不确定性推理的广义概率模型及其逻辑基础

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.

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。

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。申请人团队已建立的Borel型概率计量逻辑是本项目申请时具有代表性的研究成果，本项目旨在前期研究基础上从代数角度研究不确定性推理的广义概率模型。本项目首先分别通过语义积分化、模态形式化和代数公理化三种“概率聚合真值”方法较系统梳理和建立了概率计量逻辑理论体系，实现了王国俊团队提出的计量逻辑、美国斯坦福大学Adams团队提出的概率逻辑、捷克科学院Hajek团队提出的模糊概率逻辑以及意大利佛罗伦萨大学Mundici团队建立的逻辑代数上的态理论等间的沟通与统一；其次，着重研究了几类带有广义态算子的剩余格的Stone对偶与同态核的结构刻画；然后，系统研究了剩余偏序集的并完备化及其与核算子间的对应关系，研究了几类剩余偏序集的有限嵌入性，给出了利用并完备构造法证明有限嵌入性的统一且较为简单的证法，证明了有限生成代数簇中字问题的可判定性。此外，从为线性概率推理提供蕴涵模型角度，通过推广实质Boole蕴涵，提出了几类实质模糊蕴涵和模糊余蕴涵，研究了它们的特征刻画及相关代数性质。. 本项目按照申请书计划执行，进展顺利，深入研究了剩余格中广义态算子的代数与拓扑性质、相似收敛理论、并完备化和有限嵌入性等内容，为不确定性推理建立了具有逻辑基础、具有更高语言表达和逻辑推理能力的概率计量化模型，实现了研究目标，达到了研究预期。本项目组成员在科学出版社出版专著1部，发表标注基金资助的学术论文15篇，其中SCI论文5篇，EI论文3篇，受邀作学术报告11次，参与组织举办国内学术会议3次；项目组成员1人晋升教授，并入选2016年度陕西省青年科技新星和2017年度陕西省中青年科技创新领军人才，2人晋升副教授，2人获得博士学位；培养硕士生4人，现有在读博士2人，硕士8人。

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