基于格蕴涵代数的格值概念格及其不确定性推理与决策研究
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摘要
本文的工作属于人工智能领域中不确定性信息处理的理论与应用研究,旨在建立一种能够直接用于不确定性信息分析及处理的数学模型。研究的基本出发点在于：是客观物理世界中存在着或人类对客观物理世界反映的过程中产生了大量的不确定性；二是人类活动中不得不经常处理(如模糊性、不可比较性等)不确定性信息。相应地,研究的理论依据有：用于刻画不确定性信息的格蕴涵代数和用于形式化描述概念及概念间关系的概念格模型。
本文建立了基于格蕴涵代数的格值概念格,分别对其构成方法,约简方法以及分解计算和合并方法作了研究；针对人工智能研究领域中重要研究方向之一的不确定性推理,本文进一步研究了基于格值概念格的不确定性推理方法,进而将其运用到决策中,建立了格值单目标和多目标决策概念格,为研究带有不确定性信息的决策问题提供了数学工具,具体内容主要分为两部分：
一.关于格值概念格的理论研究方面
1.基于格蕴涵代数建立了格值概念格的数学模型,讨论了它的相关性质；
2.提出了格值概念格的矩阵构造方法,其中定义了非数值型矩阵之间的运算并给出了矩阵蕴涵运算的流程图；
3.提出了格值概念格的约简理论：分别针对于属性集和对象集给出了格值概念格的属性约简方法和对象约简方法以及各自的约简算法；
4.提出了格值概念格的分解合并原理：分别给出了格值形式背景的扩展表示方法,格值概念格的分解算法和相应的合并算法。
二.关于格值概念格的应用研究方面
1.提出了基于格值概念格的不确定性推理方法：根据组成格值形式概念的内涵和外延间的关系以及集合间的内外逼近原理,分别提出了内逼近不确定性推理方法和外逼近不确定性推理方法,并讨论了两种不确定性推理方法的合理性及它们之间的关系；
2.建立了基于格值概念格的两类决策模型：分别建立了格值单目标决策模型和多目标决策模型,相应地给出了决策规则的提取算法,并讨论了决策规则的一些性质。
This paper belongs to the theoretical and applied researches about artificial intelligence with uncertainty, which aims to establish a kind of mathematical model directly used for uncertainty information analysis and processing. The basic starting points of this research mainly include two aspects:one is that not only does there exist various kinds and large portions of uncertainty in the real world but also the uncertainty is naturally generalized in the course of outer information being reflected to human brain; the other is that human intelligence actions are always involved with the qualitative concepts with uncertainty, e.g., fuzziness and incomparability, etc. Correspondingly, the theoretical bases are the lattice implication algebra for depicting uncertainty information and the mathematical model of the concept lattice for describing the concepts and the relation among them.
This paper established the lattice-valued concept lattice and researched its constructing method, reduction method, decomposition and combination operations; uncertainty reasoning, as the one of important research directions in artificial intelligence, this paper further researched uncertainty reasoning based on lattice-valued concept lattice, and applied this uncertainty reasoning method into the decision making and established the lattice-valued single-target decision concept lattice and the multi-target decision concept lattice, which provides the mathematical tools for the research of decision making with uncertainty information. The concrete research contents include the following two parts:
Part one:The theoretical study of lattice-valued concept lattice
1. The mathematical model of lattice-valued concept lattice is established on the lattice implication algebra and some properties are talked about.
2. The matrix constructing method of lattice-valued concept lattice is proposed, where the non-numerical matrix operations are defined and the flow chart of matrix implication operation is given.
3. Reduction methods of lattice-valued concept lattice are proposed, where attribute reduction methods and object reduction methods of lattice-valued concept lattice based on attributes set and objects set are studied and their reduction algorithms are presented.
4. Decomposition and combination theories of lattice-valued concept lattice are proposed, where representation algorithm of extended lattice-valued formal context is given, and the decomposition operation and combination operation of lattice-valued concept lattice are presented.
Part two:The application study of lattice-valued concept lattice
1. Uncertainty reasoning methods based on lattice-valued concept lattice are proposed, where according to the relation between the extent and intent of the lattice-valued formal concept and approximation theory of sets, internal approximate uncertainty reasoning methods and external approximate uncertainty reasoning methods based on lattice-valued concept lattice are given and their rationality and the relationship between them are talked about.
2. Two kinds of decision models based on lattice-valued concept lattice are proposed, the single-target decision model and multi-target decision model based on lattice-valued concept lattice are given, the decision rules extracting algorithms are presented and some properties of decision rules are talked about.
本文建立了基于格蕴涵代数的格值概念格,分别对其构成方法,约简方法以及分解计算和合并方法作了研究；针对人工智能研究领域中重要研究方向之一的不确定性推理,本文进一步研究了基于格值概念格的不确定性推理方法,进而将其运用到决策中,建立了格值单目标和多目标决策概念格,为研究带有不确定性信息的决策问题提供了数学工具,具体内容主要分为两部分：
一.关于格值概念格的理论研究方面
1.基于格蕴涵代数建立了格值概念格的数学模型,讨论了它的相关性质；
2.提出了格值概念格的矩阵构造方法,其中定义了非数值型矩阵之间的运算并给出了矩阵蕴涵运算的流程图；
3.提出了格值概念格的约简理论：分别针对于属性集和对象集给出了格值概念格的属性约简方法和对象约简方法以及各自的约简算法；
4.提出了格值概念格的分解合并原理：分别给出了格值形式背景的扩展表示方法,格值概念格的分解算法和相应的合并算法。
二.关于格值概念格的应用研究方面
1.提出了基于格值概念格的不确定性推理方法：根据组成格值形式概念的内涵和外延间的关系以及集合间的内外逼近原理,分别提出了内逼近不确定性推理方法和外逼近不确定性推理方法,并讨论了两种不确定性推理方法的合理性及它们之间的关系；
2.建立了基于格值概念格的两类决策模型：分别建立了格值单目标决策模型和多目标决策模型,相应地给出了决策规则的提取算法,并讨论了决策规则的一些性质。
This paper belongs to the theoretical and applied researches about artificial intelligence with uncertainty, which aims to establish a kind of mathematical model directly used for uncertainty information analysis and processing. The basic starting points of this research mainly include two aspects:one is that not only does there exist various kinds and large portions of uncertainty in the real world but also the uncertainty is naturally generalized in the course of outer information being reflected to human brain; the other is that human intelligence actions are always involved with the qualitative concepts with uncertainty, e.g., fuzziness and incomparability, etc. Correspondingly, the theoretical bases are the lattice implication algebra for depicting uncertainty information and the mathematical model of the concept lattice for describing the concepts and the relation among them.
This paper established the lattice-valued concept lattice and researched its constructing method, reduction method, decomposition and combination operations; uncertainty reasoning, as the one of important research directions in artificial intelligence, this paper further researched uncertainty reasoning based on lattice-valued concept lattice, and applied this uncertainty reasoning method into the decision making and established the lattice-valued single-target decision concept lattice and the multi-target decision concept lattice, which provides the mathematical tools for the research of decision making with uncertainty information. The concrete research contents include the following two parts:
Part one:The theoretical study of lattice-valued concept lattice
1. The mathematical model of lattice-valued concept lattice is established on the lattice implication algebra and some properties are talked about.
2. The matrix constructing method of lattice-valued concept lattice is proposed, where the non-numerical matrix operations are defined and the flow chart of matrix implication operation is given.
3. Reduction methods of lattice-valued concept lattice are proposed, where attribute reduction methods and object reduction methods of lattice-valued concept lattice based on attributes set and objects set are studied and their reduction algorithms are presented.
4. Decomposition and combination theories of lattice-valued concept lattice are proposed, where representation algorithm of extended lattice-valued formal context is given, and the decomposition operation and combination operation of lattice-valued concept lattice are presented.
Part two:The application study of lattice-valued concept lattice
1. Uncertainty reasoning methods based on lattice-valued concept lattice are proposed, where according to the relation between the extent and intent of the lattice-valued formal concept and approximation theory of sets, internal approximate uncertainty reasoning methods and external approximate uncertainty reasoning methods based on lattice-valued concept lattice are given and their rationality and the relationship between them are talked about.
2. Two kinds of decision models based on lattice-valued concept lattice are proposed, the single-target decision model and multi-target decision model based on lattice-valued concept lattice are given, the decision rules extracting algorithms are presented and some properties of decision rules are talked about.
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