算子Rough逻辑及其归结原理

本文基于Rough集理论定义了算子η及其合成运算，并用它作用于Rough逻辑公式，从而得到了带算子的Rough逻辑．讨论了这种逻辑公式的真值、语义模型、性质、归结原理及完备性定理和它的证明．     

粒及粒计算在逻辑推理中的应用

讨论了信息粒的结构及其实例。基于Rough集方法定义了决策规则粒,构造了决策规则粒库,它被用作逻辑推理。定义了粒语言,描述了这种语言的语法、语义、粒语句的运算法则和粒之相关的几个性质。定义了粒之间的相互包含(inclusion)和相似(closeness)。基于这些概念,构造了一种逻辑推理的新模型。这种推理模式的特点在于它既是逻辑的又是集合论的。所谓逻辑的就是说推理是遵循一种逻辑运算;所谓集合论的是指这种推理可利用对应于这种逻辑公式的意义集的运算进行推理,还用实例说明了这种推理模式是可行和有效的。     

带Rough相等关系词的Rough逻辑系统及其推理

以公式的定义域集的下和上近似分别相等方法，定义了两个Rough逻辑公式Rough相等，并以此定义了Rough相等关系词“＝R”，它不仅比等值词“←→”运算有更多的直观性，而且既考虑了可定义的公式，也包含了那 边界线上不可定义或可能可定义的公式，所以，经典逻辑中的隐含量φ→ψ被移至Rough逻辑中应当解释为R.(d(φ））包含R.(d(ψ))∧R^*(d(φ))包含R^*（d(ψ））。经典逻辑中的等值式φ←→ψ被移至Rough逻辑中应当解释为R.(d(φ））＝R.(d(ψ))∧R^*(d(φ))＝R^*（d(ψ）），其中d(F)是公式F的定义区域，它可能是可定久集，也可能是不可定义集或Rough集，这是Rough逻辑与经典逻辑与其它非标准逻辑的重要区别之一，将这种Rough相等词“＝R”引入Rough逻辑中，因而得一些相关的性质和相关的推理规则。文本中建立了带Rough相等关系词“＝R”的Rough逻辑推理系统，并在这个系统下用演绎推理方法证明了几个具体的实例。     

基于粒计算的Rough集模型

上近似、下近似是Rough集的基本定义,它使我们能够用精确的集合讨论不精确的概念,Rough集利用可计算的边界域实现了G.Frege的边界思想.然而,Rough集本身的代数定义和其他各种扩展模型并没有提供简单直观的计算边界元素数目的算法.在二进制粒计算的基础上,通过定义粒矩阵和粒矩阵运算,建立了基于粒计算的知识表示方法和基于粒计算的Rough集模型,据此可以获得Rough集基本概念的粒矩阵表示和粒矩阵快速计算方法,为建立基于粒计算的知识发现算法提供了理论基础.举例证明了Rough包含与Rough相等的隶属度函数定义并非充要条件.同时给出了基于粒计算的Rough包含与Rough相等的充要条件.     

粒计算研究现状及基于Rough逻辑语义的粒计算研究

综述了粒计算的提出背景、研究现状及其发展趋势,也给出了作者的评论;论述了粒计算应用的广泛性,包括AI中的图像检索、医学诊疗系统、连续数学中的积分学及其它许多逻辑推理等方面的应用.讨论了粒计算将有希望成为处理信息和研究其它学科的理论工具和方法学.讨论了粒计算中基于Rough逻辑语义的粒及其相关性质,建立了这种粒的演绎推理.提出了基于Rough逻辑语义的粒归结原理和归结策略,包括λ-归结策略和锁归结策略.证明了这种粒归结的完全性.基于Rough逻辑语义的粒在AI的问题求解、专家系统以及机器定理证明中都将成为一种新的研究思想和新的理论工具.最后,提出了这种基于Rough逻辑语义的粒计算研究前景.     

基于Rough集的Rough数及入算子的逻辑价值*

本文在介绍Rough集基础上，提出了基于Rough集理论的Rough数概念及其运算法则．并给出了这种Rough数应用实例及其近似程度算子λ在Rough逻辑中的理论价值．     

广义模糊逻辑和锁语义归结原理

将命题的真值取在格上的模糊逻辑,我们称为广义模糊逻辑。本文讨论了这种广义模糊逻辑的性质,并证明了,对于一阶谓词公式,在广义模糊逻辑中的不可满足性和在二值逻辑中的不可满足性是等价的。还证明了原始的归结原理在广义模糊逻辑中是完备的。 最后,在模糊逻辑中讨论了涉及子句真值的语义归结原理,对于在广义模糊逻辑中的不可满足配锁子句集,在任意一个模糊解释下,使用语义归结原理,总可演绎出空子句。     

模糊逻辑的再扩充

R C.T.Lee和C.L.Chang首先讨论了真值集为单位区间的模糊逻辑,特别是证明了一阶谓词公式在二值逻辑中的不可满足性与在这种模糊逻辑中的不可满足性是等价的,从而原始的归结原理在这种模糊逻辑中是完备的。1980年,刘叙华将[1,2]的结果推广到有分界元素的有余完全分配格值逻辑,并称这种逻辑为广义模糊逻辑,容易知道,真值格有分界元素这个条件是比较苛刻的,甚至连二值逻辑的真值集B_2(二元Boole     

粒计算的α_决策逻辑语言

提出一种用于粒计算的α_决策逻辑语言．该语言是由Tarski意义下的模型和可满足性所描述的一种特殊的经典谓词逻辑．由属性值域的模糊子集代替经典的单值信息函数所得到的广义信息系统对应于模型，借助于模糊集理论的水平截集的概念，归纳地定义对泉在一定阈值水平下满足某公式．最后讨论如何利用α_决策逻辑语言描述不同的粒世界及分析形式概念和决策规则．     

领域值信息表上的邻域逻辑及其数据推理

引入了一种基于邻域值信息表的邻域逻辑,它是用邻域拓扑内点和邻域拓扑闭包作为逻辑算子的一种逻辑。其内点和闭包是先经二元关系定义了邻域系统,然后用这种邻域系统来定义它。这种逻辑被定义在信息表上,其表上的每个个体关于属性不是取单独一个值,而是扩充到取一个值的领域。公式的真值被扩充为一个区间或邻域,因此讨论一个公式可满足性的三种类型：邻域内点可满足、邻域闭包可满足和邻域可满足,即将公式的真值扩充为多值,并讨论了这种真值关于逻辑联结词的运算和公式的语义模型。最后还给出了这种逻辑的数据推理。     

-Resolution principle based on lattice-valued propositional logic LP(X)

In the present paper, resolution-based automated reasoning theory in an L-type fuzzy logic is focused. Concretely, the -resolution principle, which is based on lattice-valued propositional logic LP(X) with truth-value in a logical algebra – lattice implication algebra, is investigated. Finally, an -resolution principle that can be used to judge if a lattice-valued logical formula in LP(X) is always false at a truth-valued level (i.e., -false), is established, and the theorems of both soundness and completeness of this -resolution principle are also proved. This will become the theoretical foundation for automated reasoning based on lattice-valued logical LP(X).     

Filter-based resolution principle for lattice-valued propositional logic LP(X)

As one of most powerful approaches in automated reasoning, resolution principle has been introduced to non-classical logics, such as many-valued logic. However, most of the existing works are limited to the chain-type truth-value fields. Lattice-valued logic is a kind of important non-classical logic, which can be applied to describe and handle incomparability by the incomparable elements in its truth-value field. In this paper, a filter-based resolution principle for the lattice-valued propositional logic LP(X) based on lattice implication algebra is presented, where filter of the truth-value field being a lattice implication algebra is taken as the criterion for measuring the satisfiability of a lattice-valued logical formula. The notions and properties of lattice implication algebra, filter of lattice implication algebra, and the lattice-valued propositional logic LP(X) are given firstly. The definitions and structures of two kinds of lattice-valued logical formulae, i.e., the simple generalized clauses and complex generalized clauses, are presented then. Finally, the filter-based resolution principle is given and after that the soundness theorem and weak completeness theorems for the presented approach are proved.     

基于格值一阶逻辑LF（X）的自动推理算法

基于谓词逻辑的归结推理方法是目前理论上较为成熟、可以在计算机上实现的推理方法之一。针对格值一阶逻辑LF(X)中归结自动推理问题,以格值一阶逻辑LF(X)的α-归结原理为理论基础,通过对例子进行分析,提出了LF(X)中简单广义子句集的归结自动推理算法,并证明了该算法的可靠性和完备性。     

逻辑系统G3中命题的D3-随机真度理论

通过引入随机向量序列对赋值集进行随机化,在逻辑系统G3中提出了公式的D3-随机真度的概念,证明了全体公式的D3-随机真度之集在[0,1]中没有孤立点;提出了D3-相似度和D3-伪距离,证明了在D3-逻辑度量空间中没有孤立点;在D3-逻辑度量空间中提出3种不同类型的近似推理模式;引入公式间的相容与独立的概念,研究了其关系。为进一步研究随机推理奠定了基础。     

命题时态逻辑定理证明新方法

本文通过对近１０年命题时态逻辑定理证明方法的研究，提出了一种新的证明方法，前人的工作基于对公式的现时部分和后时部分的分解，本文的工作是基于语义反驳树构造。这种新方法为计算机自动证明命题时态逻辑定理，提供了比较好的理论框架．最后还证明了该方法的可靠性和完全性．     

随机模糊环境下的命题逻辑真度理论*

在实单位区间[0,1]具有一定概率分布的基础上,引入命题逻辑公式的随机模糊意义下的真度概念,指出随机真度是已有文献中各种命题逻辑真度的共同推广.利用随机模糊真度定义公式间的随机模糊相似度,导出全体公式集上的一种伪距离——随机模糊逻辑伪距离,证明在随机模糊逻辑伪距离空间无孤立点.利用概率论中的积分收敛定理,证明一个关于随机模糊真度的极限定理.研究已有各种真度之间的联系.证明随机逻辑伪距离空间中逻辑运算的连续性,并将概率逻辑学基本定理推广至多值命题逻辑.在随机逻辑伪距离空间中提出2种不同类型的近似推理模式并应用于实际问题的近似推理.     

Sequent forms of Herbrand theorem and their applications

New sequent forms* of the famous Herbrand theorem are proved for first-order classical logic without equality. These forms use the original notion of an admissible substitution and a certain modification of the Herbrand universe, which is constructed from constants, special variables, and functional symbols occurring only in the signature of an initial theory. Other well-known forms of the Herbrand theorem are obtained as special cases of the sequent ones. Besides, the sequent forms give an approach to the construction and theoretical investigation of computer-oriented calculi for efficient logical inference search in the signature of an initial theory. In a comparably simple way, they provide us with some technique for proving the completeness and soundness of the calculi. *A part of this investigation was performed during a visit to the University of Liverpool supported by the grant NAL/00841/G given by the Nuffield foundation.     

Linguistic truth-valued lattice-valued propositional logic system ?P(X) based on linguistic truth-valued lattice implication algebra

In the semantics of natural language, quantification may have received more attention than any other subject, and syllogistic reasoning is one of the main topics in many-valued logic studies on inference. Particularly, lattice-valued logic, a kind of important non-classical logic, can be applied to describe and treat incomparability by the incomparable elements in its truth-valued set. In this paper, we first focus on some properties of linguistic truth-valued lattice implication algebra. Secondly, we introduce some concepts of linguistic truth-valued lattice-valued propositional logic system ?P(X), whose truth-valued domain is a linguistic truth-valued lattice implication algebra. Then we investigate the semantic problem of ?P(X). Finally, we further probe into the syntax of linguistic truth-valued lattice-valued propositional logic system ?P(X), and prove the soundness theorem, deduction theorem and consistency theorem.     

An extension of the Boyer-Moore Theorem Prover to support first-order quantification

We describe an implementation of an extension to the Boyer-Moore Theorem Prover and logic that allows first-order quantification. The extension retains the capabilities of the Boyer-Moore system while allowing the increased flexibility in specification and proof that is provided by quantifiers. The idea is to Skolemize in an appropriate manner. We demonstrate the power of this approach by describing three successful proof-checking experiments using the system, each of which involves a theorem of set theory as translated into a first-order logic. We also demonstrate the soundness of our approach.This research was supported in part by ONR Contract N00014-88-C-0454. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of Computational Logic, Inc., the Office of Naval Research or the U.S. Government.     

AN EPISTEMIC LOGIC WITH QUANTIFICATION OVER NAMES

Sentential theories of belief hold that propositions (the things that agents believe and know) are sentences of a representation language. To analyze quantification into the scope of attitudes, these theories require a naming map a function that maps objects to their names in the representation language. Epistemic logics based on sentential theories usually assume a single naming map, which is built into the logic. I argue that to describe everyday knowledge, the user of the logic must be able to define new naming maps for particular problems. Since the range of a naming map is usually an infinite set of names, defining a map requires quantification over names. This paper describes an epistemic logic with quantification over names, presents a theorem-proving algorithm based on translation to first-order logic, and proves soundness and completeness. The first version of the logic suffers from the problem of logical omniscience; a second version avoids this problem, and soundness and completeness are proved for this version also.