Logic and Computability

拟议的项目包括研究可计算性理论（递归理论）和逻辑的广泛主题，包括理论和应用于数学和计算机科学的其他领域。包括在第一个领域的调查结构的集合和功能的相对复杂的计算。特别强调的是，将放在所有集合的复杂性结构的可定义性问题，以及那些有效的可定义性。包括在这一领域是分析之间的关系困难的计算功能和其他问题，如增长率，复杂性的定义，在算术和强度的公理系统需要证明其存在。纯可计算性理论方法的应用将在可计算代数和模型论领域进行。一般的方法将被开发用于将结果从任意结构转移到感兴趣的特定类中，如格，群和环。可计算性和复杂性问题的另一个研究领域是模态和直觉逻辑及其模型论。这些逻辑现在被用来模拟程序验证和计算机安全中的问题。第二个领域包括研究可由有限自动机表示的结构，决策程序，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。这里的主要重点将是混合（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主题的算法的实际实现。在这一领域有待发展的一个重要工具是无穷自动机。第一个是针对更好地理解的基本概念的可计算性和相对困难的计算不同的任务。理论工作涉及的抽象概念的可计算性，以及与应用程序的具体分支，和问题，其他领域的数学。一个重要的应用（在数学中）涉及的一般问题是，需要什么样的起始信息才能计算许多重要类别的数学结构的各个方面。实际上，结果有时表明，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所需结果的程序。这项工作的另一个方面强调的影响，具体的具体选择表示的抽象数学结构上的能力，计算特定的功能和关系。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（特别是逻辑）工具。预计其中一些工作将用于商业应用，并且已经有一家初创公司开发了几种应用程序，包括数据压缩和网络管理算法。这些技术的其他应用进行调查，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.