Abstract Rough set theory and evidence theory are two approaches to handle decision making and reduction problems of imprecise and uncertain knowledge. It is motivated that Pythagorean fuzzy set excels at describing the situation where the sum of non-membership degree and membership degree is greater than 1, and may have wider applications than intuitionistic fuzzy set. So in this paper we study probability measure of Pythagorean fuzzy sets and belief structure of Pythagorean fuzzy information systems based on rough set theory, and discuss the reduction of Pythagorean fuzzy information systems. First, we review the properties of Pythagorean fuzzy sets and the upper and lower Pythagorean fuzzy rough approximation operators on the level sets. Then using these properties, probability measure of Pythagorean fuzzy sets are constructed. And the belief and plausibility functions are studied by using the Pythagorean fuzzy rough upper and lower approximation operators. Finally, we apply the belief function to construct an attribute reduction algorithm, and an example is employed to illustrate the feasibility and validity of the algorithm.

中文翻译：

---

勾股模糊粗略逼近空间的置信函数及其应用

摘要 粗糙集理论和证据理论是处理不精确和不确定知识的决策和约简问题的两种方法。其动机是勾股模糊集擅长描述非隶属度和隶属度之和大于1的情况，可能比直觉模糊集有更广泛的应用。因此，本文基于粗糙集理论研究了勾股模糊集的概率测度和勾股模糊信息系统的置信结构，并讨论了勾股模糊信息系统的约简。首先，我们回顾了勾股模糊集的性质和水平集上的上、下勾股模糊粗近似算子。然后利用这些性质，构造毕达哥拉斯模糊集的概率测度。并利用毕达哥拉斯模糊粗略上下近似算子研究了置信度和似真度函数。最后，我们应用置信函数构造了一个属性约简算法，并通过一个例子来说明该算法的可行性和有效性。