Knowledge reductions in inconsistent information systems

Knowledge reduction is one of the most important problems in rough set theory. There are many types of knowledge reductions in the area of rough sets. It is required to provide their consistent classification. But most of information systems are not consistent because of various factors such as noise in data, compact representation, prediction capability and so on. To acquire brief decision rules from inconsistent systems, knowledge reductions are needed. The main objective of this paper is to introduce a new concept a knowledge reduction in inconsistent systems. It is referred to as maximum distribution reduction, which preserves all maximum decision rules. The maximum distribution reduction eliminates the harsh requirements of the distribution reduction and overcomes the drawback of the possible reduction that the derived decision rules may be in compatible with the ones derived from the original system. The relationships among distribution reduction, maximum distribution reduction, approximate reduction and assignment reduction are examined. The judgement theorems and discernibility matrixes with respect to those reductions are obtained, from which we can provide new approaches to knowledge reductions in inconsistent information systems.