The discernibility matrix can possess different information. In this paper, according to different discernible information quantity, four descriptions of the discernibility matrix are proposed, and the four reduct definitions, i.e., Hreduct, S-reduct, B-reduct and P-reduct are presented. The relationship and equivalence for four reducts are researched, and two general reduction algorithms are designed. In order to improve the efficiency of the reduction algorithm, the related concepts of the relative discernibility-based reduct(denoted as RD-reduct) are proposed. The equivalence between the relative discernibility-based reduct and four discernibility matrix-based reducts are revealed. Two effective reduction algorithms based on the relative discernibility are designed. Finally, the example and experiments are used to explain the feasibility and effectiveness of the proposed algorithms.