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In this paper, associated with dominance relation, lattice theory and intuitionistic fuzzy sets theory, the lattice-valued information systems with interval-valued intuitionistic fuzzy decision are proposed and some of its properties are investigated carefully. And, an approach to knowledge reduction based on discernibility matrix in consistent lattice-valued information systems with interval-valued intuitionistic fuzzy decision is constructed and an illustrative example is applied to show its validity. Moreover, extended from the idea of knowledge reduction in consistent information systems, four types of reductions and approaches to obtaining the knowledge reductions of the inconsistent lattice-valued information systems with interval-valued intuitionistic fuzzy decision are formulated via the use of discernibility matrix. Furthermore, examples are considered to show that the approaches are useful and effective. One can obtain that the research is meaningful both in theory and in application for the issue of knowledge reduction in complex information systems.

Rough set theory is emerging as a powerful tool for reasoning about data, knowledge reduction is one of the important topics in the research on rough set theory. It has been proven that finding the minimal reduct of an information system is a NP-hard problem, so is finding the minimal reduct of an incomplete information system. Main reason of causing NP-hard is combination problem of attributes. In this paper, knowledge reduction is defined from the view of information, a heuristic algorithm based on rough entropy for knowledge reduction is proposed in incomplete information systems, the time complexity of this algorithm is O(|A|²|U|). An illustrative example is provided that shows the application potential of the algorithm.

Due to issues such as noise in data, compact representation and prediction capability, many types of knowledge reduction and decision rules have been proposed and applied in inconsistent decision information systems. It is thus important to clarify the interrelationships among the existing types of knowledge reduction. In this paper, the relationships, particularly those suggested in [1], are reconsidered and rectified, and some related results are theoretically improved. In terms of two new types of reducts proposed in this paper together with other existing ones, the method for optimizing all types of decision rules is also discussed in details.

The purpose of this paper is to present a new rough set model for generating negative rules from the incomplete information system. A negative rule indicates that if an object does not satisfy the attribute-value pairs in the condition part, then we can exclude the decision part from such object. The proposed rough set model is constructed on the basis of a difference relation. Such difference relation is a binary relation without any constraints. Moreover, to simplify the negative rules generated from the difference relation-based rough approximations, the concepts of lower, upper approximate and rough reducts are also proposed. Some numerical examples are employed to substantiate the conceptual arguments.