Domain理论及Rough集理论若干相关问题研究
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摘要
随着计算机科学的飞速发展，有关计算机科学的数学基础研究越来越受到人们的关注和重视，已成为数学和计算机科学研究者共同感兴趣的领域。产生于上个世纪70年代初的Domain理论和80年代初的Rough集理论正是这样两个重要交叉领域。它们独立发展，但从共同的数学基础来看，二者均基于数学中三大基本结构之一的序结构理论，同时与拓扑、代数、范畴、逻辑等学科有着密切的联系。
它们的提出都有较强的计算机背景，但同时也是数学发展的需要。其中，Domain理论建立的初衷是为高级程序设计语言的指称语义学提供数学基础，序和拓扑的相互结合、相互作用是其基本特征，正因如此，Domain理论成为计算机科学家和数学家共同关心的领域。事实上，除了在计算机科学方面作为形式语义学的数学基础或数学框架、赋予或解释语句语义，以及应用于人工智能(AI)中知识表示及推理(KRR)、数据分析等方面之外，在若干数学分支如动力系统、分形、拓扑等方面，Domain也是一种重要的数学结构；例如，Domain环境便从序与拓扑交融的角度为某些拓扑空间提供了一种计算模型。另一方面，Rough集理论创立的目的是为处理含糊不清(vague)的概念或数据提供一个有力的数学工具，但其理论基础，依然建立在数学中的经典分支如集合论、拓扑、逻辑、代数、序结构理论之上。经过二十多年的发展，Rough集方法已被广泛应用于AI和认知科学中，尤其是在数据分析、知识表示、知识发现、机器学习、决策分析、专家系统、模式识别等领域。
作为两个数学分支，Domain理论与Rough集理论有着各自不同的研究对象和特点，但它们的研究都是在序结构的框架下进行的，在某些方面相互渗透和相互影响。本文的主旨也就是对Domain理论与Rough集理论在这些方面的共同数学基础特别是序结构方面进行讨论。文中我们运用拓扑、序结构、代数、范畴及逻辑的理论和方法，在Domain方面，研究了两类最受关注的拓扑(Hausdorff拓扑、Scott拓扑)的对偶拓扑、有限偏序集的Cartesian积、积的收缩、相关范畴和Domain方程等问题，在Rough集方面，研究了近似算子的刻画及相关范畴问题，最后建立了Domain理论与Rough集理论之间深入的内在联系。
具体而言，我们作了如下工作：
首先，我们对Domain理论中与此相关的3个公开问题进行讨论。
第一个公开问题事实上由Misloye和Lawson在拓扑学名著《Open Problems in Topology》中提出的两个关于对偶拓扑的公开问题组成；哪些拓扑它们同时也是对偶拓扑?如果对一个拓扑连续取对偶，作用有限次后所产生的拓扑之中有互为对偶的吗?对该问题，我们讨论了两类经典的拓扑即Hausdorff拓扑与Scott拓扑的相关问题。对任意Hausdorff空间(X，τ)，有τ~d=τ~(ddd)，对τ反复取对偶，连同τ本身至多产生3个不同的拓扑：τ、τ~d和τ~(dd)，由此，我们对所有Hausdorff空间做了一个分类，即三个严格递增的类，然后刻画了满足τ=τ~(dd)的Hausdorff空间。事实上，满足τ=τ~(dd)的Hausdorff空间(X，τ)恰为Hausdorffκ-空间。此外，我们研究了对偶拓扑与原拓扑之间的关系，为解决公开问题提供了一些可行的思路。另一方面，对Scott拓扑，我们的研究对象是定向完备偏序集(dcpo)D，其上Scott拓扑记为σ(D)，我们给出了σ(D)~d=ω(D)的刻画及其成立的一个较广泛的充分条件，研究了σ(D)~d=ω(D)与Scott紧集、强紧集之间的关系。此外，我们证明了对任何dcpo D有σ(D)~(dd)(?)σ(D)，进而对应于Hausdorff拓扑情形给出了σ(D)=σ(D)~(dd)的一个内在刻画，并借助于若干具体的例子讨论了σ(D)与σ(D)~(dd)之间的关系。这些结果对上述公开问题在Hausdorff拓扑与Scott拓扑这两类最受关注的重要情形下作了部分回答。
第二、三个公开问题都相关于有限偏序集的Cartesian积。
其中，第二个公开问题是Plotkin于1978年在文[90]中提出的如下猜想：对于三元真值dcpo T及任一基数，κ＞ω，函数空间[T~κ→T~κ]不是T~κ的收缩。我们知道Scott曾于1976年证明了具有可数基的连续格恰为2~ω的一个收缩以及更一般地，任一连续格恰为2的某个积的收缩[97]；随后，Plotkin于1978年证明了具有可数基的coherent domain(即两两相容的子集必有上确界的domain)恰为T~ω的收缩。然而，该结果若要推广至不可数情形，则有难以逾越的障碍，这正是Plotkin提出上述猜想的原因。文中我们构造性地证明了该猜想。这个结果不仅给出了T~κ这个基本生成结构在拓扑与序结构理论方面的一个重要性质，而且，在计算机科学中的形式语义学方面，它指出了这样一个事实：对κ＞ω，T~κ不能作为任何程序设计语言的指称语义模型；其意义在于：对于形式语义学中包涵较连续格更为广泛的基本生成结构T~κ，澄清了由其生成的coherent domain作为指称语义模型在理论上的界限。
类似地，作为对形式语义学数学模型的考虑，基于Scott的前述结论，Mislove和Lawson在[55]中更进一步提出如下公开问题即本文的第三个问题：更为一般的拓扑空间在积和收缩的作用下产生的最小封闭类具有什么性质?具体地，若选择一族有限T_0空间作为生成空间，会产生什么样的空间类?它是Cartesian Closed的吗?事实上，有限T_0空间恰为赋Scott拓扑的有限偏序集，因此，该公开问题中的有限T_0空间可直接换为有限偏序集。对该问题，我们具体讨论了生成的空间含T的任何一族有限偏序集(具有最小元)F，F在积和收缩的作用下产生的最小封闭类不是Cartesian Closed的，从而说明了：
在由任何一族非平凡有限偏序集作为生成空间在积和收缩的作用下产生的最小封闭类中，
(1)连续格范畴是其最小的关于任意积和收缩封闭的Cartesian Closed满子范畴，
(2)若生成空间中每一个对象均为有限L-domain，则连续格范畴是其唯一的关于任意积和收缩封闭的Cartesian Closed满子范畴。
综上，我们澄清了Domain理论中关于对偶拓扑、有限偏序集所生成的空间及相关范畴的一些疑问。
本文另一工作是研究Rough集理论中与序结构相关的2个问题。
正如前面所提到的，Rough集理论与Domain理论一样具有知识表示及推理、数据分析等功能，而其中所用到的核心概念是近似算子(approximation operator)。首先，我们从整体角度刻画该理论中的近似算子，然后，进一步从范畴角度考虑所有近似空间。
对一个近似空间(U，R)，由于上、下近似算子从整体看是对偶的，所以我们主要讨论了上近似算子在R为各种可能关系情形下的刻画。这些刻画表明了Rough集理论与序结构理论之间的密切联系，同时我们得到了该算子的一些重要而新颖的性质。另一方面，基于前述刻画及应用之需，我们建立了近似空间与完备原子布尔格、素代数格之间的联系，从不同知识间关系的角度，定义了近似空间范畴AS_E，用范畴的语言(构造)解释了对信息系统的不同处理。比如，pullback可用来解释两个或两个以上源信息系统的融合；pushout可用来解释两个或两个以上源信息系统的共有的背景知识。在Rough集理论中引入范畴打破了原有的对知识处理的单一的方式，从范畴层次对已有的相关或不相关的知识进行处理。以上对近似算子的刻画以及引入范畴理论为Rough集的研究提供了一个新的途径。
最后，我们从Domain理论与Rough集理论共有的知识表示及推理功能的角度，借助于形式概念分析的方法，在概念格这个共同的基点上，通过模态算子◇和□构成的Galois联络以及引入逼近结构，在两个理论之间建立了深入的内在联系．具体而言，我们通过Galois联络讨论了Rough集理论中已有的两种概念格结构，并将其推广至无穷情形，得出在Rough集框架下的完备格的概念格表示，由此说明了：Rough集框架下的概念格与标准的概念格具有同等的表达力。然后，从逻辑的角度，我们讨论了Rough概念格与信息系统(由Scott定义)之间的联系，证明了对任何形式上下文(U，V，(?))，当属性集V有限时，其面向对象的Rough概念恰和相应导出信息系统的状态一致。一般地，对无限情形，我们在Rough概念格中引入逼近结构，定义了Rough可逼近概念，得到任何形式上下文的Rough可逼近概念和相应导出信息系统的状态一致，然后构造了Rough集框架下的Domain结构，并给出完备代数格的Rough可逼近概念格表示。以上结果不仅建立了Domain理论和Rough集理论之间深入的内在联系，而且也表明了Domain中蕴含非经典逻辑结构。
With the rapid development of computer science, people are paying moreattention to the research about its mathematical foundations which have beenthe common field of mathematicians and computer scientists. Domain Theory(DT) and Rough Set Theory (RST), established in 1970's and 1980's, respec-tively, are just two very important cross-discipline fields. They developed in-dependently, but they are both based on order theory and are simultaneouslyrelated to topology, algebra, category, logic and so on.
They both have stronger backgrounds of computer science, in fact, it isalso necessary for the development of mathematics. On the one hand, the orig-inal purpose of Domain Theory is to provide the mathematica foundation fordenotational semantics of programming languages, where the interaction be-tween topology and order is essential. So DT becomes the research field thatmathematicians and computer scientists are commonly interested in. Actually,besides the formal semantics of programming languages, knowledge represen-tation and reasoning (KRR), and data analysis in AI, domain is still a veryvital mathematical structure in topology, dynamical systems, fractal,, measure,integral and so on. For example, from the viewpoint of the interaction betweentopology and order, domain environment provides a computational model forsome important topological spaces. On the other hand, establishing Rough Set Theory is to process the vague concept or data. But its theoretical foundationis still based on the classical mathematical branches such as topology, ordertheory, set theory, algebra, logic and so on. In the past more than 20 years,rough approach has been broadly applied in AI and cognitive science, especially,data analysis, KRR, KDD, machine learning, decision analysis, expert systems,pattern recognition, etc.
As two mathematical branches, DT and RST have different research objectsand characteristics, but their studies are both done within the framework oforder and are affected each other in some aspects. The main purpose of thepaper is to discuss some aspects about their common mathematical foundation,particularly order structure. In this paper, we will use the theories of topology,order, algebra, category and logic. On the DT side, we study the duals of twoimportant and classical topologies (Hausdorff topology, Scott toplogy), retractsof Cartesian product of finite posets, related categories and domain equations.On the RST side, we give a characterization of approximation operators andstudy the related category. At last, the deeply internal relationship betweenDT and RST is established.
In detail, our work is as follows.
First of all, we discuss three related problems in Domain Theory.
The first one is about dual topology. That is a problem in "Open Problemsin Topology" raised by Mislove and Lawson as the following: Characterize thosetopologies that arise as dual topologies; if one continues the process of takingduals, does the process terminate after finitely many steps with topologies thatare duals of each other? For this problem, we discuss the related results abouttwo classical topologies, i.e., Hausdorff topology and Scott topology. Let (X,т)be a Hausdorff topological space. We haveт~d=т~(ddd). Therefore, if one contin-ues taking duals forт, at most 3 topologies can arise:т,т~d andт~(dd). Accordingto the number of topologies generated by taking duals, Hausdorff spaces can be classified to three increasing classes. Then we give a characterization of Haus-dorff spaces withт=т~(dd) which are precisely Hausdorff k-spaces. In addition, westudy the properties of topologies that arise as dual topologies. On the otherhand, for Scott topology, let D be a dcpo, endowed with the Scott topologyσ(D). We show a characterization and a sufficient condition ofσ(D)~d=ω(D)and study the relationship betweenσ(D)~d=ω(D), Scott compact sets andstrongly compact sets. For a dcpo D,σ(D)~(dd)(?)σ(D)holds. Furthermore, wegive a characterization ofσ(D)~(dd)=σ(D) and a sufficient condition to decideσ(D)~(dd)=σ(D) easily. The above results give a partial answer to the proceedingopen problem.
The second and third problems are about Cartesian products of finiteposets.
The second one is the following conjecture by Plotkin in [90]: for the three-element truthvalue cpo T, ifκ＞ω, then the function space [T~κ→T~κ] is nota retract of T~κ. We constructively give a positive answer to this conjecture.This result not only shows a topological and order property of the basic gen-erating structure T~κ, but also expresses a following fact: forκ＞ω, T~κcannotprovide denotational semantics for any programming languages that incorpo-rateλ-calculus, which makes clear the theoretical limit of coherent domainsgenerated by T~κas the models of denotational semantics.
Similarly, based on Scott's result that any continuous lattice is preciselysome product of 2, further, Mislove and Lawson in [55] raised the followingproblem i.e. the third one of the paper: Investigate those "varieties" of topolog-ical spaces that are generated by a certain class of spaces by taking the smallestclass closed under retracts and products. When do all members of the varietyarise as a retract of a product of generating spaces? What classes arise whenone starts with a set of finite T_0 spaces? In the latter case is the variety gen-erated Cartesian Closed? Is it finitely generated? In fact, finite T_0 spaces areexactly finite posets with Scott topology, so the finite T_0 spaces can be replaced by finite posets. For this open problem, we concretely discuss any family F offinite posets whose generated spaces include T. We show that the class gen-erated by F is not Cartesian Closed. Thus the result shows that the categoryCONT of continuous lattices and continuous functions is the least CartesianClosed full subcategory (closed under products and retracts) of the variety gen-erated by any family of nontrivial finite posets. If F consists of finite nontrivialL-domains, then CONT is the unique Cartesian Closed Category of the variety.
The above results of three aspects make clear some douts about dual topol-ogy, retracts of Cartesian product of finite posets and related categories.
The other work of this paper is to study three problems related to orderstructure in Rough Set Theory.
As mentioned previously, like DT, RST has the functions of knowledgerepresentation and reasoning, data analysis, etc., where the used key conceptis approximation operator. Firstly, we characterize the approximation opera-tor. Secondly, further we consider all approximation spaces from the angle ofcategory.
Let (U, R) be an approximation space. We focus on upper approximationoperators, because lower approximation operators and upper approximationoperators are dual in some sense. We give the characterizations of upper ap-proximation operators under possible cases, which shows the deep relationshipbetween Rough Set Theory and Order Theory. Moreover, some important andnovel properties of this operator are obtained. On the other hand, the rela-tionship between approximation spaces, complete atomic boolean lattices, andcomplete prime algebraic lattices are established. Then, to process differentknowledge bases, we define a category AS_E of approximation spaces. We canuse the category-theoretic language to interpret the different processing of in-formation systems. For example, pullback can express the merging of two (ormore) source approximation spaces and pushout can express the common back- ground knowledge of two (or more) source approximation spaces, where thepullback and the pushout lie in the same square. Introducing category theoryinto Rough Set Theory breaks up the original approach that knowledge basesare processed singly. From the categorical point of view, we can process therelated or unrelated knowledge. The above studies open a new way to researchRough Set Theory.
At last, based on a common research object, we establish the deep andinternal relationship between Domain Theory and Rough Set Theory throughGalois connection (◇,□) and introducing approximation structures. First ofall, through Galois connections(◇,□), we discuss the structures of partial or-ders on two known rough concept lattices. They are extended to infinite case,thus a rough concept representation of complete lattices is obtained. Then wediscuss the relationship between rough concept lattice and information system(in the sense of Scott) and prove that for any formal context (U, V,(?)), if Vis finite, all object-oriented concepts are precisely states of derived informationsystem. Generally, in the infinite case, we introduce rough approximable con-cept, then rough approximable concepts coincide with states of correspondingderived information system for every formal context. Furthermore, we constructa domain from any formal context, i.e., rough approximable concept lattice andshow the important representation theorem—a rough approximable concept lat-tice representation of every complete algebraic lattice. The above results notonly establish the deep relationship between Domain Theory and Rough SetTheory, but also show that some class of domains implies non-classical logicalstructure.
它们的提出都有较强的计算机背景，但同时也是数学发展的需要。其中，Domain理论建立的初衷是为高级程序设计语言的指称语义学提供数学基础，序和拓扑的相互结合、相互作用是其基本特征，正因如此，Domain理论成为计算机科学家和数学家共同关心的领域。事实上，除了在计算机科学方面作为形式语义学的数学基础或数学框架、赋予或解释语句语义，以及应用于人工智能(AI)中知识表示及推理(KRR)、数据分析等方面之外，在若干数学分支如动力系统、分形、拓扑等方面，Domain也是一种重要的数学结构；例如，Domain环境便从序与拓扑交融的角度为某些拓扑空间提供了一种计算模型。另一方面，Rough集理论创立的目的是为处理含糊不清(vague)的概念或数据提供一个有力的数学工具，但其理论基础，依然建立在数学中的经典分支如集合论、拓扑、逻辑、代数、序结构理论之上。经过二十多年的发展，Rough集方法已被广泛应用于AI和认知科学中，尤其是在数据分析、知识表示、知识发现、机器学习、决策分析、专家系统、模式识别等领域。
作为两个数学分支，Domain理论与Rough集理论有着各自不同的研究对象和特点，但它们的研究都是在序结构的框架下进行的，在某些方面相互渗透和相互影响。本文的主旨也就是对Domain理论与Rough集理论在这些方面的共同数学基础特别是序结构方面进行讨论。文中我们运用拓扑、序结构、代数、范畴及逻辑的理论和方法，在Domain方面，研究了两类最受关注的拓扑(Hausdorff拓扑、Scott拓扑)的对偶拓扑、有限偏序集的Cartesian积、积的收缩、相关范畴和Domain方程等问题，在Rough集方面，研究了近似算子的刻画及相关范畴问题，最后建立了Domain理论与Rough集理论之间深入的内在联系。
具体而言，我们作了如下工作：
首先，我们对Domain理论中与此相关的3个公开问题进行讨论。
第一个公开问题事实上由Misloye和Lawson在拓扑学名著《Open Problems in Topology》中提出的两个关于对偶拓扑的公开问题组成；哪些拓扑它们同时也是对偶拓扑?如果对一个拓扑连续取对偶，作用有限次后所产生的拓扑之中有互为对偶的吗?对该问题，我们讨论了两类经典的拓扑即Hausdorff拓扑与Scott拓扑的相关问题。对任意Hausdorff空间(X，τ)，有τ~d=τ~(ddd)，对τ反复取对偶，连同τ本身至多产生3个不同的拓扑：τ、τ~d和τ~(dd)，由此，我们对所有Hausdorff空间做了一个分类，即三个严格递增的类，然后刻画了满足τ=τ~(dd)的Hausdorff空间。事实上，满足τ=τ~(dd)的Hausdorff空间(X，τ)恰为Hausdorffκ-空间。此外，我们研究了对偶拓扑与原拓扑之间的关系，为解决公开问题提供了一些可行的思路。另一方面，对Scott拓扑，我们的研究对象是定向完备偏序集(dcpo)D，其上Scott拓扑记为σ(D)，我们给出了σ(D)~d=ω(D)的刻画及其成立的一个较广泛的充分条件，研究了σ(D)~d=ω(D)与Scott紧集、强紧集之间的关系。此外，我们证明了对任何dcpo D有σ(D)~(dd)(?)σ(D)，进而对应于Hausdorff拓扑情形给出了σ(D)=σ(D)~(dd)的一个内在刻画，并借助于若干具体的例子讨论了σ(D)与σ(D)~(dd)之间的关系。这些结果对上述公开问题在Hausdorff拓扑与Scott拓扑这两类最受关注的重要情形下作了部分回答。
第二、三个公开问题都相关于有限偏序集的Cartesian积。
其中，第二个公开问题是Plotkin于1978年在文[90]中提出的如下猜想：对于三元真值dcpo T及任一基数，κ＞ω，函数空间[T~κ→T~κ]不是T~κ的收缩。我们知道Scott曾于1976年证明了具有可数基的连续格恰为2~ω的一个收缩以及更一般地，任一连续格恰为2的某个积的收缩[97]；随后，Plotkin于1978年证明了具有可数基的coherent domain(即两两相容的子集必有上确界的domain)恰为T~ω的收缩。然而，该结果若要推广至不可数情形，则有难以逾越的障碍，这正是Plotkin提出上述猜想的原因。文中我们构造性地证明了该猜想。这个结果不仅给出了T~κ这个基本生成结构在拓扑与序结构理论方面的一个重要性质，而且，在计算机科学中的形式语义学方面，它指出了这样一个事实：对κ＞ω，T~κ不能作为任何程序设计语言的指称语义模型；其意义在于：对于形式语义学中包涵较连续格更为广泛的基本生成结构T~κ，澄清了由其生成的coherent domain作为指称语义模型在理论上的界限。
类似地，作为对形式语义学数学模型的考虑，基于Scott的前述结论，Mislove和Lawson在[55]中更进一步提出如下公开问题即本文的第三个问题：更为一般的拓扑空间在积和收缩的作用下产生的最小封闭类具有什么性质?具体地，若选择一族有限T_0空间作为生成空间，会产生什么样的空间类?它是Cartesian Closed的吗?事实上，有限T_0空间恰为赋Scott拓扑的有限偏序集，因此，该公开问题中的有限T_0空间可直接换为有限偏序集。对该问题，我们具体讨论了生成的空间含T的任何一族有限偏序集(具有最小元)F，F在积和收缩的作用下产生的最小封闭类不是Cartesian Closed的，从而说明了：
在由任何一族非平凡有限偏序集作为生成空间在积和收缩的作用下产生的最小封闭类中，
(1)连续格范畴是其最小的关于任意积和收缩封闭的Cartesian Closed满子范畴，
(2)若生成空间中每一个对象均为有限L-domain，则连续格范畴是其唯一的关于任意积和收缩封闭的Cartesian Closed满子范畴。
综上，我们澄清了Domain理论中关于对偶拓扑、有限偏序集所生成的空间及相关范畴的一些疑问。
本文另一工作是研究Rough集理论中与序结构相关的2个问题。
正如前面所提到的，Rough集理论与Domain理论一样具有知识表示及推理、数据分析等功能，而其中所用到的核心概念是近似算子(approximation operator)。首先，我们从整体角度刻画该理论中的近似算子，然后，进一步从范畴角度考虑所有近似空间。
对一个近似空间(U，R)，由于上、下近似算子从整体看是对偶的，所以我们主要讨论了上近似算子在R为各种可能关系情形下的刻画。这些刻画表明了Rough集理论与序结构理论之间的密切联系，同时我们得到了该算子的一些重要而新颖的性质。另一方面，基于前述刻画及应用之需，我们建立了近似空间与完备原子布尔格、素代数格之间的联系，从不同知识间关系的角度，定义了近似空间范畴AS_E，用范畴的语言(构造)解释了对信息系统的不同处理。比如，pullback可用来解释两个或两个以上源信息系统的融合；pushout可用来解释两个或两个以上源信息系统的共有的背景知识。在Rough集理论中引入范畴打破了原有的对知识处理的单一的方式，从范畴层次对已有的相关或不相关的知识进行处理。以上对近似算子的刻画以及引入范畴理论为Rough集的研究提供了一个新的途径。
最后，我们从Domain理论与Rough集理论共有的知识表示及推理功能的角度，借助于形式概念分析的方法，在概念格这个共同的基点上，通过模态算子◇和□构成的Galois联络以及引入逼近结构，在两个理论之间建立了深入的内在联系．具体而言，我们通过Galois联络讨论了Rough集理论中已有的两种概念格结构，并将其推广至无穷情形，得出在Rough集框架下的完备格的概念格表示，由此说明了：Rough集框架下的概念格与标准的概念格具有同等的表达力。然后，从逻辑的角度，我们讨论了Rough概念格与信息系统(由Scott定义)之间的联系，证明了对任何形式上下文(U，V，(?))，当属性集V有限时，其面向对象的Rough概念恰和相应导出信息系统的状态一致。一般地，对无限情形，我们在Rough概念格中引入逼近结构，定义了Rough可逼近概念，得到任何形式上下文的Rough可逼近概念和相应导出信息系统的状态一致，然后构造了Rough集框架下的Domain结构，并给出完备代数格的Rough可逼近概念格表示。以上结果不仅建立了Domain理论和Rough集理论之间深入的内在联系，而且也表明了Domain中蕴含非经典逻辑结构。
With the rapid development of computer science, people are paying moreattention to the research about its mathematical foundations which have beenthe common field of mathematicians and computer scientists. Domain Theory(DT) and Rough Set Theory (RST), established in 1970's and 1980's, respec-tively, are just two very important cross-discipline fields. They developed in-dependently, but they are both based on order theory and are simultaneouslyrelated to topology, algebra, category, logic and so on.
They both have stronger backgrounds of computer science, in fact, it isalso necessary for the development of mathematics. On the one hand, the orig-inal purpose of Domain Theory is to provide the mathematica foundation fordenotational semantics of programming languages, where the interaction be-tween topology and order is essential. So DT becomes the research field thatmathematicians and computer scientists are commonly interested in. Actually,besides the formal semantics of programming languages, knowledge represen-tation and reasoning (KRR), and data analysis in AI, domain is still a veryvital mathematical structure in topology, dynamical systems, fractal,, measure,integral and so on. For example, from the viewpoint of the interaction betweentopology and order, domain environment provides a computational model forsome important topological spaces. On the other hand, establishing Rough Set Theory is to process the vague concept or data. But its theoretical foundationis still based on the classical mathematical branches such as topology, ordertheory, set theory, algebra, logic and so on. In the past more than 20 years,rough approach has been broadly applied in AI and cognitive science, especially,data analysis, KRR, KDD, machine learning, decision analysis, expert systems,pattern recognition, etc.
As two mathematical branches, DT and RST have different research objectsand characteristics, but their studies are both done within the framework oforder and are affected each other in some aspects. The main purpose of thepaper is to discuss some aspects about their common mathematical foundation,particularly order structure. In this paper, we will use the theories of topology,order, algebra, category and logic. On the DT side, we study the duals of twoimportant and classical topologies (Hausdorff topology, Scott toplogy), retractsof Cartesian product of finite posets, related categories and domain equations.On the RST side, we give a characterization of approximation operators andstudy the related category. At last, the deeply internal relationship betweenDT and RST is established.
In detail, our work is as follows.
First of all, we discuss three related problems in Domain Theory.
The first one is about dual topology. That is a problem in "Open Problemsin Topology" raised by Mislove and Lawson as the following: Characterize thosetopologies that arise as dual topologies; if one continues the process of takingduals, does the process terminate after finitely many steps with topologies thatare duals of each other? For this problem, we discuss the related results abouttwo classical topologies, i.e., Hausdorff topology and Scott topology. Let (X,т)be a Hausdorff topological space. We haveт~d=т~(ddd). Therefore, if one contin-ues taking duals forт, at most 3 topologies can arise:т,т~d andт~(dd). Accordingto the number of topologies generated by taking duals, Hausdorff spaces can be classified to three increasing classes. Then we give a characterization of Haus-dorff spaces withт=т~(dd) which are precisely Hausdorff k-spaces. In addition, westudy the properties of topologies that arise as dual topologies. On the otherhand, for Scott topology, let D be a dcpo, endowed with the Scott topologyσ(D). We show a characterization and a sufficient condition ofσ(D)~d=ω(D)and study the relationship betweenσ(D)~d=ω(D), Scott compact sets andstrongly compact sets. For a dcpo D,σ(D)~(dd)(?)σ(D)holds. Furthermore, wegive a characterization ofσ(D)~(dd)=σ(D) and a sufficient condition to decideσ(D)~(dd)=σ(D) easily. The above results give a partial answer to the proceedingopen problem.
The second and third problems are about Cartesian products of finiteposets.
The second one is the following conjecture by Plotkin in [90]: for the three-element truthvalue cpo T, ifκ＞ω, then the function space [T~κ→T~κ] is nota retract of T~κ. We constructively give a positive answer to this conjecture.This result not only shows a topological and order property of the basic gen-erating structure T~κ, but also expresses a following fact: forκ＞ω, T~κcannotprovide denotational semantics for any programming languages that incorpo-rateλ-calculus, which makes clear the theoretical limit of coherent domainsgenerated by T~κas the models of denotational semantics.
Similarly, based on Scott's result that any continuous lattice is preciselysome product of 2, further, Mislove and Lawson in [55] raised the followingproblem i.e. the third one of the paper: Investigate those "varieties" of topolog-ical spaces that are generated by a certain class of spaces by taking the smallestclass closed under retracts and products. When do all members of the varietyarise as a retract of a product of generating spaces? What classes arise whenone starts with a set of finite T_0 spaces? In the latter case is the variety gen-erated Cartesian Closed? Is it finitely generated? In fact, finite T_0 spaces areexactly finite posets with Scott topology, so the finite T_0 spaces can be replaced by finite posets. For this open problem, we concretely discuss any family F offinite posets whose generated spaces include T. We show that the class gen-erated by F is not Cartesian Closed. Thus the result shows that the categoryCONT of continuous lattices and continuous functions is the least CartesianClosed full subcategory (closed under products and retracts) of the variety gen-erated by any family of nontrivial finite posets. If F consists of finite nontrivialL-domains, then CONT is the unique Cartesian Closed Category of the variety.
The above results of three aspects make clear some douts about dual topol-ogy, retracts of Cartesian product of finite posets and related categories.
The other work of this paper is to study three problems related to orderstructure in Rough Set Theory.
As mentioned previously, like DT, RST has the functions of knowledgerepresentation and reasoning, data analysis, etc., where the used key conceptis approximation operator. Firstly, we characterize the approximation opera-tor. Secondly, further we consider all approximation spaces from the angle ofcategory.
Let (U, R) be an approximation space. We focus on upper approximationoperators, because lower approximation operators and upper approximationoperators are dual in some sense. We give the characterizations of upper ap-proximation operators under possible cases, which shows the deep relationshipbetween Rough Set Theory and Order Theory. Moreover, some important andnovel properties of this operator are obtained. On the other hand, the rela-tionship between approximation spaces, complete atomic boolean lattices, andcomplete prime algebraic lattices are established. Then, to process differentknowledge bases, we define a category AS_E of approximation spaces. We canuse the category-theoretic language to interpret the different processing of in-formation systems. For example, pullback can express the merging of two (ormore) source approximation spaces and pushout can express the common back- ground knowledge of two (or more) source approximation spaces, where thepullback and the pushout lie in the same square. Introducing category theoryinto Rough Set Theory breaks up the original approach that knowledge basesare processed singly. From the categorical point of view, we can process therelated or unrelated knowledge. The above studies open a new way to researchRough Set Theory.
At last, based on a common research object, we establish the deep andinternal relationship between Domain Theory and Rough Set Theory throughGalois connection (◇,□) and introducing approximation structures. First ofall, through Galois connections(◇,□), we discuss the structures of partial or-ders on two known rough concept lattices. They are extended to infinite case,thus a rough concept representation of complete lattices is obtained. Then wediscuss the relationship between rough concept lattice and information system(in the sense of Scott) and prove that for any formal context (U, V,(?)), if Vis finite, all object-oriented concepts are precisely states of derived informationsystem. Generally, in the infinite case, we introduce rough approximable con-cept, then rough approximable concepts coincide with states of correspondingderived information system for every formal context. Furthermore, we constructa domain from any formal context, i.e., rough approximable concept lattice andshow the important representation theorem—a rough approximable concept lat-tice representation of every complete algebraic lattice. The above results notonly establish the deep relationship between Domain Theory and Rough SetTheory, but also show that some class of domains implies non-classical logicalstructure.
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