【作者】 潘小东；

【导师】 徐扬；

【作者基本信息】 西南交通大学 ， 交通信息工程及控制， 2010， 博士

【摘要】 近四十年来,由于不确定性推理在控制系统中的广泛应用,关于不确定性推理的逻辑基础-非经典逻辑理论的研究更加引起了人们广泛的关注。自从Pavelka等人在二十世纪七十年代末初步建立命题模糊逻辑理论以来,非经典逻辑已经成为人工智能研究领域里的一个重要研究方向,同时也是不确定性推理研究的一个关键问题。一方面,非经典逻辑在机器自动证明理论、多智能体系统、程序验证等领域获得了广泛的应用。另一方面,它也丰富和发展了纯数学理论的研究。格值逻辑是一种重要的非经典逻辑,它不仅能刻画全序性信息,而且还能刻画非全序(即不可比较)的不确定性信息。本文的主要工作是建立基于格蕴涵代数的程度化的格值命题逻辑的语义和语法理论,并建立基于格值命题逻辑系统的语言真值不确定性推理理论和推理方法,同时构造相应的不确定性推理算法。第一部分,本文得到了由格蕴涵代数所诱导的格蕴涵序半群的代数特性。引入了两个新的概念：格蕴涵n-序半群和格蕴涵p-序半群；证明了一个格蕴涵n-序半群是剩余半群,一个格蕴涵p-序半群是算术格序半群；定义了格蕴涵n-序半群之间的同态映射,并在此基础之上刻画了格蕴涵n-序半群和格蕴涵p-序半群中的滤子和s1理想的代数性质。本文还给出了格蕴涵n-序半群中s1理想的几类典型的扩张。这些讨论有望为进一步研究格蕴涵代数的性质和基于格蕴涵代数的格值逻辑提供了一种新的思路。本文也给出了语言真值格蕴涵代数L18的所有的子代数、滤子以及LI理想。第二部分,本文在基于格蕴涵代数的格值命题逻辑系统Lp中建立了广义重言式理论。定义了系统Lp中的L-型重言式、L-型矛盾式以及α-重言式等概念,给出了几类广义重言式之间的关系定理。本文定义了L-型模糊逻辑公式集的可满足性概念,在此基础之上,给出了语义闭包算子的概念,并基于语义结论算子定义了信息的相容性、理论等概念。给出了语义闭包算子的紧致性、逻辑紧致性定理以及相应的闭包系统,同时也建立由某个给定的信息所诱导的Fp上的同余关系,并建立了相应的商代数理论。结合模糊集和L-模糊集合理论,本文也得到了由语义闭包算子所诱导的P(Fp)上的闭包算子的性质定理。第三部分,本文建立了一种程度化的基于格蕴涵代数的格值命题逻辑系统Lp的语法理论。定义了程度化的形式证明和语法结论算子,证明了一些常用的定理,定义了基于某种信息的可证等价关系并证明了关于可证等价关系的几个重要定理。研究了所建立的语法理论与第三章所建立的语义理论的协调性问题,建立了广义演绎定理和某些特殊情况下的完备性定理。这些研究拟为构造基于格值逻辑系统的不确定性推理方法提供必要的理论准备。第四部分,本文刻画了基于语言真值格蕴涵代数L18的格值命题逻辑系统的一些性质,并针对三类典型的不确定性推理模型,建立了基于程度化的格值命题逻辑系统的语言真值不确定性推理理论和推理方法,同时,也构造了相应的不确定性推理算法。从逻辑语义和语法上,本文也详细分析了所建立的不确定性推理方法和推理算法的合理性。更多还原

【Abstract】 In the recent forty years, since the theory of uncertain reasoning has been applied broadly in control systems, the logical foundation of it-non-classical logic have attracted a considerable deal of attention much more, In the late of 1970’s, Pavelka etc. established elementarily fuzzy propositional logic, hereafter non-classical logics have been developed into an important research direction in Artificial Intelligence field. On the one hand, non-classical logics have been applied broadly in machine automatic prove, multi-agent system and program validation and so forth research fields. On the other hand, non-classical logics have also enriched and developed the theory of pure mathematics. Lattice-valued logic is a kind of important non-classical logic, it not only can characterizes the information with linearly ordered, but also the information with nonlinearly ordered, that is to say, the incomparable information. The main aim of this paper is to establish the lattice-valued propositional logic system with degree based on lattice implication algebras, which includes the semantic theory and syntactic theory, and further establish the theory and methods of uncertain reasoning with linguistic truth-valued, and set up the corresponding algorithms.In section one, the algebraic properties of lattice implication ordered semigroups induced by lattice implication algebras is obtained. We introduce two new concepts, i.e. lattice implication n-ordered semigroup and lattice implication p-ordered semigroup, prove that a lattice implication n-ordered semigroup is a residuated semigroup, and a lattice implication p-ordered semigroup is an arithmetic lattice-ordered semigroup. We also define the notion of lattice implication n-ordered semigroup homomorphism, and based on it, we characterize the algebraic properties of filters and sl ideals in lattice implication n-ordered semigroups and lattice implication p-ordered semigroups. At one time, we present several typical expansions of sl ideals in lattice implication n-ordered semigroups. It should be hopeful that these investigations can provide a kind of new train of thought investigations for further researching into the properties of lattice implication algebra and the theory of lattice-valued logic based on lattice implication algebras. Likewise we find out all subalgebras, filters and LI-ideals in linguistic truth-valued lattice implication algebra L18.In section two, the theory of generalized tautology is established in lattice-valued propositional system (?)P based on lattice implication algebras. We define several notions of L-type tautology, L-type contradiction andα-tautology etc., present several theorems about the relations among these generalized tautologies. We define the notion of satisfiability for L-fuzzy logic formulas set, based on it, define the semantic closure operation, and further define the notion of consistency of information and theory based on semantic closure operation. We also present several theorems about the compact property, logic compact property of semantic closure operation and the corresponding closure systems, establish the congruence relation on (?)P induced by certain given information and the corresponding quotient lattice implication algebra. Combining with the theory of fuzzy set and L-fuzzy set, we also obtained theorems about the properties of closure operation on P((?)p) induced by the semantic closure operation.In section three, we establish a kind of syntactic theory with some degree in the lattice-valued propositional logic system based on lattice implication algebras. We define form proof with some degree and syntactic closure operation, prove some theorems used often, and define the notion of provable equivalent relation, prove several important theorems about provable equivalent relations. We also investigate the consistency of the syntactic theory and the semantic theory established in above chapter, and establish generalized deduction theorem and completeness theorem. These results are useful to provide necessary academic preparation for establishing uncertain reasoning methods based on lattice-valued logic system.In section four, we characterized some properties of lattice-valued propositional logic system based on linguistic truth-valued lattice implication algebra L18, and then establish the theory and methods of uncertain reasoning with linguistic truth-valued, at one time, set up the corresponding reasoning algorithms. From the perspective of logic semantics and syntax, we also analyze the rationality of uncertain reasoning methods and algorithms which have been established ahead.更多还原