Pub Date : 2014-05-01DOI: 10.1017/CBO9781107338579.014
R. Soare
Abstract . In §1 we give a short overview for a general audience of Godel, Church, Turing, and the discovery of computability in the 1930s. In the later sections we mention a series of our previous papers where a more detailed analysis of computability, Turing's work, and extensive lists of references can be found. The sections from §2—§9 challenge the conventional wisdom and traditional ideas found in many books and papers on computability theory. They are based on a half century of my study of the subject beginning with Church at Princeton in the 1960s, and on a careful rethinking of these traditional ideas. The references in all my papers and books are given in the format, author [year], as in Turing [1936], in order that the references are easily identified without consulting the bibliography and are uniform over all papers. A complete bibliography of historical articles from all my books and papers on computabilityis given on the page as explained in §10. §1. A very brief overview of computability . 1.1. Hilbert's programs . Around 1880 Georg Cantor, a German mathematician, invented naive set theory. A small fraction of this is sometimes taught to elementary school children. It was soon discovered that this naive set theory was inconsistent because it allowed unbounded set formation, such as the set of all sets. David Hilbert, the world's foremost mathematician from 1900 to 1930, defended Cantor's set theory but suggested a formal axiomatic approach to eliminate the inconsistencies. He proposed two programs.
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Pub Date : 2013-02-12DOI: 10.1017/CBO9781107338579.008
T. Hales
The article gives a survey of mathematical proofs that rely on computer calculations and formal proofs.
本文综述了依赖计算机计算和形式证明的数学证明。
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Pub Date : 2012-06-15DOI: 10.1017/CBO9781107338579.002
J. Avigad, V. Brattka
For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems,” which assert that a construction can be carried out to meet a given specification, and “theorems,” which assert that some property holds of a particular geometric configuration. For example, Proposition 10 of Book I reads “To bisect a given straight line.” Euclid’s “proof” gives the construction, and ends with the (Greek equivalent of) quod erat faciendum, or Q.E.F., “that which was to be done.” Proofs of theorems, in contrast, end with quod erat demonstrandum, or “that which was to be shown”; but even these typically involve the construction of auxiliary geometric objects in order to verify the claim. Similarly, algebra was devoted to discovering algorithms for solving equations. This outlook characterized the subject from its origins in ancient Egypt and Babylon, through the ninth century work of al-Khwarizmi, to the solutions to the cubic and quadratic equations in Cardano’s Ars magna of 1545, and to Lagrange’s study of the quintic in his Reflexions sur la resolution algebrique des equations of 1770. The theory of probability, which was born in an exchange of letters between Blaise Pascal and Pierre de Fermat in 1654 and developed further by Christian Huygens and Jakob Bernoulli, provided methods for calculating odds related to games of chance. Abraham de Moivre’s 1718 monograph on the subject was
在数学的大部分历史中，数学本质上是算法。欧几里得《几何原理》中的几何命题分为两大类:“问题”，它断言一个构造可以满足给定的规范;“定理”，它断言某些性质适用于特定的几何构型。例如，第一册的第十项提案是“将一条给定的直线平分。”欧几里得的“证明”给出了这个结构，并以(相当于希腊语的)quod erat faciatum或q.e.f.结束，“要做的事情”。相反，定理的证明以quod erat demonstrandum或“要证明的东西”结束;但即使是这些通常也涉及到辅助几何物体的构建，以验证断言。同样，代数致力于发现解方程的算法。这一观点使这门学科从其起源于古埃及和巴比伦，到9世纪的al-Khwarizmi的作品，到1545年卡尔达诺的Ars magna中三次方程和二次方程的解，再到1770年拉格朗日在他的《reflections sur la resolution algebrique des equations》中对五次方程的研究。概率论于1654年诞生于布莱兹·帕斯卡和皮埃尔·德·费马的书信往来中，并由克里斯蒂安·惠更斯和雅各布·伯努利进一步发展，它提供了计算与机会游戏有关的赔率的方法。亚伯拉罕·德·莫弗尔1718年关于这个主题的专著是
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Pub Date : 2011-12-20DOI: 10.1017/CBO9781107338579.009
S. Homer, A. Selman
Turing’s beautiful capture of the concept of computability by the “Turing machine” linked computability to a device with explicit steps of operations and use of resources. This invention led in a most natural way to build the foundations for computational complexity.
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Pub Date : 1900-01-01DOI: 10.1017/CBO9781107338579.004
H. Buhrman
We revisit the notion of a quantum Turing-machine, whose design is based on the laws of quantum mechanics. It turns out that such a machine is not more powerful, in the sense of computability, than the machine originally constructed by Turing. Quantum Turingmachines do not violate the Church-Turing thesis. The benefit of quantum computing lies in efficiency. Quantum computers appear to be more efficient, in time, than classical Turing-machines, however its exact additional computational power is unclear, as this question ties in with deep open problems in complexity theory. We will sketch where BQP, the quantum analogue of the complexity class P, resides in the realm of complexity classes.
{"title":"Turing in Quantumland","authors":"H. Buhrman","doi":"10.1017/CBO9781107338579.004","DOIUrl":"https://doi.org/10.1017/CBO9781107338579.004","url":null,"abstract":"We revisit the notion of a quantum Turing-machine, whose design is based on the laws of quantum mechanics. It turns out that such a machine is not more powerful, in the sense of computability, than the machine originally constructed by Turing. Quantum Turingmachines do not violate the Church-Turing thesis. The benefit of quantum computing lies in efficiency. Quantum computers appear to be more efficient, in time, than classical Turing-machines, however its exact additional computational power is unclear, as this question ties in with deep open problems in complexity theory. We will sketch where BQP, the quantum analogue of the complexity class P, resides in the realm of complexity classes.","PeriodicalId":139105,"journal":{"name":"Turing's Legacy","volume":"348 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121701976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.1017/CBO9781107338579.015
P. Welch
In recent years there has emerged the study of discrete computational models which are allowed to act transfinitely . By ‘discrete’ we mean that the machine models considered are not analogue machines, but compute by means of distinct stages or in units of time. The paradigm of such models is, of course, Turing’s original machine model. If we concentrate on this for a moment, the machine is considered to be running a program P perhaps on some natural number input n ∈ N and is calculating P (n). Normally we say this is a successful computation if the machine halts after a finite number of stages and we may read off some designated form of output: ‘P (n)↓’. However if the machine fails to halt after a finite time it may be exhibiting a variety of behaviours on its tape. Mathematically we may ask what happens ‘in the limit’ as the number of stages approaches ω. The machine may of course go haywire, and simply be rewriting a particular cell infinitely often, or else the Read/Write head may go ‘off to infinity’ as it moves inexorably down the tape. These kind of considerations are behind the notion of ‘computation in the limit’ which we consider below. Or, it may only rewrite finitely often to any cell on the tape, and leave something meaningful behind: an infinite string of 0, 1’s and thus an element of Cantor space 2. What kind of elements could be there? Considerations of what may lay on an output tape at an infinite stage first surface in the notion of ‘computation in the limit’ or ‘limit decidable’. Whilst the first publication on the matter seems to be two papers coincidentally appearing in the same year, 1965, as Martin Davis has commented, surely this was already known to Post?
近年来出现了允许超有限作用的离散计算模型的研究。通过“离散”，我们的意思是所考虑的机器模型不是模拟机器，而是通过不同的阶段或时间单位来计算。当然，这些模型的范例就是图灵最初的机器模型。如果我们把注意力集中在这一点上，机器被认为是在运行一个程序P，也许是在某个自然数输入n∈n上，并且正在计算P (n)。通常我们说这是一个成功的计算，如果机器在有限的阶段后停止，我们可以读出一些指定的输出形式:' P (n)↓'。然而，如果机器在有限的时间后没有停止，它可能会在其磁带上表现出各种行为。从数学上讲，我们可能会问，当阶段数接近ω时，“极限”会发生什么。当然，机器可能会失控，只是无限频繁地重写一个特定的单元，或者读/写磁头可能会在磁带上无情地向下移动时“走向无限”。这些考虑是在“极限计算”的概念背后，我们将在下面讨论。或者，它可能只对磁带上的任何单元格进行有限频率的重写，并留下一些有意义的东西:一个无限的0和1字符串，因此是康托空间2的一个元素。那里可能有什么样的元素?“极限计算”或“极限可决定”的概念首先涉及到在无限级上输出纸带上的内容。正如马丁·戴维斯(Martin Davis)所评论的那样，关于这个问题的第一次出版物似乎是1965年同一年巧合出现的两篇论文，但波斯特肯定已经知道这一点了吧?
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Pub Date : 1900-01-01DOI: 10.1017/CBO9781107338579.006
E. Fokina, V. Harizanov, A. Melnikov
In the last few decades there has been increasing interest in computable model theory. Computable model theory uses the tools of computability theory to explores algorithmic content (e¤ectiveness) of notions, theorems, and constructions in various areas of ordinary mathematics. In algebra this investigation dates back to van der Waerden who in his 1930 book Modern Algebra de ned an explicitly given eld as one the elements of which are uniquely represented by distinguishable symbols with which we can perform the eld operations algorithmically. In his pioneering paper on non-factorability of polynomials from 1930, van der Waerden essentially proved that an explicit eld (F;+; ) does not necessarily have an algorithm for splitting polynomials in F [x] into their irreducible factors. Gödels incompleteness theorem from 1931 is an astonishing early result of computable model theory. Gödel showed that there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms.The work of Turing, Gödel, Kleene, Church, Post, and others in the mid-1930s established the rigorous mathematical foundations for the computability theory. In the 1950s, Fröhlich and Shepherdson used the precise notion of a computable function to obtain a collection of results and examples about explicit rings and elds. For example, Fröhlich and
在过去的几十年里，人们对可计算模型理论的兴趣越来越大。可计算模型理论使用可计算理论的工具来探索普通数学各个领域的概念、定理和结构的算法内容(有效性)。在代数中，这项研究可以追溯到van der Waerden，他在1930年的著作《现代代数 》中需要一个明确给定的 场，作为其中的一个元素，用可区分的符号唯一地表示，我们可以用算法来执行 场运算。van der Waerden在1930年关于多项式不可因式的开创性论文中，实质上证明了一个显式 场(F;+;)不一定有将F [x]中的多项式分解为其不可约因子的算法。1931年Gödel的不完备定理是可计算模型理论的一个惊人的早期结果。Gödel表明“事实上，在普通整数理论中存在一些相对简单的问题，这些问题不能由公理来决定。”图灵、Gödel、克莱因、丘奇、波斯特和其他人在20世纪30年代中期的工作为可计算理论建立了严格的数学基础。在20世纪50年代，Fröhlich和Shepherdson使用了可计算函数的精确概念，获得了关于显环和 场的一系列结果和例子。例如:Fröhlich和
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Pub Date : 1900-01-01DOI: 10.1017/CBO9781107338579.005
R. Downey
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.
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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Pub Date : 1900-01-01DOI: 10.1017/CBO9781107338579.012
D. Normann
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Pub Date : 1900-01-01DOI: 10.1017/CBO9781107338579.010
Charles F. Miller
We trace the emergence of unsolvable problems in algebra and topology from the unsolvable halting problem for Turing machines. §
我们从图灵机的不可解停止问题追溯了代数和拓扑中不可解问题的出现。§
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