不确定性推理的广义概率模型及其逻辑基础

项目名称： 不确定性推理的广义概率模型及其逻辑基础

项目编号： No.61473336

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 其他

项目作者： 周红军

作者单位： 陕西师范大学

项目金额： 60万元

中文摘要： 信息的不确定性是现实生活中普遍存在的一个基本特征，将概率论和命题逻辑交叉融合是不确定性推理领域多年来的研究热点之一。申请人团队已分别从语义计量化和语构公理化角度建立了概率计量逻辑和有界整剩余格中的广义态理论。在上述工作的基础上，本项目拟进一步以一般剩余格表示全体非经典事件的代数结构，以有界剩余格取代单位区间表示事件概率的取值域，通过引入几类广义态算子表示非经典事件的概率来实现相应子结构命题逻辑的概率计量化研究，并探讨相应模糊概率逻辑的完备代数语义、(强)有限模型性质及可判定性等问题，以期建立不确定性推理的广义概率模型。本项目拟包括以下专题研究：(i)剩余格中广义态算子的代数及拓扑性质；(ii)剩余格中基于广义态算子的相似收敛理论及其柯西完备化；(iii)带有内部广义态算子的剩余格(简称内广态剩余格)的各种格完备化，如并完备、典型完备、核完备及DM-完备等；(iv)内广态剩余格的有限嵌入性。

中文关键词： 概率计量逻辑；广义态理论；剩余格；完备化；有限嵌入性

英文摘要： Uncertainty of information is a fundamental and unavoidable feature of our real life. Investigation on interactions between Probability Theory and Propositional Logics so as to model uncertainty of non-classical events is one of the hot research topics in the field of Reasoning about Uncertainty in recent decades.?The applicant's research group has established?two closely-related inter-disciplines about interactions between Information Science and Mathematics, which we call Probabilistically Quantitative Logic and Generalized State Theory on bounded and integral residuated lattices, from the two points of view of semantic quantification and syntactical axiomatization, respectively.?On the basis of our previous work, the present project aims to introduce, by means of using general (not necessarily bounded or integral) residuated lattices to represent algebraic structures of non-classical events being considered, and of replacing the unit interval with an abitrary bounded residuated lattice to serve as the range of probabilities of non-classical events, several kinds of generalized states as probabilities of non-classical events to realize the probabilistic quantification of Substructural Propositional Logics. Then we will discuss some properties of the obtained Fuzzy Probabilistic Logics such as complete algebraic semantics, (strong) finite model property and decidability to establish a generalized probabilistic model for Reasoning about Uncertainty. The contents of this project include: (i) algebraic and topological properties of generalized states on residuated lattices; (ii) similarity convergence and its Cauchy completion of residuated lattices with respect to generalized states; (iii) various lattice completions, such as join completion, canonical completion, nuclear completion and Dedekind-MacNeille completion, of residuated lattices with internal generalized states, which provide complete algebraic semantics for the associated Fuzzy Probabilistic Logics; (iv) finite embeddability property of residuated lattices with internal generalized states, which implies the (strong) finite model property and decidability of universal theory of the corresponding Fuzzy Probabilistic Logics.

英文关键词： Probabilistically Quantitative Logic;Generalized State Theory;Residuated Lattice;Completion;Finite Embeddability Property