Logic and Computability

该项目包括对可计算性理论（递归理论）和逻辑的广泛主题的研究，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研究是由计算的相对复杂性的不同概念排序的集合和函数的结构。特别强调的是可有效枚举的集合的复杂性结构，以及由不受限制的图灵机计算定义的相对可计算性的最一般概念。还将研究对oracle信息访问或计算运行时间设置有效边界的计算过程，以及基于实数有效近似的相对收敛率的实数之间的缩减。纯可计算理论的方法将应用于可计算代数和模型理论领域。第二个领域包括研究可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，以及基于所做理论工作的这些主题的算法的实际实现。在第一个领域提出的工作是为了更好地理解可计算性的基本概念和不同任务的相对计算难度。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具体分支和问题的应用。一个重要的应用（在数学中）涉及到一个一般问题，即需要什么起始信息才能计算许多重要的数学结构类的其他方面。实际上，结果有时表明，对于某些重要任务没有算法，或者编写计算期望结果的程序需要比预期更多的信息。第二个领域更直接地处理开发程序验证、数据管理和现实世界复杂系统的自动控制等关键领域所需的数学（特别是逻辑）工具。预计这项工作的一部分将用于商业应用，并且已经有一家初创公司开发了一些应用程序，包括数据压缩和网络管理算法。

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 different notions of relative complexity of computation. Particular emphasis will be placed on the complexity structures of sets which are effectively enumerable and on the most general notion of relative computability as defined by unrestricted Turing machine computations. Computation procedures that place effective bounds on the access to oracle information or the run time of the computations will also be investigated, as well as reductions between real numbers based on the relative rates of convergence of effective approximations to the real numbers. Applications of the methods of pure computability theory will be made in the areas of computable algebra and model theory. 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.The work proposed in the first area 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 notions 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 other 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. The second area 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.