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Computational Logic of Euclidean Spaces

欧几里得空间的计算逻辑

基本信息

批准号：
EP/E034942/1
负责人：
Michael Zakharyaschev
金额：
$ 31.48万
依托单位：
Birkbeck, University of London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2007
资助国家：
英国
起止时间：
2007 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FE034942%2F1
关键词：
Computational Logic Euclidean Spaces

项目摘要

Much of the spatial information we encounter in everyday situations isqualitative, rather than quantitative, in character. Thus, forinstance, we may know which of two objects is the closer withoutmeasuring their distances; we may perceive an object to be convexwithout being able to describe its precise shape; or we may identifytwo areas on a map as sharing a boundary without knowing the equationthat describes it. This observation has prompted the development,within Artificial Intelligence, of various formalisms for reasoningwith qualitative spatial information.Although substantial progress has been made in analysing themathematical foundations and computational characteristics of suchformalisms, most of that progress has centred on systems for reasoningabout highly abstract problems concerning (typically) arbitraryregions in very general classes of topological spaces. But of course,the geometrical entities of interest for practical problems are notarbitrary subsets of general topological spaces, but rathermathematically very well-behaved regions of 2 and 3-dimensionalEuclidean space; moreover, the geometrical properties and relationsthese problems are concerned with are typically not merely topological, butrather affine or even metric in character. Together, these factorsseverly limit the practical usefulness of current qualitative spatialreasoning formalisms. Overcoming this limitation represents anexciting mathematical and computational challenge.We propose to meet this challenge by drawing on developments inmathematical logic, geometrical topology, and algebraic geometry thatthe spatial reasoning literature in AI has so far failed fully toexploit. Specifically, we shall investigate the computationalproperties of spatial and spatio-temporal logics for reasoning aboutmathematically well-behaved regions of 2- and 3-dimensional Euclideanspace. We shall develop and implement algorithms for reasoning with these logics. This investigation will illuminate the important relationships betweenhitherto separate research traditions, provide new techniques foraddressing challenging problems in the mathematical geometry, andyield logics of direct relevance to practical spatial reasoningproblems.

我们在日常生活中遇到的许多空间信息在性质上都是定性的，而不是定量的。因此，例如，我们可以知道两个物体中哪一个更近，而无需测量它们的距离；我们可能会认为一个物体是凸的，但无法描述它的精确形状；或者我们可能会在不知道描述边界的方程的情况下将地图上的两个区域识别为共享边界。这一观察促进了人工智能领域各种利用定性空间信息进行推理的形式主义的发展。尽管在分析此类形式主义的数学基础和计算特征方面已经取得了实质性进展，但大部分进展都集中在用于推理涉及非常一般的拓扑空间类别中（通常）任意区域的高度抽象问题的系统上。当然，实际问题中感兴趣的几何实体不是一般拓扑空间的任意子集，而是数学上表现良好的 2 维和 3 维欧几里得空间区域；此外，这些问题所涉及的几何性质和关系通常不仅仅是拓扑的，而且具有仿射甚至度量的特征。这些因素共同严重限制了当前定性空间推理形式主义的实际用途。克服这一限制代表着令人兴奋的数学和计算挑战。我们建议通过利用数学逻辑、几何拓扑和代数几何的发展来应对这一挑战，而人工智能中的空间推理文献迄今为止未能充分利用这些发展。具体来说，我们将研究空间和时空逻辑的计算属性，以推理 2 维和 3 维欧几里德空间的数学良好区域。我们将开发并实现用这些逻辑进行推理的算法。这项研究将阐明迄今为止不同的研究传统之间的重要关系，提供解决数学几何中具有挑战性的问题的新技术，并产生与实际空间推理问题直接相关的逻辑。