Logic and Computability

拟议的项目包括对可计算性理论(递归理论)和逻辑的广泛专题的研究，既有理论上的，也有应用于数学和计算机科学的其他领域的。第一个领域包括对集合和函数的结构的研究，这些结构按计算的相对复杂性排序。将特别强调所有集合和那些有效可列举集合的复杂性结构中的可定义性问题。这一领域包括分析计算函数的难度与其他问题之间的关系，例如增长速度、它们在算术中定义的复杂性以及证明它们存在所需的公理系统的强度。纯可计算性理论的方法将在可计算代数和模型论领域得到应用。将开发出将结果从任意结构转换到特定感兴趣类别的结果的通用方法，例如格子、群和环。另一个要研究可计算性和复杂性问题的领域是模态逻辑和直觉逻辑及其模型理论。这些逻辑现在被用于对程序验证和计算机安全中的问题进行建模。第二个领域包括研究由有限自动机表示的结构，决策过程，非单调逻辑的分析和发展，并发编程模型，以及线性规划思想和算法在数据结构和逻辑编程中的应用。这里的主要焦点将是混合(连续和离散)控制理论的逻辑和数学基础，以及基于正在进行的理论工作的这些学科的算法的实际实施。在这一领域有待开发的一个重要工具是无限自动机。第一个目的是为了更好地理解可计算性的基本概念和不同任务的计算的相对难度。理论工作既涉及可计算性的抽象概念，也涉及数学其他领域的特定分支和问题的应用。一个重要的应用(在数学中)涉及到一个普遍的问题，即需要什么样的起始信息才能计算许多重要的数学结构的各个方面。在实践中，结果有时表明，对于某些重要任务没有算法，或者需要比预期更多的信息来编写计算期望结果的程序。这项工作的另一个方面强调了对抽象数学结构的具体表示的具体选择对一个人计算特定函数和关系的能力的影响。第二种类型更直接地涉及开发程序验证、数据管理和现实世界复杂系统的自动化控制等关键领域所需的数学(尤其是逻辑)工具。其中一些工作预计会有商业应用，而且已经有一家初创公司开发了几个应用程序，包括数据压缩和网络管理算法。这些技术的其他有待研究的应用包括多媒体应用和分布式连续系统。

The proposed project includes research into a broad range of topics in computability theory (recursion theory) and logic both theoretical and applied to other areas of mathematics and computer science. Included in the first area are investigations of the structures of sets and functions ordered by relative complexity of computation. Particular emphasis will be placed on the issue of definability in the complexity structures of all sets and those which are effectively enumerable. Included in this area is the analysis of the relations between difficulty of computing functions and other issues such as rates of growth, complexity of their definitions in arithmetic and the strength of axiom systems needed to prove their existence. Applications of the methods of pure computability theory will be made in the areas of computable algebra and model theory. General methods will be developed for transferring results from arbitrary structures to ones in specific classes of interest such as lattices, groups and rings. Another area where issues of computability and complexity are to be studied is that of modal and intuitionistic logics and their model theory. These logics are now used to model issues in program verification and computer security. The second area includes the study of structures representable by finite automata, decision procedures, the analysis and development of nonmonotonic logic, concurrent programming models, and applications of linear programming ideas and algorithms to data structures and logic programming. A primary focus here will be the logical and mathematical foundations of hybrid (continuous and discrete) control theory as well as the practical implementation of algorithms for these subjects based on the theoretical work being done. An important tool to be developed in this area is that of infinitary automata.The research proposed is of two types. The first is directed at a better understanding of the fundamental notions of computability and relative difficulty of computation for different tasks. The theoretical work deals both with the abstract notion of computability as well as with applications to specific branches of, and questions in, other areas of mathematics. One important application (within mathematics) concerns the general question of what starting information is needed to be able to compute various aspects of many important classes of mathematical structures. In practical terms, the results at times indicate that there are no algorithms for certain important tasks or that more information than might have been expected is needed to write programs calculating the desired results. Another aspect of this work stresses the impact that the specific concrete choice of representation of an abstract mathematical structure has on one's ability to compute specific functions and relations. The second type deals more directly with developing the mathematical (and especially logical) tools needed for the crucial areas of program verification, data management and automated control of real-world complex systems. Commercial applications are expected for some of this work and there has already been a spin off to a start-up company developing several applications including data compression and network management algorithms. Other applications of these techniques to be investigated include multimedia applications and distributed continuous systems.