可计算理论,算法随机性和图灵的预测

R. Downey
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摘要

本文着眼于图灵遗产在计算中的应用,特别是算法随机性理论,其中经典数学概念(如度量)可以计算化。它还追溯了图灵在早期手稿中对这一理论的预测。从丘奇、克莱因、波斯特,尤其是图灵的工作开始,特别是在神奇的1936年,我们知道计算意味着什么。图灵的理论在递归理论和可计算理论的名义下得到了实质性的发展。图灵的工作可以被看作是1936年思想的高潮。这篇论文,以及图灵1939年的论文(141)(基于他的同名博士论文),为纯计算理论(现在称为可计算性或递归理论)奠定了坚实的基础。本文简要介绍了图灵遗产的主要组成部分——可计算性理论的一些主要研究方向。可计算性理论及其用于对计算任务进行分类的工具已在许多领域得到应用,如分析、代数、逻辑、计算机科学等。这些应用将在本卷的文章中讨论。这个理论甚至可以应用到所谓的逆向数学中的证明理论中。逆数学试图通过对二阶数学中理解公理的校准来校准数学定理的逻辑强度。一般来说,大多数分离,即定理在一个系统中成立而在另一个系统中不成立的证明,是在正常模型中而不是在非标准模型中实现的。因此,一般来说?这项研究得到了新西兰马斯登基金的支持。本文中的一些工作是作者在2012年作为艾伦·图灵“语义和语法”项目的一部分在英国剑桥艾萨克·牛顿研究所访问时完成的。其中一些工作已在Becher(7)和Downey(42)的CiE 2012上发表。非常感谢Veronica Becher, Carl Jockusch, Paul Schupp, Ted Slaman和Richard Shore的无数更正。
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Computability theory, algorithmic randomness and Turing's anticipation
This article looks at the applications of Turing's Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing's anticipation of this theory in an early manuscript. Beginning with the work of Church, Kleene, Post and particularly Turing, es- pecially in the magic year of 1936, we know what computation means. Turing's theory has substantially developed under the names of recursion theory and computability theory. Turing's work can be seen as perhaps the high point in the conuence of ideas in 1936. This paper, and Turing's 1939 paper (141) (based on his PhD Thesis of the same name), laid solid foundations to the pure theory of computation, now called computability or recursion theory. This article gives a brief history of some of the main lines of investigation in computability theory, a major part of Turing's Legacy. Computability theory and its tools for classifying computational tasks have seen applications in many areas such as analysis, algebra, logic, computer science and the like. Such applications will be discussed in articles in this volume. The theory even has applications into what is thought of as proof theory in what is called reverse mathematics. Reverse mathematics attempts to claibrate the logi- cal strength of theorems of mathematics according to calibrations of comprehen- sion axioms in second order mathematics. Generally speaking most separations, that is, proofs that a theorem is true in one system but not another, are per- formed in normal \!" models rather than nonstandard ones. Hence, egnerally ? Research supported by the Marsden Fund of New Zealand. Some of the work in this paper was done whilst the author was a visiting fellow at the Isaac Newton Institute, Cambridge, UK, as part of the Alan Turing \Semantics and Syntax" programme, in 2012. Some of this work was presented at CiE 2012 in Becher (7) and Downey (42). Many thanks to Veronica Becher, Carl Jockusch, Paul Schupp, Ted Slaman and Richard Shore for numerous corrections.
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