分析中的可计算性

工程和自然科学中大量的科学计算问题,需要可靠的计算方法和计算软件,通常的实数浮点表示方法只提供了一种逼近实数的基本方法.对实数的不同逼近方法诱导不同的可计算性.对实数的重新理解和认识是研究分析中可计算性的重要基础.Klaus Weihrauch基于第二型图灵计算模型,引入了基于无穷串的可计算函数的概念,并建立了第二型能行计算理论.有关可计算实数理论的几种经典模型可以统一在第二型能行计算理论框架中.对于一般的集合,为研究其中元素的可计算性,引入基于字母集Σ的表示系统.形式上是一个部分函数υ∶Σ＊→M或δ∶Σω→M（称为命名系统）,不同命名系统下刻画不同的逼近方法,诱导出不同的可计算性,在能行拓扑空间中诱导出不同的拓扑.拓扑与命名系统之间的内在联系,使得抽象空间中可计算性的研究得到自然延伸.

Computability in Analysis

Credible computational methods and softwares are needed for plenty of computation problems in engineering and science.The usual floating-point representation is a basic approximation method of real numbers only.The different approximation methods would induced different computabilities for real numbers.The representation and knowledge of real numbers are the important basises researching computability in analysis.Klaus Weihrauch finds type-2 effective theory by introducing computable functions on infinite strings based on the type-2 Turing computation model.Many classical computable models on real numbers can be represented in type-2 effective theory.The representation system based on the alphabetis introduced to investigate the computability of elements in Abstract set.For the representation of elements is normally a partly function or,called naming system.The different naming systems can induce different computabilities and effective topology spaces.The researches of computability in Abstract spaces are naturally extended based on relations of topologies and naming systems.