EAGER: Computational Models, Topological Games, and Classical Information

现代计算机系统过于复杂，无法直接分析，因此通常采用数学模型来分析这些计算机系统，以了解它们的功能如何以及它们是否正确运行。计算模型领域致力于构建这样的模型，并将其应用于解决计算中的问题，例如指定用于解决特定问题的计算设备。与此截然不同的是，数论中的丢芬图近似领域研究的是有理数如何很好地近似实数——有理数可以表示为两个整数的商。尽管可以追溯到亚历山大的丢番图时代，但这里仍有许多深层次的问题尚未解决。最后，对经典传播渠道的研究构成了经典信息论领域的基础，在日常生活的许多方面起着重要作用。本项目将探索建立计算模型的新兴技术与经典信息和丢番图近似研究之间的关系。这些代表了数学、计算机科学和信息理论的非常不同的领域，该项目的目标是在它们之间建立联系，揭示可以应用于解决每个领域问题的共同基本原则。这些原则将导致可以从一个领域使用的方法来解决其他领域的问题。提议的工作跨越了许多研究领域：领域理论和计算模型；拓扑游戏;经典信息与经典渠道家族；复杂系统分析；以及在施密特博弈和丢番图近似中的应用。支持这项工作的通道容量的最新结果是基于通道及其容量的拓扑视图，该视图提供了一种独特的方法来分析相关的通道族。对现有的构建空间域模型的方法的分析——与基于Choquet游戏的方法相反的共代数方法——将探索这些方法之间的关系，并将为它们在上述领域的新问题和有趣问题的适用性提供见解。使用领域理论方法分析Schmidt游戏是一种新颖的想法，它将有助于理解这些游戏以及它们与Choquet游戏的不同之处。

Modern computer systems are too complex to be analyzed directly, and often mathematical models are employed to analyze such computer systems to understand how well they function and whether they operate correctly. The area of computational models is devoted to constructing such models, and to their application to solve problems in computing, such as specifying computational devices designed to solve specific problems. Quite distinct from this, the area of Diophantine approximation from number theory studies how well real numbers can be approximated by rational numbers - ones that can be expressed a quotient of two integers. Despite dating back to the time of Diophantus of Alexandria, there are many deep questions here that remain unsolved. Finally, the study of classical communication channels forms the basis for the area of classical information theory that plays a fundamental role in many aspects of everyday life. This project will explore relationships between emerging techniques for building computational models, and research in classical information and in Diophantine approximation. These represent very diverse areas of mathematics, computer science and information theory, and a goal of the project is to develop links between them that reveal common underlying principles that can be applied to solve problems in each area. Such principles will lead to methods that can be used from one area to solve problems in the other areas. The proposed work spans a number of research areas: domain theory and computational models; topological games; classical information and families of classical channels; analysis of complex systems; and applications to Schmidt games and Diophantine approximation. Recent results on channel capacity that underpin the work are based on a topological view of channels and their capacity that offers a unique way to analyze related families of channels. The proposed analysis of existing approaches to constructing domain models of spaces - the co-algebraic approach as opposed to one based on Choquet games - will explore the relationship between these approaches, and will provide insights into their applicability to new and interesting problems in the range of areas listed above. Using domain-theoretic approaches to analyze Schmidt games is a novel idea that should prove useful in understanding these games and how they differ from ones such as Choquet games.