As generalizations of single-valued information systems, interval-valued information systems (IVISs) can better express the real data with uncertainty in some applications. Attribute reduction methods for complete IVISs or complete interval-valued decision systems (IVDSs) have been developed. However, there are few researches on attribute reduction for incomplete interval-valued information systems (IIVISs). The paper aims to investigate the attribute reduction issue in IIVISs. Firstly, the maximal and minimal distances, which characterize the difference between two interval values, are defined, and the maximal and minimal similarity degrees are given. Secondly, the fuzzy $\alpha$-similarity relation is defined based on similarity between interval values, and the concept of $\alpha$-equivalence relation is raised. Thirdly, entropy measures are investigated for IIVISs in view of $\alpha$-equivalence relations. Fourthly, a new attribute reduction approach for IIVISs is proposed by using conditional entropy, and its corresponding algorithm is given. Finally, experiments to verify the effectiveness and feasibility of the newly proposed approach for attribute reduction in IIVISs are presented. These results will be helpful to perfect the uncertainty measurement model, and provide an approach for attribute reduction in IIVISs.

作为单值信息系统的推广，区间值信息系统（IVIS）在某些应用中可以更好地表达具有不确定性的真实数据。已经开发了完整 IVIS 或完整区间值决策系统 (IVDS) 的属性约简方法。然而，关于不完全区间值信息系统（IIVISs）的属性约简的研究很少。本文旨在研究 IIVIS 中的属性约简问题。首先定义表征两个区间值差异的最大和最小距离，并给出最大和最小相似度。其次，模糊$\alpha$-基于区间值之间的相似性定义相似性关系，以及 $\alpha$- 提出了等价关系。第三，考虑到 IIVIS 的熵测度$\alpha$- 等价关系。第四，提出了一种新的基于条件熵的IIVISs属性约简方法，并给出了相应的算法。最后，通过实验来验证新提出的 IIVIS 属性约简方法的有效性和可行性。这些结果将有助于完善不确定度测量模型，并为IIVISs中的属性约简提供一种方法。