基于软集的不确定计算理论融合及其应用研究

Mathematical modeling, analysis and computing of problems with uncertainty is one of the hottest areas in interdisciplinary research involving applied mathematics, computational intelligence and decision science. Soft sets provide us a new way of coping with uncertainty from the viewpoint of parameterization. This research project focuses on the fusion of uncertain computing theories based on soft sets. By virtue of lattice theory, universal algebra and fuzzy sets, we shall establish a fundamental theory of lattice-valued soft sets in a systematical way. We shall further develop the theories as well as methodologies with regard to soft rough computing models by incorporating rough sets and granular computing. This will give rise to some hybrid uncertain computing models in a justifiable sense. The relationships among these uncertain computing models will be ascertained and some properties of related algebraic, topological and order-theoretical structures will be investigated. In addition, we shall study uncertain decision making and data analysis methods based on lattice-valued soft sets and related uncertain computing models. We shall propose algorithms for high speed data mining in lattice-valued information systems and extend the applications of uncertain computing theories to areas such as intelligent information processing of web and complex data. This study is conducive to a better understanding of the essence of uncertainty, which can strengthen the theoretical basis of uncertain computing, and provide some mathematical foundations for intelligent processing of complex heterogeneous data. This can also promote research as well as applications in areas including uncertain mathematics, decision analysis, data science and computational intelligence.

不确定问题的数学建模、分析与计算是应用数学、计算智能和决策科学交叉研究的热点之一。软集从参数化角度为处理不确定性提供了一种新思路。本课题将以软集理论为基础，探索不确定计算理论的有效融合，借助格论、泛代数和模糊集构建格值软集的系统化基础理论，融合粗糙集和粒计算完善基于软集的粗糙计算理论及方法，力图获得一些合理的混合不确定计算模型，明确各种不确定计算模型之间的关系，并深入分析模型相关的代数、拓扑和序结构性质。在此基础上，发展基于格值软集及相关不确定计算模型的不确定决策和数据分析方法，提出格值信息系统中的高速海量数据挖掘算法，并拓展不确定计算理论在面向网络和复杂数据的智能信息处理等领域的应用。该项研究有利于丰富不确定性的数学内涵，深化不确定计算的理论基础，为不确定环境下复杂异构数据的智能化处理提供数学依据，并促进不确定数学、决策分析、数据科学和计算智能等方面的研究及应用。

不确定性是客观和主观现实中广泛存在的一种本质要素。对不确定性的探索最早可追溯至古希腊时代的哲学思辨，是学术界长期关注的根本性问题之一。软集理论强调从参数化的角度认识不确定性和复杂性，为研究不确定性提供了一种新思路。本项目以软集理论为基础，融合程度化、粒度化和参数化等不确定计算中的重要思想，深入分析了不确定性的数学内涵，综合运用软集、格论、泛代数、模糊集、软计算、数理逻辑、决策分析、数据挖掘、最优化理论等领域的相关知识和方法，详细研究了模糊软集的分解和表示、软集逻辑公式及其真度、软关系和半群上的软同余、区间值模糊软集的非经典代数性质及重要不等式、基于参数类化软集的极大关联规则挖掘等重要问题，初步构建了基于软集的不确定计算、数据挖掘和决策分析的理论框架。本项目所取得的研究成果紧跟学科发展前沿，这些成果在发展软集等不确定性数学理论的同时，也为决策支持、数据挖掘和商务智能等领域的技术开发奠定了理论基础，具有广阔的应用前景。