Turing's Legacy最新文献

Turing and the discovery of computability 图灵和可计算性的发现

Turing's Legacy Pub Date : 2014-05-01 DOI: 10.1017/CBO9781107338579.014

R. Soare

{"title":"Turing and the discovery of computability","authors":"R. Soare","doi":"10.1017/CBO9781107338579.014","DOIUrl":"https://doi.org/10.1017/CBO9781107338579.014","url":null,"abstract":"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.","PeriodicalId":139105,"journal":{"name":"Turing's Legacy","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125207589","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}

引用次数: 2

Mathematics in the age of the Turing machine 图灵机时代的数学

Turing's Legacy Pub Date : 2013-02-12 DOI: 10.1017/CBO9781107338579.008

T. Hales

引用次数: 26

Computability and analysis: the legacy of Alan Turing 可计算性和分析:图灵的遗产

Turing's Legacy Pub Date : 2012-06-15 DOI: 10.1017/CBO9781107338579.002

J. Avigad, V. Brattka

{"title":"Computability and analysis: the legacy of Alan Turing","authors":"J. Avigad, V. Brattka","doi":"10.1017/CBO9781107338579.002","DOIUrl":"https://doi.org/10.1017/CBO9781107338579.002","url":null,"abstract":"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","PeriodicalId":139105,"journal":{"name":"Turing's Legacy","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121392116","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}

引用次数: 79

Turing and the development of computational complexity 图灵和计算复杂性的发展

Turing's Legacy Pub Date : 2011-12-20 DOI: 10.1017/CBO9781107338579.009

S. Homer, A. Selman

引用次数: 2

Turing in Quantumland 图灵在量子世界

Turing's Legacy Pub Date : 1900-01-01 DOI: 10.1017/CBO9781107338579.004

H. Buhrman

引用次数: 0

Transfinite machine models 超限机器模型

Turing's Legacy Pub Date : 1900-01-01 DOI: 10.1017/CBO9781107338579.015

P. Welch

{"title":"Transfinite machine models","authors":"P. Welch","doi":"10.1017/CBO9781107338579.015","DOIUrl":"https://doi.org/10.1017/CBO9781107338579.015","url":null,"abstract":"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?","PeriodicalId":139105,"journal":{"name":"Turing's Legacy","volume":"34 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":"131312633","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}

引用次数: 8

Computable model theory 可计算模型理论

Turing's Legacy Pub Date : 1900-01-01 DOI: 10.1017/CBO9781107338579.006

E. Fokina, V. Harizanov, A. Melnikov

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引用次数: 20

Computability theory, algorithmic randomness and Turing's anticipation 可计算理论，算法随机性和图灵的预测

Turing's Legacy Pub Date : 1900-01-01 DOI: 10.1017/CBO9781107338579.005

R. Downey

{"title":"Computability theory, algorithmic randomness and Turing's anticipation","authors":"R. Downey","doi":"10.1017/CBO9781107338579.005","DOIUrl":"https://doi.org/10.1017/CBO9781107338579.005","url":null,"abstract":"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.","PeriodicalId":139105,"journal":{"name":"Turing's Legacy","volume":"2 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":"121867619","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}

引用次数: 0

Higher generalizations of the Turing Model

Turing's Legacy Pub Date : 1900-01-01 DOI: 10.1017/CBO9781107338579.012

D. Normann

引用次数: 1

Turing machines to word problems 图灵机到文字问题

Turing's Legacy Pub Date : 1900-01-01 DOI: 10.1017/CBO9781107338579.010

Charles F. Miller

引用次数: 8