Rational approximations are introduced and studied in granular graded sets and generalizations thereof by the first author in recent research papers. The concept of rationality is determined by related ontologies and coherence between granularity, parthood perspective and approximations used in the context. In addition, a framework is introduced by her in the mentioned paper(s). Granular approximations constructed as per the procedures of VPRS are likely to be more rational than those constructed from a classical perspective under certain conditions. This may continue to hold for some generalizations of the former; however, a formal characterization of such conditions is not available in the previously published literature. In this research, theoretical aspects of the problem are critically examined, uniform generalizations of granular VPRS are introduced, new connections with granular graded rough sets are proved, appropriate concepts of substantial parthood are introduced, and their extent of compatibility with the framework is accessed. Furthermore, meta applications to cluster validation, image segmentation and dynamic sorting are invented. Basic assumptions made are explained, and additional examples are constructed for readability.

翻译：第一位作者在最近的研究论文中将理性近似引入颗粒分级集,并对其进行研究。合理性的概念由相关主理学以及颗粒性、局部性观点和背景中所用近近似之间的一致性决定。此外,她还在上述论文中引入了一个框架。根据VPRS程序构建的颗粒近近近近可能比在某些条件下从古典角度构建的更合理。这可能继续维持对前者的某种概括性;然而,以前出版的文献中没有正式描述这种条件。在这一研究中,对问题的理论方面进行了严格审查,对颗粒性VPRS进行了统一的一般化,对颗粒性VPRS进行了统一化,证明了与颗粒性分级粗糙组的新联系,提出了适当的实质性部分概念,并了解了与框架兼容的程度。此外,还发明了对集群验证、图像分化和动态排序的元应用。还解释了基本假设,并构建了其他可读性实例。