Certain Answers in a Rough World

One of the main challenges in knowledge representation and reasoning is still to cope with vague and imprecise information in an adequate manner. This imprecision is found in many knowledge domains, particularly medicine and life sciences. A typical source of imprecision in these domains arises from the level of detail in which the knowledge is described. For example, a disease is usually diagnosed through a series of symptoms that a patient presents, but two individuals, say Ana and Bob, showing the same symptoms might in fact suffer from different maladies. Thus, while these individuals might be equivalent from a symptomatic point of view, they might be classified into different illness classes. One of the many approaches suggested for handling imprecise knowledge is based on rough approximations. Generally speaking, the individuals in a domain are partitioned into equivalence classes, based on their indiscernibility according to the current level of detail. An individual belongs to the upper approximation of a class C (denoted C), if it is indiscernible from some element of C. For instance, Ana and Bob are in the same symptomatic equivalence class. If Bob is diagnosed with, say the Cooties, then Ana potentially has the Cooties, too. In rough terminology, Ana is in the upper approximation of Cooties (Cooties). An analogous lower approximation of a class can be defined, too. Intuitively, C contains the prototypical elements of the class C: if an element x belongs to C, then every element indiscernible from x is guaranteed to belong to C. Rough extensions of Description Logics (DLs) [1] have been proposed as a formalism for handling these upper and lower approximations [5]. An important example is the rough DL ELρ, which extends EL with two new rough constructors. Formally, ELρ concepts are built from concept names A and role names r through the grammar rule C ::= A | > | C u C | ∃r.C | C | C. The semantics of this logic is based on interpretations I = (∆I , ·I , ρI) that extend standard interpretations by an equivalence relation ρI over the elements of ∆I . The interpretation function is extended to the classical constructors in the usual way, and to the rough constructors by setting