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However, in reallife situations, it is sometimes impossible to de ne a classifying concept in a precise (crisp) manner. For example, given a new case, it may not be possible to know to which class it belongs. The best knowledge derived from past cases may only give us enough information to say that this new case belongs to a boundary between certain cases, which may consist of various possible solutions. The formulation of these lower and upperset approximations can be generalized to some arbitrary level of precision which forms the basis of rough concept approximations. Some basic de nitions and examples of rough set theory are described in Appendix D. Let us now look at an example (see Table 2.2) that illustrates the concept of rough sets. Let us consider a patient information system consisting of six patients, where each one is described by three health conditions: C1 , C2 , and C3 , and an illness by y1 or y2. Now the question is how to describe an illness uniquely (i.e., y1 or y2 ) in terms of the health conditions. After analyzing the data, it is clear that this question cannot be answered uniquely. For example, patients 3 and 6 have the same conditions, but patient 3 suffers from y1 , whereas patient 6 suffers from y2 . Hence, it is impossible to give a unique description of illness y1 or y2. The best we could decide is that patients 1, 2, and 5 surely suffer from illness y1 , whereas patients 1, 2, 3, 5, and 6 possibly suffer from illness y1 . Similarly, patient 4 surely suffers from illness y2 , whereas patients 3, 4, and 6 possibly suffer from illness y2 . Since it is impossible to give the unique characteristics of y1 or y2, a rough set based technique can be used. Two sets corresponding to the concepts of surely and possibly, the lower and upper approximations for each illness, respectively, describe the characteristics of y1 and y2 . The formal de nition of rough sets given by Pawlak [4] is as follows: Any subset B of A determines a binary relation IB on U, called an indiscernibility relation, and de ned as x IB y if and only if Attr(x) Attr(y) for every Attr 2 B, where Attr(x) denotes the value of attribute a for element x. IB is obviously an equivalence relation. The family of all equivalence classes of IB (i.e., the partition determined by B) will be denoted by U/IB, or simply U/B; an equivalence class of IB (i.e., the block of the partition U/B) containing x will be denoted by [x]B. If (x,y) belongs to IB, x and y will be considered as Bindiscernible. 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In the third step it is only ensured that no directed cycles (first rule of the third step) and no additional vstructures are introduced (second rule of the third step). That is, these two rules fix edge directions that are implied by the vstructures found in the second step. Hence, if we cancel the third rule and thus implicitly accept that the algorithm may stop with a graph with some undirected edges, it yields a compact representation of the class of all Markov equivalent graphs that are perfect maps. Although this algorithm appears to be simple and convenient, there are some problems connected with it, which we are going to discuss, together with attempts at their solution, in the remainder of this section. 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