Computable functions are the basic objects of study in computability theory. Computable functions are the formalized analogue of the intuitive notion of algorithm. They are used to discuss computability without referring to any concrete model of computation such as Turing machines or register machines. Any definition, however, must make reference to some specific model of computation but all valid definitions yield the same class of functions. Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions.

Before the precise definition of computable function, mathematicians often used the informal term ''effectively calculable''. This term has since come to be identified with the computable functions. Note that the effective computability of these functions does not imply that they can be ''efficiently'' computed (i.e. computed within a reasonable amount of time). In fact, for some effectively calculable functions it can be shown that any algorithm that computes them will be very inefficient in the sense that the running time of the algorithm increases exponentially (or even superexponentially) with the length of the input. The fields of feasible computability and computational complexity study functions that can be computed efficiently.

According to the Church-Turing thesis, computable functions are exactly the functions that can be calculated using a mechanical calculation device given unlimited amounts of time and storage space. Equivalently, this thesis states that any function which has an algorithm is computable. Note that an algorithm in this sense is understood to be a sequence of steps a person with unlimited time and an infinite supply of pen and paper could follow.

The Blum axioms can be used to define an abstract computational complexity theory on the set of computable functions. In computational complexity theory, the problem of determining the complexity of a computable function is known as a function problem.

Definition

The class of computable functions can be defined in many equivalent models, including

Turing machines

μ-recursive functions

Lambda calculus

Post machines (Post-Turing machines and tag machines).

Register machines

Although those models use different representations for the functions, their inputs and their outputs, translations exist between any two models. In the remainder of this article, functions from natural numbers to natural numbers are used (as is the case for, e.g., the μ-recursive functions).

Each computable function takes a fixed, finite number of natural numbers as arguments. Because the functions are partial in general, they may not be defined for every possible choice of input. If a computable function is defined for a certain input, then it returns a single natural number as output (this output can be interpreted as a list of numbers using a pairing function). These functions are also called partial recursive functions. In computability theory, the domain of a function is taken to be the set of all inputs for which the function is defined.

A function which is defined for all possible arguments is called total. If a computable function is total, it is called a total computable function or total recursive function.

The notation indicates that the partial function is defined on arguments , and the notation indicates that is defined on the arguments and the value returned is . The case that a function is undefined for arguments is denoted by .

Characteristics of computable functions

The basic characteristic of a computable function is that there must be a finite procedure (an algorithm) telling how to compute the function. The models of computation listed above give different interpretations of what a procedure is and how it is used, but these interpretations share many properties. The fact that these models give equivalent classes of computable functions stems from the fact that each model is capable of reading and mimicking a procedure for any of the other models, much as a compiler is able to read instructions in one computer language and emit instructions in another language.

Enderton [1977] gives the following characteristics of a procedure for computing a computable function; similar characterizations have been given by Turing [1936], Rogers [1967], and others.

“There must be exact instructions (i.e. a program), finite in length, for the procedure.”

Thus every computable function must have a finite program that completely describes how the function is to be computed. It is possible to compute the function by just following the instructions; no guessing or special insight is required.

“If the procedure is given a ''k''-tuple x in the domain of ''f'', then after a finite number of discrete steps the procedure must terminate and produce ''f(x)''.”

Intuitively, the procedure proceeds step by step, with a specific rule to cover what to do at each step of the calculation. Only finitely many steps can be carried out before the value of the function is returned.

“If the procedure is given a ''k''-tuple x which is not in the domain of ''f'', then the procedure might go on forever, never halting. Or it might get stuck at some point, but it must not pretend to produce a value for ''f'' at x.”

Thus if a value for ''f(x)'' is ever found, it must be the correct value. It is not necessary for the computing agent to distinguish correct outcomes from incorrect ones because the procedure is always correct when it produces an outcome.

Enderton goes on to list several clarifications of these requirements of the procedure for a computable function:

The procedure must theoretically work for arbitrarily large arguments. It is not assumed that the arguments are smaller than the number of atoms in the Earth, for example.

The procedure is required to halt after finitely many steps in order to produce an output, but it may take arbitrarily many steps before halting. No time limitation is assumed.

Although the procedure may use only a finite amount of storage space during a successful computation, there is no bound on the amount of space that is used. It is assumed that additional storage space can be given to the procedure whenever the procedure asks for it.

The field of computational complexity studies functions with prescribed bounds on the time and/or space allowed in a successful computation.

Computable sets and relations

A set of natural numbers is called computable (synonyms: recursive, decidable) if there is a computable, total function such that for any natural number , if is in and if is not in .

A set of natural numbers is called computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function such that for each number , is defined if and only if is in the set. Thus a set is computably enumerable if and only if it is the domain of some computable function. The word ''enumerable'' is used because the following are equivalent for a nonempty subset of the natural numbers: is the domain of a computable function. is the range of a total computable function. If is infinite then the function can be assumed to be injective. If a set is the range of a function then the function can be viewed as an enumeration of , because the list will include every element of .

Because each finitary relation on the natural numbers can be identified with a corresponding set of finite sequences of natural numbers, the notions of computable relation and computably enumerable relation can be defined from their analogues for sets.

Formal languages

''

In computability theory in computer science, it is common to consider formal languages. An alphabet is an arbitrary set. A word on an alphabet is a finite sequence of symbols from the alphabet; the same symbol may be used more than once. For example, binary strings are exactly the words on the alphabet . A language is a subset of the collection of all words on a fixed alphabet. For example, the collection of all binary strings that contain exactly 3 ones is a language over the binary alphabet.

A key property of a formal language is the level of difficulty required to decide whether a given word is in the language. Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input; this is usually considered routine. A language is called computable (synonyms: recursive, decidable) if there is a computable function such that for each word over the alphabet, if the word is in the language and if the word is not in the language. Thus a language is computable just in case there is a procedure that is able to correctly tell whether arbitrary words are in the language.

A language is computably enumerable (synonyms: recursively enumerable, semidecidable) if there is a computable function such that is defined if and only if the word is in the language. The term ''enumerable'' has the same etymology as in computably enumerable sets of natural numbers.

Examples

The following functions are computable:

Each function with a finite domain; e.g., any finite sequence of natural numbers.

Each constant function ''f'' : N^''k'' → N, ''f''(''n''₁,...''n''_''k'') := ''n''.

Addition ''f'' : N²→ N, ''f''(''n''₁,''n''_''2'') := ''n''₁ + ''n''₂

The function which gives the list of prime factors of a number.

The greatest common divisor of two numbers is a computable function.

Bézout's identity, a linear Diophantine equation

If ''f'' and ''g'' are computable, then so are: ''f + g'', ''f * g'', $f\; \backslash circ\; g$ if ''f'' is unary, max(''f'',''g''), min(''f'',''g''), arg max{''y''≤''f(x)''} and many more combinations.

The following examples illustrate that a function may be computable though it is not known which algorithm computes it.

The function ''f'' such that ''f(n) = 1'' if there is a sequence of ''at least n'' consecutive fives in the decimal expansion of π, and ''f(n) = 0'' otherwise, is computable. (The function ''f'' is either the constant 1 function, which is computable, or else there is a ''k'' such that ''f(n) = 1'' if ''n < k'' and ''f(n) = 0'' if ''n ≥ k''. Every such function is computable. It is not known whether there are arbitrarily long runs of fives in the decimal expansion of π, so we don't know ''which'' of those functions is ''f''. Nevertheless, we know that the function ''f'' must be computable.)

Each finite segment of an ''in''computable sequence of natural numbers (such as the Busy Beaver function ''Σ'') is computable. E.g., for each natural number ''n'', there exists an algorithm that computes the finite sequence ''Σ(0), Σ(1), Σ(2), ..., Σ(n)'' — in contrast to the fact that there is no algorithm that computes the ''entire'' Σ-sequence, i.e. ''Σ(n)'' for all ''n''. Thus, "Print 0, 1, 4, 6, 13" is a trivial algorithm to compute ''Σ(0), Σ(1), Σ(2), Σ(3), Σ(4)''; similarly, for any given value of ''n'', such a trivial algorithm ''exists'' (even though it may never be ''known'' or produced by anyone) to compute ''Σ(0), Σ(1), Σ(2), ..., Σ(n)''.

Church-Turing thesis

The Church-Turing thesis states that any function computable from a procedure possessing the three properties listed above is a computable function. Because these three properties are not formally stated, the Church-Turing thesis cannot be proved. The following facts are often taken as evidence for the thesis:

Many equivalent models of computation are known, and they all give the same definition of computable function (or a weaker version, in some instances).

No stronger model of computation which is generally considered to be effectively calculable has been proposed.

The Church-Turing thesis is sometimes used in proofs to justify that a particular function is computable by giving a concrete description of a procedure for the computation. This is permitted because it is believed that all such uses of the thesis can be removed by the tedious process of writing a formal procedure for the function in some model of computation.

Incomputable functions and unsolvable problems

Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it. Furthermore, this procedure has to be encoded in the finite alphabet used by the computational model, so there are only countably many computable functions. For example, functions may be encoded using a string of bits (the alphabet ).

The real numbers are uncountable so most real numbers are not computable. See computable number. The set of finitary functions on the natural numbers is uncountable so most are not computable. Concrete examples of such functions are Busy beaver, Kolmogorov complexity, or any function that outputs the digits of noncomputable number, such as Chaitin's constant.

Similarly, most subsets of the natural numbers are not computable. The Halting problem was the first such set to be constructed. The Entscheidungsproblem, proposed by David Hilbert, asked whether there is an effective procedure to determine which mathematical statements (coded as natural numbers) are true. Turing and Church independently showed in the 1930s that this set of natural numbers is not computable. According to the Church-Turing thesis, there is no effective procedure (with an algorithm) which can perform these computations.

Extensions of computability

Relative Computability

The notion of computability of a function can be relativized to an arbitrary set of natural numbers ''A''. A function ''f'' is defined to be computable in ''A'' (equivalently ''A''-computable or computable relative to ''A'') when it satisfies the definition of a computable function with modifications allowing access to ''A'' as an oracle. As with the concept of a computable function relative computability can be given equivalent definitions in many different models of computation. This is commonly accomplished by supplementing the model of computation with an additional primitive operation which asks whether a given integer is a member of ''A''. We can also talk about ''f'' being computable in ''g'' by identifying ''g'' with its graph.

Higher Recursion Theory

Hyperarithmetical theory studies those sets that can be computed from a computable ordinal number of iterates of the Turing jump of the empty set. This is equivalent to sets defined by both a universal and existential formula in the language of second order arithmetic and to some models of Hypercomputation. Even more general recursion theories have been studied, such as E-recursion theory in which any set can be used as an argument to an E-recursive function.

Hyper-computation

Although the Church-Turing thesis states that the computable functions include all functions with algorithms, it is possible to consider broader classes of functions that relax the requirements that algorithms must possess. The field of Hypercomputation studies models of computation that go beyond normal Turing computation. These don't violate the Church-Turing thesis since they allow operations that, whether or not they can be implemented in a physical device, couldn't be performed by a human working with pencil and paper.

See also

Computable number

Effective method

Theory of computation

Recursion theory

Turing degree

Arithmetical hierarchy

Hypercomputation

Super-recursive algorithm

Semicomputable function

References

Enderton, H.B. Elements of recursion theory. ''Handbook of Mathematical Logic'' (North-Holland 1977) pp. 527–566.

Rogers, H. ''Theory of recursive functions and effective computation'' (McGraw-Hill 1967).

Turing, A. (1936), On Computable Numbers, With an Application to the Entscheidungsproblem. ''Proceedings of the London Mathematical Society'', Series 2, Volume 42 (1936). Reprinted in M. Davis (ed.), ''The Undecidable'', Raven Press, Hewlett, NY, 1965.

Category:Computability theory Category:Theory of computation

ar:دوال حسابية ca:Funció computable es:Función computable fr:Fonction calculable it:Funzione calcolabile ja:計算可能関数 pl:Funkcja obliczalna pt:Função computável ru:Вычислимая функция simple:Computable function zh:可计算函数

Coordinates	19°45′″N96°6′″N
name	Gordon D. Plotkin
birth date	September 09, 1946
birth place	Glasgow
residence	Scotland
nationality	British
field	logician, computer scientist, mathematician
work institution	University of Edinburgh
alma mater	Edinburgh
doctoral advisor	Rod Burstall
doctoral students	Luca Cardelli (1982)Marcelo Fiore (1994)Philippa Gardner (1992)Martin Hofmann (1995)John Longley (1995)Eugenio Moggi (1988)Mohammad Reza Mousavi (2005)Michael D. Pedersen (2010)David Pym (1990)Alex Simpson (1994)Lǐ WèiGlynn Winskel (1980) }}

Gordon D. Plotkin, FRS, FRSE (born 9 September 1946, in Glasgow) is a Scottish computer scientist.

Gordon Plotkin is best-known for his introduction of structural operational semantics (SOS) and his work on denotational semantics. In particular, his notes on ''A Structural Approach to Operational Semantics'' of 1981 were very influential. He has contributed to many other areas of computer science.

Plotkin is now Professor of Theoretical Computer Science in the School of Informatics at The University of Edinburgh.

Biography

Plotkin received his PhD in 1972 from the University of Edinburgh, where he studied under Rod Burstall. He has remained at Edinburgh, and was, with Burstall and Milner, a co-founder of the Laboratory for Foundations of Computer Science (LFCS).

Works

1975: "Call-by-Name, Call-by Value and the Lambda Calculus".

''A Structural Approach to Operational Semantics'' by G.D. Plotkin (1981)

''Program Verification and Semantics: Further Work'' (2004)

Awards

He was elected a Fellow of the Royal Society in 1992, is a Fellow of the Royal Society of Edinburgh and a Member of the Academia Europæa. He is also a winner of the Royal Society Wolfson Research Merit Award.

See also

Church-Rosser theorem

Domain theory

Golem (ILP)

Informatics

LFCS

Operational semantics

Programming language for Computable Functions

Unbounded nondeterminism

University of Edinburgh

References

External links

Official home page

Personal home page

Online publications

Symposium for Gordon Plotkin

Category:1946 births Category:Living people Category:British computer scientists Category:Fellows of the Royal Society Category:Members of Academia Europaea Category:Royal Society Wolfson Research Merit Award holders Category:Formal methods people Category:Programming language researchers Category:British Jews Category:Jewish scientists Category:Alumni of the University of Edinburgh Category:Academics of the University of Edinburgh Category:Fellows of the Royal Society of Edinburgh

it:Gordon Plotkin

Coordinates	19°45′″N96°6′″N
name	Robin Milner
birth date	January 13, 1934
birth place	Yealmpton, Plymouth, England
death date	March 20, 2010
death place	Cambridge, England
field	Computer Science
work institution	FerrantiCity University, LondonSwansea UniversityStanford UniversityUniversity of EdinburghUniversity of Cambridge
doctoral advisor	None, as Milner never did a PhD
doctoral students	George MilneAvra CohnRaymond AubinMike SandersonAlan MycroftLuis DamasBrian MonahanKevin MitchellKim Larsen (1986)Mads Tofte (1988)K.V.S. Prasad (1989)Faron MollerDave BerryChris ToftsPeter SewellDavide Sangiorgi (1993)David N. Turner (1995)Alex MifsudJames J. Leifer (2001)
known for	LCFMLCalculus of communicating systemsPi-calculusHindley-Milner type inference
prizes	Turing Award
footnotes	}}

Arthur John ''Robin'' Gorell Milner FRS FRSE (Robin Milner or A.J.R.G. Milner, born 13 January 1934 near Plymouth, died 20 March 2010 in Cambridge) was a prominent British computer scientist.

Life, education and career

Milner was born in Yealmpton, near Plymouth, England into a military family. He was awarded a scholarship to Eton College in 1947, and subsequently served in the Royal Engineers, attaining the rank of Second Lieutenant. He then enrolled at King's College, Cambridge, graduating in 1957, Milner first worked as a schoolteacher then as a programmer at Ferranti, before entering academia at City University, London, then Swansea University, Stanford University, and from 1973 at the University of Edinburgh, where he was a co-founder of the Laboratory for Foundations of Computer Science (LFCS). He returned to Cambridge as the head of the Computer Laboratory in 1995 from which he eventually stepped down, although he was still at the laboratory. From 2009, Milner was a SICSA Advanced Research Fellow and held (part-time) the Chair of Computer Science at the University of Edinburgh.

Milner died of a heart attack on 20 March 2010 in Cambridge. His wife, Lucy, died shortly before him.

Contributions

Milner is generally regarded as having made three major contributions to computer science. He developed LCF, one of the first tools for automated theorem proving. The language he developed for LCF, ML, was the first language with polymorphic type inference and type-safe exception handling. In a very different area, Milner also developed a theoretical framework for analyzing concurrent systems, the calculus of communicating systems (CCS), and its successor, the pi-calculus. At the time of his death, he was working on bigraphs, a formalism for ubiquitous computing subsuming CCS and the pi-calculus.

Honors and awards

He was made a Fellow of the Royal Society in 1988 and received the ACM Turing Award in 1991. In 1994 he was inducted as a Fellow of the ACM. In 2004, the Royal Society of Edinburgh awarded Milner with a Royal Medal for his "bringing about public benefits on a global scale". In 2008, he was elected a Foreign Associate of the National Academy of Engineering for "fundamental contributions to computer science, including the development of LCF, ML, CCS, and the pi-calculus."

Selected publications

''A Calculus of Communicating Systems'', Robin Milner. Springer-Verlag (LNCS 92), 1980. ISBN 3-540-10235-3

''Communication and Concurrency'', Robin Milner. Prentice Hall (International Series in Computer Science), 1989. ISBN 0-131-15007-3

''The Definition of Standard ML'', Robin Milner, Mads Tofte, Robert Harper, MIT Press 1990

''The Definition of Standard ML'' (Revised), Robin Milner, Mads Tofte, Robert Harper, David MacQueen, MIT Press 1997. ISBN 0-262-63181-4

''Commentary on Standard ML'', Robin Milner, Mads Tofte, MIT Press 1997. ISBN 0-262-63137-7

''Communicating and Mobile Systems: the Pi-Calculus'', Robin Milner. Cambridge University Press, 1999. ISBN 0-521-65869-1

''The Space and Motion of Communicating Agents'', Robin Milner, Cambridge University Press, 2009. ISBN 9780521738330

Publications by Robin Milner in DBLP

Bibliography

''Proof, Language, and Interaction: Essays in Honour of Robin Milner'', edited by Gordon Plotkin, Colin Stirling and Mads Tofte. The MIT Press, 2000. ISBN 0-262-16188-5.

The Royal Society of Edinburgh: ''Royal Gold Medals for Outstanding Achievement'' (2004 press release). http://www.royalsoced.org.uk/rse_press/2004/medals.htm

A brief biography of and speech by Robin Milner

A Brief Scientific Biography of Robin Milner (from Proof, Language, and Interaction: Essays in Honour of Robin Milner)

References

External links

Address in Bologna, a short address by Milner on receiving Laurea Honoris Causa in Computer Science from the University of Bologna, summarising some of his main works, 9 July 1997

Is informatics a science?, conference at ENS, 10 December 2007

Category:1934 births Category:2010 deaths Category:People from Devon Category:Old Etonians Category:Alumni of King's College, Cambridge Category:British computer scientists Category:Fellows of the Royal Society Category:Turing Award laureates Category:Academics of City University London Category:Academics of Swansea University Category:Stanford University faculty Category:Academics of the University of Edinburgh Category:Formal methods people Category:Members of the University of Cambridge Computer Laboratory Category:Fellows of the Association for Computing Machinery Category:Programming language designers Category:Programming language researchers Category:Fellows of the Royal Society of Edinburgh Category:Fellows of the British Computer Society Category:Royal Engineers officers Category:Computer science writers Category:Deaths from myocardial infarction Category:Cardiovascular disease deaths in England Category:Members of IFIP Technical Committee 1

de:Robin Milner es:Robin Milner fr:Robin Milner id:Robin Milner nl:Robin Milner ja:ロビン・ミルナー pl:Robin Milner pt:Robin Milner ro:Robin Milner ru:Милнер, Робин sk:Robin Milner sr:Робин Милнер vi:Robin Milner zh:罗宾·米尔纳

Coordinates	19°45′″N96°6′″N
name	Alan Mathison Turing
birth date	June 23, 1912
birth place	Maida Vale, London, England, United Kingdom
death date	June 07, 1954
death place	Wilmslow, Cheshire, England, United Kingdom
residence	United Kingdom
nationality	British
field	Mathematics, Cryptanalysis, Computer science
work institutions	University of CambridgeGovernment Code and Cypher SchoolNational Physical LaboratoryUniversity of Manchester
alma mater	King's College, CambridgePrinceton University
doctoral advisor	Alonzo Church
doctoral students	Robin Gandy
known for	Halting problemTuring machineCryptanalysis of the EnigmaAutomatic Computing EngineTuring AwardTuring TestTuring patterns
prizes	Officer of the Order of the British EmpireFellow of the Royal Society }}

Alan Mathison Turing, OBE, FRS ( ; 23 June 1912 – 7 June 1954), was an English mathematician, logician, cryptanalyst and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a significant role in the creation of the modern computer. Turing is widely considered to be the father of computer science and artificial intelligence.

During the Second World War, Turing worked for the Government Code and Cypher School at Bletchley Park, Britain's codebreaking centre. For a time he was head of Hut 8, the section responsible for German naval cryptanalysis. He devised a number of techniques for breaking German ciphers, including the method of the bombe, an electromechanical machine that could find settings for the Enigma machine. After the war he worked at the National Physical Laboratory, where he created one of the first designs for a stored-program computer, the ACE.

Towards the end of his life Turing became interested in mathematical biology. He wrote a paper on the chemical basis of morphogenesis, and he predicted oscillating chemical reactions such as the Belousov–Zhabotinsky reaction, which were first observed in the 1960s.

Turing's homosexuality resulted in a criminal prosecution in 1952, when homosexual acts were still illegal in the United Kingdom. He accepted treatment with female hormones (chemical castration) as an alternative to prison. He died in 1954, just over two weeks before his 42nd birthday, from cyanide poisoning. An inquest determined it was suicide; his mother and some others believed his death was accidental. On 10 September 2009, following an Internet campaign, British Prime Minister Gordon Brown made an official public apology on behalf of the British government for the way in which Turing was treated after the war.

Childhood and youth

Alan Turing was conceived in India. His father, Julius Mathison Turing, was a member of the Indian Civil Service. Julius and wife Ethel Sara Stoney (1881–1976, daughter of Edward Waller Stoney, chief engineer of the Madras Railways) wanted Alan to be brought up in England, so they returned to Maida Vale, London, where Alan Turing was born on 23 June 1912, as recorded by a blue plaque on the outside of the building, later the Colonnade Hotel. He had an elder brother, John. His father's civil service commission was still active, and during Turing's childhood years his parents travelled between Hastings, England and India, leaving their two sons to stay with a retired Army couple. Very early in life, Turing showed signs of the genius he was to later prominently display.

His parents enrolled him at St Michael's, a day school at 20 Charles Road, St Leonards on Sea, at the age of six. The headmistress recognised his talent early on, as did many of his subsequent educators. In 1926, at the age of 14, he went on to Sherborne School, a famous independent school in the market town of Sherborne in Dorset. His first day of term coincided with the General Strike in Britain, but so determined was he to attend his first day that he rode his bicycle unaccompanied more than from Southampton to school, stopping overnight at an inn.

Turing's natural inclination toward mathematics and science did not earn him respect with some of the teachers at Sherborne, whose definition of education placed more emphasis on the classics. His headmaster wrote to his parents: "I hope he will not fall between two stools. If he is to stay at Public School, he must aim at becoming ''educated''. If he is to be solely a ''Scientific Specialist'', he is wasting his time at a Public School". Despite this, Turing continued to show remarkable ability in the studies he loved, solving advanced problems in 1927 without having even studied elementary calculus. In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but he extrapolated Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.

Turing's hopes and ambitions at school were raised by the close friendship he developed with a slightly older fellow student, Christopher Morcom, who was Turing's first love interest. Morcom died suddenly only a few weeks into their last term at Sherborne, from complications of bovine tuberculosis, contracted after drinking infected cow's milk as a boy. Turing's religious faith was shattered and he became an atheist. He adopted the conviction that all phenomena, including the workings of the human brain, must be materialistic, but he still believed in the survival of the spirit after death.

University and work on computability

After Sherborne, Turing went to study at King's College, Cambridge. He was an undergraduate there from 1931 to 1934, graduating with first-class honours in Mathematics, and in 1935 was elected a fellow at King's on the strength of a dissertation on the central limit theorem.

In 1928, German mathematician David Hilbert had called attention to the ''Entscheidungsproblem'' (decision problem). In his momentous paper "On Computable Numbers, with an Application to the ''Entscheidungsproblem''" (submitted on 28 May 1936 and delivered 12 November), Turing reformulated Kurt Gödel's 1931 results on the limits of proof and computation, replacing Gödel's universal arithmetic-based formal language with what became known as Turing machines, formal and simple devices. He proved that some such machine would be capable of performing any conceivable mathematical computation if it were representable as an algorithm. He went on to prove that there was no solution to the ''Entscheidungsproblem'' by first showing that the halting problem for Turing machines is undecidable: it is not possible to decide, in general, algorithmically whether a given Turing machine will ever halt. While his proof was published subsequent to Alonzo Church's equivalent proof in respect to his lambda calculus, Turing was unaware of Church's work at the time.

Turing's approach is considerably more accessible and intuitive. It was also novel in its notion of a 'Universal (Turing) Machine', the idea that such a machine could perform the tasks of any other machine, or in other words, is provably capable of computing anything that is computable. Turing machines are to this day a central object of study in theory of computation.

In his memoirs Turing wrote that he was disappointed about the reception of this 1936 paper and that only two people had reacted – these being Heinrich Scholz and Richard Bevan Braithwaite.

The paper also introduces the notion of definable numbers.

From September 1936 to July 1938 he spent most of his time at the Institute for Advanced Study, Princeton, New Jersey, studying under Alonzo Church. In addition to his purely mathematical work, he studied cryptology and also built three of four stages of an electro-mechanical binary multiplier. In June 1938 he obtained his PhD from Princeton; his dissertation (''Systems of Logic Based on Ordinals'') introduced the concept of ordinal logic and the notion of relative computing, where Turing machines are augmented with so-called oracles, allowing a study of problems that cannot be solved by a Turing machine.

Back in Cambridge, he attended lectures by Ludwig Wittgenstein about the foundations of mathematics. The two argued and disagreed, with Turing defending formalism and Wittgenstein arguing that mathematics does not discover any absolute truths but rather invents them. He also started to work part-time with the Government Code and Cypher School (GCCS).

Cryptanalysis

During the Second World War, Turing was a main participant in the efforts at Bletchley Park to break German ciphers. Building on cryptanalysis work carried out in Poland by Marian Rejewski, Jerzy Różycki and Henryk Zygalski from Cipher Bureau before the war, he contributed several insights into breaking both the Enigma machine and the Lorenz SZ 40/42 (a Teleprinter (Teletype) cipher attachment codenamed ''Tunny'' by the British), and was, for a time, head of Hut 8, the section responsible for reading German naval signals.

From September 1938, Turing had been working part-time (notionally for the British Foreign Office) with the Government Code and Cypher School (GCCS), the British code breaking organisation. He worked on the problem of the German Enigma machine, and collaborated with Dilly Knox, a senior GCCS codebreaker. On 4 September 1939, the day after the UK declared war on Germany, Turing reported to Bletchley Park, the wartime station of GCCS.

In 1945, Turing was awarded the OBE for his wartime services, but his work remained secret for many years.

Turing had something of a reputation for eccentricity at Bletchley Park. Jack Good, a cryptanalyst who worked with him, is quoted by Ronald Lewin as having said of Turing:

in the first week of June each year he would get a bad attack of hay fever, and he would cycle to the office wearing a service gas mask to keep the pollen off. His bicycle had a fault: the chain would come off at regular intervals. Instead of having it mended he would count the number of times the pedals went round and would get off the bicycle in time to adjust the chain by hand. Another of his eccentricities is that he chained his mug to the radiator pipes to prevent it being stolen.

While working at Bletchley, Turing, a talented long-distance runner, occasionally ran the to London when he was needed for high-level meetings, and he was capable of world-class marathon standards.

Turing–Welchman bombe

Within weeks of arriving at Bletchley Park, Turing had specified an electromechanical machine which could help break Enigma faster than bomba from 1938, the bombe, named after and building upon the original Polish-designed bomba. The bombe, with an enhancement suggested by mathematician Gordon Welchman, became one of the primary tools, and the major automated one, used to attack Enigma-protected message traffic.

Jack Good opined:

Turing's most important contribution, I ''think'', was of part of the design of the bombe, the cryptanalytic machine. He had the idea that you could use, in effect, a theorem in logic which sounds to the untrained ear rather absurd; namely that from a contradiction, you can deduce ''everything.''

The bombe searched for possibly correct settings used for an Enigma message (i.e., rotor order, rotor settings, etc.), and used a suitable ''crib'': a fragment of probable plaintext. For each possible setting of the rotors (which had of the order of 10¹⁹ states, or 10²² for the four-rotor U-boat variant), the bombe performed a chain of logical deductions based on the crib, implemented electrically. The bombe detected when a contradiction had occurred, and ruled out that setting, moving onto the next. Most of the possible settings would cause contradictions and be discarded, leaving only a few to be investigated in detail. Turing's bombe was first installed on 18 March 1940. More than two hundred bombes were in operation by the end of the war.

Hut 8 and Naval Enigma

Turing decided to tackle the particularly difficult problem of German naval Enigma "because no one else was doing anything about it and I could have it to myself". In December 1939, Turing solved the essential part of the naval indicator system, which was more complex than the indicator systems used by the other services. The same night that he solved the naval indicator system, he conceived the idea of ''Banburismus'', a sequential statistical technique (what Abraham Wald later called sequential analysis) to assist in breaking naval Enigma, "though I was not sure that it would work in practice, and was not in fact sure until some days had actually broken". For this he invented a measure of weight of evidence that he called the ''Ban''. Banburismus could rule out certain orders of the Enigma rotors, substantially reducing the time needed to test settings on the bombes.

In 1941, Turing proposed marriage to Hut 8 co-worker Joan Clarke, a fellow mathematician, but their engagement was short-lived. After admitting his homosexuality to his fiancée, who was reportedly "unfazed" by the revelation, Turing decided that he could not go through with the marriage.

In July 1942, Turing devised a technique termed ''Turingery'' (or jokingly ''Turingismus'') for use against the Lorenz cipher messages produced by the Germans' new Geheimschreiber machine (''secret writer''). This was codenamed ''Tunny '' at Bletchley Park. He also introduced the Tunny team to Tommy Flowers who, under the guidance of Max Newman, went on to build the Colossus computer, the world's first programmable digital electronic computer, which replaced a simpler prior machine (the Heath Robinson) and whose superior speed allowed the brute-force decryption techniques to be applied usefully to the daily changing cyphers. A frequent misconception is that Turing was a key figure in the design of Colossus; this was not the case.

Turing travelled to the United States in November 1942 and worked with U.S. Navy cryptanalysts on Naval Enigma and bombe construction in Washington, and assisted at Bell Labs with the development of secure speech devices. He returned to Bletchley Park in March 1943. During his absence, Hugh Alexander had officially assumed the position of head of Hut 8, although Alexander had been ''de facto'' head for some time—Turing having little interest in the day-to-day running of the section. Turing became a general consultant for cryptanalysis at Bletchley Park.

Alexander wrote as follows about his contribution:

There should be no question in anyone's mind that Turing's work was the biggest factor in Hut 8's success. In the early days he was the only cryptographer who thought the problem worth tackling and not only was he primarily responsible for the main theoretical work within the Hut but he also shared with Welchman and Keen the chief credit for the invention of the Bombe. It is always difficult to say that anyone is absolutely indispensable but if anyone was indispensable to Hut 8 it was Turing. The pioneer's work always tends to be forgotten when experience and routine later make everything seem easy and many of us in Hut 8 felt that the magnitude of Turing's contribution was never fully realised by the outside world.

In the latter part of the war he moved to work at Hanslope Park, where he further developed his knowledge of electronics with the assistance of engineer Donald Bayley. Together they undertook the design and construction of a portable secure voice communications machine codenamed ''Delilah''. It was intended for different applications, lacking capability for use with long-distance radio transmissions, and in any case, Delilah was completed too late to be used during the war. Though Turing demonstrated it to officials by encrypting/decrypting a recording of a Winston Churchill speech, Delilah was not adopted for use. Turing also consulted with Bell Labs on the development of SIGSALY, a secure voice system that was used in the later years of the war.

Early computers and the Turing test

From 1945 to 1947 Turing lived in Church Street, Hampton while he worked on the design of the ACE (Automatic Computing Engine) at the National Physical Laboratory. He presented a paper on 19 February 1946, which was the first detailed design of a stored-program computer. Although ACE was a feasible design, the secrecy surrounding the wartime work at Bletchley Park led to delays in starting the project and he became disillusioned. In late 1947 he returned to Cambridge for a sabbatical year. While he was at Cambridge, the Pilot ACE was built in his absence. It executed its first program on 10 May 1950.

In 1948, he was appointed Reader in the Mathematics Department at Manchester (now part of The University of Manchester). In 1949, he became Deputy Director of the computing laboratory at the University of Manchester, and worked on software for one of the earliest stored-program computers—the Manchester Mark 1. During this time he continued to do more abstract work, and in "Computing machinery and intelligence" (Mind, October 1950), Turing addressed the problem of artificial intelligence, and proposed an experiment which became known as the Turing test, an attempt to define a standard for a machine to be called "intelligent". The idea was that a computer could be said to "think" if a human interrogator could not tell it apart, through conversation, from a human being. In the paper, Turing suggested that rather than building a program to simulate the adult mind, it would be better rather to produce a simpler one to simulate a child's mind and then to subject it to a course of education. A reversed form of the Turing test is widely used on the Internet; the CAPTCHA test is intended to determine whether the user is a human or a computer.

In 1948, Turing, working with his former undergraduate colleague, D. G. Champernowne, began writing a chess program for a computer that did not yet exist. In 1952, lacking a computer powerful enough to execute the program, Turing played a game in which he simulated the computer, taking about half an hour per move. The game was recorded. The program lost to Turing's colleague Alick Glennie, although it is said that it won a game against Champernowne's wife.

His Turing test was a significant and characteristically provocative and lasting contribution to the debate regarding artificial intelligence, which continues after more than half a century.

He also invented the LU decomposition method in 1948, used today for solving an equations matrix.

Pattern formation and mathematical biology

Turing worked from 1952 until his death in 1954 on mathematical biology, specifically morphogenesis. He published one paper on the subject called ''The Chemical Basis of Morphogenesis'' in 1952, putting forth the Turing hypothesis of pattern formation. His central interest in the field was understanding Fibonacci phyllotaxis, the existence of Fibonacci numbers in plant structures. He used reaction–diffusion equations which are central to the field of pattern formation. Later papers went unpublished until 1992 when ''Collected Works of A.M. Turing'' was published. His contribution is considered a seminal piece of work in this field.

Conviction for indecency

In January 1952, Turing met Arnold Murray outside a cinema in Manchester. After a lunch date, Turing invited Murray to spend the weekend with him at his house, an invitation which Murray accepted although he did not show up. The pair met again in Manchester the following Monday, when Murray agreed to accompany Turing to the latter's house. A few weeks later Murray visited Turing's house again, and apparently spent the night there.

After Murray helped an accomplice to break into his house, Turing reported the crime to the police. During the investigation, Turing acknowledged a sexual relationship with Murray. Homosexual acts were illegal in the United Kingdom at that time, and so both were charged with gross indecency under Section 11 of the Criminal Law Amendment Act 1885, the same crime for which Oscar Wilde had been convicted more than fifty years earlier.

Turing was given a choice between imprisonment or probation conditional on his agreement to undergo hormonal treatment designed to reduce libido. He accepted chemical castration via oestrogen hormone injections.

Turing's conviction led to the removal of his security clearance, and barred him from continuing with his cryptographic consultancy for GCHQ. His British passport was not revoked, though he was denied entry to the United States after his conviction. At the time, there was acute public anxiety about spies and homosexual entrapment by Soviet agents, because of the recent exposure of the first two members of the Cambridge Five, Guy Burgess and Donald Maclean, as KGB double agents. Turing was never accused of espionage but, as with all who had worked at Bletchley Park, was prevented from discussing his war work.

Death

On 8 June 1954, Turing's cleaner found him dead; he had died the previous day. A post-mortem examination established that the cause of death was cyanide poisoning. When his body was discovered an apple lay half-eaten beside his bed, and although the apple was not tested for cyanide, it is speculated that this was the means by which a fatal dose was delivered. An inquest determined that he had committed suicide, and he was cremated at Woking Crematorium on 12 June 1954. Turing's mother argued strenuously that the ingestion was accidental, caused by her son's careless storage of laboratory chemicals. Biographer Andrew Hodges suggests that Turing may have killed himself in an ambiguous way quite deliberately, to give his mother some plausible deniability. David Leavitt has suggested that Turing was re-enacting a scene from the 1937 film ''Snow White'', his favourite fairy tale, pointing out that he took "an especially keen pleasure in the scene where the Wicked Witch immerses her apple in the poisonous brew."

Epitaph

Recognition and tributes

A biography published by the Royal Society shortly after Turing's death (and while his wartime work was still subject to the Official Secrets Act) recorded:

Since 1966, the Turing Award has been given annually by the Association for Computing Machinery to a person for technical contributions to the computing community. It is widely considered to be the computing world's highest honour, equivalent to the Nobel Prize.

''Breaking the Code'' is a 1986 play by Hugh Whitemore about Alan Turing. The play ran in London's West End beginning in November 1986 and on Broadway from 15 November 1987 to 10 April 1988. There was also a 1996 BBC television production. In all cases, Derek Jacobi played Turing. The Broadway production was nominated for three Tony Awards including Best Actor in a Play, Best Featured Actor in a Play, and Best Direction of a Play, and for two Drama Desk Awards, for Best Actor and Best Featured Actor.

On 23 June 1998, on what would have been Turing's 86th birthday, Andrew Hodges, his biographer, unveiled an official English Heritage Blue Plaque at his birthplace and childhood home in Warrington Crescent, London, later the Colonnade hotel. To mark the 50th anniversary of his death, a memorial plaque was unveiled on 7 June 2004 at his former residence, Hollymeade, in Wilmslow, Cheshire.

On 13 March 2000, Saint Vincent and the Grenadines issued a set of stamps to celebrate the greatest achievements of the twentieth century, one of which carries a recognisable portrait of Turing against a background of repeated 0s and 1s, and is captioned: "1937: Alan Turing's theory of digital computing".

On 28 October 2004, a bronze statue of Alan Turing sculpted by John W Mills was unveiled at the University of Surrey in Guildford, marking the 50th anniversary of Turing's death; it portrays him carrying his books across the campus.

In 2006, Boston Pride named Turing their Honorary Grand Marshal.

Turing was one of four mathematicians examined in the 2008 BBC documentary entitled "Dangerous Knowledge".

The Princeton Alumni Weekly named Turing the second most significant alumnus in the history of Princeton University, second only to President James Madison.

A 1.5-ton, life-size statue of Turing was unveiled on 19 June 2007 at Bletchley Park. Built from approximately half a million pieces of Welsh slate, it was sculpted by Stephen Kettle, having been commissioned by the late American billionaire Sidney Frank.

Turing has been honoured in various ways in Manchester, the city where he worked towards the end of his life. In 1994, a stretch of the A6010 road (the Manchester city intermediate ring road) was named Alan Turing Way. A bridge carrying this road was widened, and carries the name Alan Turing Bridge. A statue of Turing was unveiled in Manchester on 23 June 2001. It is in Sackville Park, between the University of Manchester building on Whitworth Street and the Canal Street gay village. The memorial statue, depicts the "father of Computer Science" sitting on a bench at a central position in the park. The statue was unveiled on Turing's birthday.

Turing is shown holding an apple—a symbol classically used to represent forbidden love, the object that inspired Isaac Newton's theory of gravitation, and the means of Turing's own death. The cast bronze bench carries in relief the text 'Alan Mathison Turing 1912–1954', and the motto 'Founder of Computer Science' as it would appear if encoded by an Enigma machine: 'IEKYF ROMSI ADXUO KVKZC GUBJ'.

A plinth at the statue's feet says 'Father of computer science, mathematician, logician, wartime codebreaker, victim of prejudice'. There is also a Bertrand Russell quotation saying 'Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture.' The sculptor buried his old Amstrad computer, which was an early popular home computer, under the plinth, as a tribute to "the godfather of all modern computers".

In 1999, ''Time Magazine'' named Turing as one of the 100 Most Important People of the 20th Century for his role in the creation of the modern computer, and stated: "The fact remains that everyone who taps at a keyboard, opening a spreadsheet or a word-processing program, is working on an incarnation of a Turing machine." Turing is featured in the 1999 Neal Stephenson novel "Cryptonomicon."

In 2002, Turing was ranked twenty-first on the ''BBC'' nationwide poll of the 100 Greatest Britons.

The logo of Apple computer is often erroneously referred to as a tribute to Alan Turing, with the bite mark a reference to his method of suicide. Both the designer of the logo and the company deny that there is any homage to Turing in the design of the logo.

In 2010, actor/playwright Jade Esteban Estrada portrayed Turing in the solo musical, "ICONS: The Lesbian and Gay History of the World, Vol. 4."

Turing is mentioned several times in the DLC for the 2K Games "Bioshock 2" He is mentioned several times by the voice actor who portrays C.M. Porter. There is also a telegram in Porter's office requesting he come to London and work with Turing. He is also mentioned in Assassin's Creed: Brotherhood, when an employee of Abstergo Industries orders him to be killed and commands the would-be executioner to "make it look biblical".

In February 2011, Turing's papers from the Second World War were bought for the nation with an 11th-hour bid by the National Heritage Memorial Fund, allowing them to stay at Bletchley Park.

Government apology

In August 2009, John Graham-Cumming started a petition urging the British Government to posthumously apologise to Alan Turing for prosecuting him as a homosexual. The petition received thousands of signatures. Prime Minister Gordon Brown acknowledged the petition, releasing a statement on 10 September 2009 apologising and describing Turing's treatment as "appalling":

Thousands of people have come together to demand justice for Alan Turing and recognition of the appalling way he was treated. While Turing was dealt with under the law of the time and we can't put the clock back, his treatment was of course utterly unfair and I am pleased to have the chance to say how deeply sorry I and we all are for what happened to him ... So on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better.

Tributes by universities

A celebration of Turing's life and achievements arranged by the British Logic Colloquium and the British Society for the History of Mathematics was held on 5 June 2004.

The University of Surrey has a statue of Turing on their main piazza.

Istanbul Bilgi University organises an annual conference on the theory of computation called "Turing Days". The University of Texas at Austin has an honours computer science programme named the Turing Scholars. In the early 1960s Stanford University named the sole lecture room of the Polya Hall Mathematics building "Alan Turing Auditorium". One of the amphitheatres of the Computer Science department (LIFL) at the University of Lille in northern France is named in honour of Alan M. Turing (the other amphitheatre is named after Kurt Gödel).

The Department of Computer Science at Pontifical Catholic University of Chile, the Polytechnic University of Puerto Rico, Los Andes University in Bogotá, Colombia, King's College, Cambridge and Bangor University in Wales have computer laboratories named after Turing.

The University of Manchester, The Open University, Oxford Brookes University and Aarhus University (in Århus, Denmark) all have buildings named after Turing.

Alan Turing Road in the Surrey Research Park is named for Alan Turing.

Carnegie Mellon University has a granite bench, situated in The Hornbostel Mall, with the name "A. M. Turing" carved across the top, "Read" down the left leg, and "Write" down the other.

The École Internationale des Sciences du Traitement de l'Information has named its recently acquired third building "Turing".

The University of Ghent has one of its main computer rooms (in a building used mostly by mathematicians and computing scientists) named for Alan Turing.

The University of Oregon has a bust of Turing on the side of the Deschutes Hall, the computer science building.

Centenary commemoration

180px|border|rightTo mark the 100th anniversary of Turing's birth, the Turing Centenary Advisory Committee (TCAC) is coordinating the Alan Turing Year, a year-long programme of events around the world honouring Turing's life and achievements. The TCAC working with The University of Manchester faculty members and a broad spectrum of people from Cambridge University and Bletchley Park, is chaired by S. Barry Cooper, with Alan Turing's nephew Sir John Dermot Turing acting as TCAC Honorary President.

Events are scheduled in many countries around the world including the USA, Brazil, China, Czech Republic, the Philippines, New Zealand, Israel, Spain, Norway, Italy, Portugal and Germany. The keystone events will be a three-day conference in Manchester, UK in June examining Turing's mathematical and code-breaking achievements, and a Turing Centenary Conference in Cambridge organised by King's College, Cambridge and the association Computability in Europe.

See also

Good–Turing frequency estimation

Turing degree

Turing machine examples

Turing switch

Unorganized machine

Notes

References

Petzold, Charles (2008). "The Annotated Turing: A Guided Tour through Alan Turing's Historic Paper on Computability and the Turing Machine". Indianapolis: Wiley Publishing. ISBN 978-0-470-22905-7

Smith, Roger (1997). ''Fontana History of the Human Sciences''. London: Fontana.

Weizenbaum, Joseph (1976). ''Computer Power and Human Reason''. London: W.H. Freeman. ISBN 0-7167-0463-3

Turing's mother, who survived him by many years, wrote this 157-page biography of her son, glorifying his life. It was published in 1959, and so could not cover his war work. Scarcely 300 copies were sold (Sara Turing to Lyn Newman, 1967, Library of St John's College, Cambridge). The six-page foreword by Lyn Irvine includes reminiscences and is more frequently quoted. This 1986 Hugh Whitemore play tells the story of Turing's life and death. In the original West End and Broadway runs, Derek Jacobi played Turing and he recreated the role in a 1997 television film based on the play made jointly by the BBC and WGBH, Boston. The play is published by Amber Lane Press, Oxford, ASIN: B000B7TM0Q

Williams, Michael R. (1985) ''A History of Computing Technology'', Englewood Cliffs, New Jersey: Prentice-Hall, ISBN 0-8186-7739-2

Further reading

Gleick, James, ''The Information: A History, A Theory, A Flood'', New York: Pantheon, 2011, ISBN 9780375423727

Leavitt, David, ''The Man Who Knew Too Much: Alan Turing and the Invention of the Computer'', W. W. Norton, 2006

External links

Alan Turing RKBExplorer

Alan Turing Year

CiE 2012: Turing Centenary Conference

Visual Turing

Turing Machine calculators at Wolfram Alpha

Alan Turing site maintained by Andrew Hodges including a short biography

AlanTuring.net – Turing Archive for the History of Computing by Jack Copeland

The Turing Archive – contains scans of some unpublished documents and material from the Kings College, Cambridge archive

Papers

An extensive list of Turing's papers, reports and lectures, plus translated versions and collections BibNetWiki

Oral history interview with Donald W. Davies, Charles Babbage Institute, University of Minnesota; Davies describes computer projects at the U.K. National Physical Laboratory, from the 1947 design work of Alan Turing to the development of the two ACE computers

Oral history interview with Nicholas C. Metropolis, Charles Babbage Institute, University of Minnesota. Metropolis was the first director of computing services at Los Alamos National Laboratory; topics include the relationship between Alan Turing and John von Neumann

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