees. It allows to express, e.g. that a newborn child is in the perinatal period with degree 1, but a threeweek-old belongs to this period only with degree 0.3. In Mathematical Fuzzy Logic, the interpretation of the logical constructors, such as conjunction, disjunction, and implication, is determined by the choice of a binary triangular norm (or t-norm). Fuzzy Description Logics combine DLs with Mathematical Fuzzy Logic as a means to formally represent and reason with vague conceptual knowledge [18,19]. So far, research on fuzzy DLs was mainly focused on fuzzy extensions of propositionally closed DLs. Unfortunately, in fuzzy DLs a negation constructor often leads to undecidability [7,11]. To the best of our knowledge, the only fuzzy extensions of EL studied so far are based on the Gödel t-norm [16,20]. In these logics, fuzzy subsumption between concepts can be decided in polynomial time. Beyond this tractable case, very little is known about the complexity of subsumption with general t-norms. If we restrict the set of membership degrees to be finite, subsumption can be decided in exponential time [3,8], but for the interval [0, 1] nothing is known, even for expressive fuzzy DLs in which consistency is decidable [5].