权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Computability of discontinuous functions-Towards its paradigm

不连续函数的可计算性——走向它的范式

基本信息

批准号：
16340028
负责人：
YASUGI Mariko
金额：
$ 4.86万
依托单位：
Kyoto Sangyo University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2004
资助国家：
日本
起止时间：
2004 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16340028/
关键词：
Computable analysis Effective continuity Effective sequence of uniformities limit Effective sequence of Fine continuous functions Effective Fine convergence of function sequences Limit recursion Proof animation Fractals with infinite bases

项目摘要

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.

在研究一些相对于欧几里德度量而言不连续的实数函数的可计算性时，最有用和最自然的方法是通过隔离不连续点来统一函数的域。阐明了有效均匀空间特别是精细度量空间的一般理论。针对具有不同间断点的函数序列的可计算性问题，提出了有效一致序列及其极限理论。另一种理论是，在用不连续函数来表征可计算元素序列的图像时，承认限制递归函数。在自然环境中，这两种理论是等效的。可以说，有效均匀性理论（有效均匀性序列）是某些间断函数可计算性的基本方法。限制递归相当于Sigma^0_1排除中间。计算出了Sigma^0_1排除中值和其他半构造性原理之间的相对强度。至于泛函分析，各种定理的有效性，主要是巴纳赫空间，已经取得了进展。研究了用{0,1,Bottom}表示实数，表征实数的可计算性。构造了一些反例：可顺序计算但无有效连续点的函数序列和Banach-Mazur可计算但不可马尔可夫的函数。在应用中，研究了双旋转图的动力学、多扇区增长理论中逆问题的描述、具有无限多个收缩图的分形的可计算性问题、用于发现非平凡结的硬件和软件的重组。