Multilateral Researches on Computability Problems on the Continuum

这个项目的目的是在Pour-E1风格的连续体上的可计算性结构;它的扩展，应用和形式化。大部分目标已稳步实现。本项目的主要研究目标是真实的间断函数，特别是分段连续函数的可计算性问题，即间断点处函数值的计算基础.极限计算：这是一种通过取递归函数的极限的计算方法。我们已经证明了许多分段连续函数都可以用这种方法计算.有效一致空间：通过隔离间断点得到的一致空间上的可计算性结构的理论已经发展。许多分段连续函数在这一理论中已被证明是可计算的。函数的有效收敛分别关于一致性和度量化的等价性.极限计算与一致空间：在一定条件下，分别从极限计算和一致性两个方面研究分段连续函数的序列可计算性.沃尔什分析方法：发展了用精细度量表示可计算性概念的理论，并定义了各种可计算性概念。极限可计算数学的形式系统：定义了一个可执行极限可计算数学的形式系统，并对其进行了函数解释.泛函分析中的可计算性：求解无粘偏微分方程的可计算性和插值空间上某些线性算子的有效性得到了肯定的解决.表示真实的数的理论：用一个一致的区域表示一个完全一致的空间的理论，以及分别用格雷码和某个范畴表示真实的数的理论已经发展起来。

The purpose of this project is the computability structure on the continuum in Pour-E1 style; its extension, application, and formalization. Most of the objectives have been steadily achieved. We here report our results.The major research target of this project is the computability problems of real discontinuous functions, especially piecewise continuous functions, that is, the foundations of computation of function values at discontinuous points.1. Limit computation: This is a computation method by taking the limits of recursive functions. We have shown that many of piecewise continuous functions are computable with this method.2. Effective uniform space: The theory of the computability structure on the uniform space obtained by isolating discontinuous points has been developed. Many of piecewise continuous functions have been shown to be computable in this theory. The equivalence of effective convergences of a function with regards to respectively uniformity and its metrization.3. Limit computation and uniform space: Under a certain condition, sequential computabilities of a piecewise continuous function with regards to respectively limit computation and uniformity.4. Method of Walsh analysis: The theory of representing computability notions in terms of Fine metric has been developed, and various notions of computability have been defined.5. A formal system of limit computable mathematics: A formal system in which limiting computable mathematics can be executed has been defined, and its functional interpretation has been carried out.6. Computability in functional analysis: Effectivity of solving the invisid partial differential equation and effectivity of some linear operators on the interpolation space have been solved affirmatively.7. Theories of representing real numbers: Theories of representing a complete uniform space by a uniform domain, and representations of real numbers by respectively Gray codes and a certain category have been developed.