Computability of discontinuous functions-Towards its paradigm

在研究欧几里得度规不连续实函数的可计算性时，最实用和最自然的方法是通过分离不连续点来统一函数的定义域。阐明了有效均匀空间，特别是细度量空间的一般理论。针对具有不同不连续点的函数序列的可计算性问题，提出了一致有效序列及其极限理论。承认极限递归函数用不连续函数来表示可计算元素序列的象是另一种理论。在自然环境中，这两种理论是等价的。我们可以说，有效均匀性理论（有效均匀性序列）是研究某些不连续函数可计算性的基本方法。极限递归等价于剔除中间的∑^0_1。计算出了σ ^0_1不含中间和其它半建设性原理的相对强度。在泛函分析方面，各种定理的实效性，主要是在巴拿赫空间上取得了进展。研究了实数的{0,1，Bottom}表示形式，表征了实数的可计算性。构造了一些反例：一个函数序列是顺序可计算的，但没有有效连续点；一个函数是Banach-Mazur可计算的，但不是Markov可计算的。在应用方面，研究了双旋转映射的动力学，多扇区增长理论中逆问题的描述，具有无限多收缩映射的分形的可计算性问题，非平凡结点发现的硬件重组和软件重组。

In studying computability of some real functions which are discontinuous with respect to the Euclidean metric, the most useful and natural METHOD is uniformization of the domain of a function by isolating the discontinuous points. A general theory of the effective uniform space, especially the space of Fine metric, has been clarified. For the computability problem of a function sequence whose functions have different discontinuity points, a theory of the effective sequence of uniformities and its limit has been developed. Admitting limiting recursive functions in characterizing the image of a computable sequence of elements by a discontinuous function is another theory. In a natural setting, these two theories are equivalent. We can claim that the theory of the effective uniformity (the sequence of effective uniformities) is the fundamental method for the computability of some discontinuous functions. Limiting recursion is equivalent with Sigma^0_1 excluded middle. The relative strength between Sigma^0_1 excluded middle and other semi-constructive principles have been worked out. As for functional analysis, effectivization of various theorems, mainly on the Banach space, has made progress. A representation of real numbers in terms of {0,1,Bottom},characterizing the computability of real numbers has been studied. Some counter-examples have been constructed :a function sequence which is sequentially computable but has no effectively continuous points and a function which is Banach-Mazur computable but is not Markov computable. In applications, dynamics of double rotation maps, description of the inverse problem in multi-sectorial growth theory, the computability problem of fractals with infinitely many contraction maps, reorganization of hardware and software for discovery of non-trivial knots have been studied.