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，这些结构按相对计算复杂性的不同概念排序。重点将特别放在有效可列举的集合的复杂性结构上，以及由无限制图灵机计算定义的最一般的相对可计算性概念。还将研究对甲骨文信息的访问或计算的运行时间设置有效界限的计算程序，以及基于有效近似与实数的相对收敛速度的实数之间的减缩。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。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，以及基于已有理论工作的这些学科的算法的实际实施。第一个领域中提出的工作旨在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，也涉及对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其他方面。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。第二个领域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。

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.