Domain结构的内蕴拓扑刻画和信息系统表示

Domain theory aims for constructing mathematical basis for functional computer language studies. It is an interaction research area of mathematics and computer science. This project intends to employ intrinsic topologies and information systems to study domain structures. More specifically, we use the Scott topology, the Lawson topology and the measure topology to characterize quasi continuous domains and meet continuous domains. We also explore representations of domain structures by formal topologies and study various properties of domain structures and their functional spaces, to obtain more types of domains and feature of function spaces. We introduce more kinds of information systems to represent various domains by states of information systems, and study their relationships. Domain structures will be generalized to countable approximation posets, and the countable Scott topology, the countable measure topology, etc, will be introduced. Some characterizations by these new intrinsic topologies of countable approximation property will be given. As applications, by employing the relations of topology and partial orders, we investigate properties of approximation operators in rough set theory and give some conditions for two dcpos to be isomorphic whenever their intrinsic topologies as lattices are isomorphic.

Domain理论旨在为计算机函数式语言研究奠定数学基础，是数学与计算机理论的交叉研究领域。 本项目利用内蕴拓扑和信息系统研究多种Domain结构。具体是利用Scott拓扑, Lawson拓扑和测度拓扑来刻画拟连续和交连续等Domain结构, 研究Domain结构的形式拓扑表示，由此获得更多Domain结构类型及各类函数空间的诸多特性; 引入多种信息系统类型, 利用信息系统的状态集来表达多种类型Domain并探讨它们间的关系; 推广Domain的研究到可数逼近集，引入可数测度拓扑， 可数Scott拓扑等并刻画可数逼近性。作为应用，从拓扑和序的紧密关系入手研究粗糙近似算子的序论性质和拓扑特性；利用Domain结构的内蕴拓扑刻画研究有同构内蕴拓扑的两dcpo的同构问题。

Domain理论旨在为计算机函数式语言奠定数学基础, 是数学与计算机理论的交叉研究领域. 本项目主要研究domain及连续偏序集的内蕴拓扑刻画和信息系统表示. 经4年努力获得如下主要结果: 利用Scott拓扑, e-拓扑和测度拓扑等刻画了拟连续和交连续偏序集, 提出了更多广义domain结构类型，获得了与DOM范畴等价的新范畴wADom, 沟通了Domain理论和信息系统的研究; 证明了S-超连续偏序集与一致连续偏序集是等价的, 建立了原本看上去不同的两个概念间的联系, 将不同的研究归于一路; 证明了个偏序集P是连续偏序集当且仅当对任意CD-格L, Scott函数空间[P\to L]是CD-格当且仅当对某非平凡CD-格L, [P\to L]是CD-格. 引入了可数测度拓扑, 可数Scott拓扑等并刻画了可数逼近集. 与多人合作构造了反例说明存在不是Scott拓扑决定的dcpo, 解决了Ho-Zhao问题, 从而将相关研究重点转移到寻找Scott拓扑决定dcpo的充分和充要条件上..与赵东升合作构造了反例说明存在Scott拓扑决定dcpo, 它不是dominated dcpo, 说明dominated dcpo类不是极大Scott闭集格忠实类. 也部分解决了Amadio和Curien关于stable domain范畴的最大Cartesain闭子范畴的公开问题，将这一持续20多年的开问题向前推进了一大步..从拓扑和序的紧密关系入手, 作为理论应用, 还研究了抽象知识库的多种约简存在性, 证明了有限非空抽象知识库总存在唯一的并饱和约简且给出了具体算法.