Rule Induction Based on Rough Sets from Possibilistic Information under Lipski's Approach

How rules are induced on the basis of rough sets under Lipski's approach has been examined in a possibilistic information system where attribute values in information tables are expressed by normal possibility distributions. In Lipski's approach possible tables are created from the original information table and each possible table has a possibilistic degree with which it is the actual information table. In each possible table, rough approximations are obtained for a set of objects. The possible membership degree of an object is the maximum of possibilistic degrees of possible tables where the object is included in the rough approximations. The certain membership degree is one minus the maximum of possibilistic degrees of possible tables where the object is not included in the rough approximations. Therefore, the certain and possible membership degrees are the lower and upper bounds of the membership degree that the object actually has. This leads an object to have membership degrees expressed by not a single, but an interval value for rough approximations, which is essential in possibilistic information systems. For the rough approximations, the complementarity property holds, as is so in complete information systems. This gives justification for dealing with possibilistic information under Lipski's approach. It is required to use rough approximations whose element consists of a pair of an object and a rule that it supports in order to induce rules from an information table. Criterion are introduced for whether or not objects validly supports rules, because they support lots of rules with low degrees in possibilistic information systems. By using the criterions, only rules that are validly supported can be selected from lots of rules.