Alan Turing

How to use this course

Alan Turing worked for barely twenty years, and in that time he did four kinds of work. He gave the definition of computability that all of computer science rests on. He led the attack on German naval Enigma and created statistical methods that were secret for decades. He designed an electronic stored-program computer and wrote some of its first programs. In his last years he founded the mathematical theory of biological pattern formation. This course covers each, with the life around them, including his prosecution in 1952 and his death in 1954.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. Lessons 04 to 06 follow his logic papers of 1936 and 1939. Lessons 07 and 08 treat the codebreaking, using what has been declassified. Lessons 09 and 10 cover his computing and his biology.

Timeline

London and Sherborne, 1912–1931

Alan Mathison Turing was born on 23 June 1912 in Maida Vale, London, while his father was on leave from the Indian Civil Service. His parents, Julius Mathison Turing and Ethel Sara Stoney, had married in Dublin in 1907. To keep the boys in England while the parents were in India, Alan and his elder brother John were boarded with a family in St Leonards-on-Sea. From 1922 Alan attended Hazelhurst preparatory school in Sussex.

In 1926 he entered Sherborne School in Dorset. The first day of term fell during the General Strike, with no trains running, and the 13-year-old cycled about 60 miles to get there. Sherborne valued the classics. Turing’s mathematics and science were eccentric and self-directed, and his masters complained about his untidy work, but he won nearly every mathematics prize the school offered. He read Einstein’s popular account of relativity on his own and worked out calculus ahead of the syllabus.

Christopher Morcom

In 1928 Turing formed an intense friendship with Christopher Morcom, a gifted older boy who shared his passion for science and astronomy. Morcom died on 13 February 1930 of bovine tuberculosis. The loss shook Turing deeply. His letters to Morcom’s mother show him turning over questions about mind, matter and whether a mind could survive the body, questions that reappear transformed in his later writing on machines and thought.

He won a scholarship to King’s College, Cambridge in 1930 and went up in 1931.

King’s College, 1931–1936

Turing read mathematics at King’s from 1931 to 1934 and graduated with first-class honours. King’s suited him: tolerant, liberal and intellectually serious. He rowed, ran, and read John von Neumann’s book on the mathematical foundations of quantum mechanics.

His fellowship dissertation, On the Gaussian Error Function, proved a form of the central limit theorem: suitably normalized sums of independent random variables tend to the normal distribution. He discovered afterward that Jarl Lindeberg had proved it in 1922. The King’s electors judged the work on its merits, and in 1935, aged 22, Turing became a fellow of King’s.

The Entscheidungsproblem

In 1935 he attended Max Newman’s Part III course on the foundations of mathematics. Newman explained Kurt Gödel’s incompleteness theorems and the open Entscheidungsproblem of David Hilbert and Wilhelm Ackermann: is there a definite method that decides, for any formula of first-order logic, whether it is provable? To answer “no”, one needs a precise definition of “definite method”. Turing found one by analysing what a human computer does with pencil and paper, and from it built an abstract machine.

He completed “On Computable Numbers, with an Application to the Entscheidungsproblem” in spring 1936 and submitted it in late May. Shortly before, Alonzo Church at Princeton had published a negative answer using his λ-calculus. Newman supported publication anyway, since Turing’s approach was entirely different and more convincing as an analysis of computation. Turing added an appendix proving that his computable functions are exactly Church’s λ-definable ones. The paper appeared in the Proceedings of the London Mathematical Society (series 2, vol. 42), with a short correction in 1937. In 1936 he also won the Smith’s Prize.

Princeton and the eve of war, 1936–1939

In September 1936 Turing went to Princeton to work with Church. He stayed until July 1938, the second year supported by a Procter Fellowship. In Princeton he wrote his doctoral thesis, Systems of Logic Based on Ordinals, introducing oracle machines and relative computability (Lesson 06). He received the PhD in June 1938. Von Neumann offered him a post as his assistant at the Institute for Advanced Study; Turing declined and returned to Cambridge.

He also built things. In Princeton he designed and partly built an electric multiplier from relays, interested in its use for ciphers. Back in Cambridge he obtained a small Royal Society grant in 1939 for a machine of gears to compute the Riemann zeta function, hoping to find zeros off the critical line. He cut many of the gears before war stopped the project. He attended Ludwig Wittgenstein’s 1939 lectures on the foundations of mathematics and argued with him about contradiction and the use of mathematics in building bridges.

From September 1938 he worked part-time for the Government Code and Cypher School (GC&CS). When Britain declared war, he reported to its wartime home at Bletchley Park, on 4 September 1939.

Machines and computable numbers

For the general reader. What can be calculated by following rules? Turing imagined a clerk with an unlimited strip of squared paper, who looks at one square at a time, remembers only a fixed finite amount, and at each step writes a symbol, moves one square, and changes state of mind. He argued that any mechanical calculation can be broken down into such steps. The resulting “Turing machine” is still the standard definition of an algorithm.

Definitions

Turing’s machines of 1936 ran forever, printing binary digits on alternate squares. A machine was circle-free if it printed infinitely many digits, and a real number in \([0,1]\) was computable if some circle-free machine printed its binary expansion. Today one usually speaks of partial functions \(\mathbb N\to\mathbb N\) computed by halting machines; the two formulations are equivalent.

Why this definition

Section 9 of the paper argues that the definition captures everything a human computer can do: the human sees a bounded number of symbols at once, has finitely many states of mind, and changes only a bounded amount at each step. Church proposed λ-definability for the same role. The claim that either formalism captures exactly the effectively calculable functions is now called the Church–Turing thesis. It is a claim about an informal notion, so it cannot be proved; the equivalence of all proposed formalizations is strong evidence for it.

The universal machine

Proof sketch. Write each transition of \(M\) as a string of symbols; the list of transitions is the standard description. \(U\) keeps on its tape a copy of \(M\)’s tape together with a mark for \(M\)’s state and head position. To simulate one step, \(U\) shuttles back and forth: it reads the marked state and symbol, searches the description for the matching transition, and updates the copy. Each of these is a finite search, programmable with finitely many states. ∎

The universal machine is the stored-program idea in its purest form: one fixed device that does any computation when given its description as data.

Machines are finite strings over a finite alphabet, so there are countably many, and countable sets have measure zero. Turing also noted in his 1937 correction that defining computable reals through their binary digits causes trouble with arithmetic, and proposed a more robust representation (Exercise 4.3).

Undecidability

For the general reader. Could a program inspect any other program and tell whether it will run forever? Turing proved no such program can exist. The proof turns the hypothetical checker against itself, like the liar paradox. From this he concluded that Hilbert’s dream of a mechanical procedure to settle every question of logic is impossible.

Turing’s own formulations were that no machine decides whether an arbitrary machine is circle-free, and no machine decides whether an arbitrary machine ever prints the symbol 0. The cleaner “halting problem” formulation was introduced later, in the 1950s, and is what we prove.

Proof. Suppose \(H\) decides \(K\). Build \(D\): on input \(\ulcorner M\urcorner\), run \(H\); if it outputs 1, loop forever; if 0, halt. Then \(D\) halts on \(\ulcorner D\urcorner\) iff \(H\) outputs 0 on \(\ulcorner D\urcorner\) iff \(D\) does not halt on \(\ulcorner D\urcorner\), a contradiction. To enumerate \(K\), run all \(M\) on \(\ulcorner M\urcorner\) in parallel (dovetailing) and list those that halt. ∎

Proof sketch (Turing). For each machine \(M\), Turing wrote down a first-order sentence \(\mathrm{Un}(M)\) using predicates for “at moment \(n\) the machine is in configuration such and such” and a successor relation, with axioms describing the transitions. He showed that \(\mathrm{Un}(M)\) is provable if and only if \(M\) eventually prints 0. A decision procedure for provability would therefore decide the printing problem, which is impossible. ∎

Oracles and ordinal logics

For the general reader. Gödel showed that any consistent formal system strong enough for arithmetic leaves some true statements unproved, for example its own consistency. A natural repair is to add the missing statement as a new axiom, and repeat, transfinitely. Turing’s Princeton thesis studied how far this goes. Along the way he imagined a machine allowed to consult an “oracle” for an uncomputable answer, an idea that became the basis of relative computability.

Oracle machines

An o-machine with oracle \(A\subseteq\mathbb N\) is a Turing machine with a special query state: it writes \(n\) on the tape, enters the query state, and in one step receives the answer to “is \(n\in A\)?”. A set \(B\) is Turing reducible to \(A\), written \(B\le_T A\), if some o-machine with oracle \(A\) decides \(B\). Sets with \(A\le_T B\le_T A\) have the same Turing degree.

The proof of Lesson 05 goes through verbatim with every machine given access to \(A\). So there is no “most uncomputable” set: \(\emptyset<_T\emptyset'<_T\emptyset''<_T\cdots\). Emil Post took up Turing’s definition in 1944 and founded degree theory.

Ordinal logics

Start from a theory \(T_0\) such as Peano arithmetic. Set \(T_{\alpha+1}=T_\alpha+\mathrm{Con}(T_\alpha)\), and at a limit take unions. To make this effective, ordinals must be given by notations, and Turing used Church and Kleene’s system \(\mathcal O\) of notations for constructive ordinals. Different notations for the same ordinal may give different theories.

The catch is that finding the right notation is as hard as deciding \(\varphi\): the notation encodes the truth of \(\varphi\). Solomon Feferman extended the analysis in 1962, showing that progressions based on reflection principles prove every true arithmetic sentence along suitable paths through \(\mathcal O\). Turing’s thesis is an early and careful study of what transfinite iteration can and cannot buy.

Enigma and the bombe

For the general reader. The German armed forces enciphered radio traffic with the Enigma, a typewriter whose rotating wheels scrambled each letter differently. The number of possible settings was astronomically large. Polish mathematicians broke it before the war. When the Germans made it harder, Turing designed an electromechanical machine, the bombe, that used guessed words of plaintext to eliminate almost all settings by pure logic.

The machine as permutations

Let \(P\) be the plugboard, an involution of the 26 letters swapping some pairs. At position \(i\), the current through the rotors, reflector and back is a permutation \(S_i\), also an involution and with no fixed points, because the reflector \(R\) pairs all letters and \(S_i=\rho_i^{-1}R\rho_i\). The encipherment at position \(i\) is

So the same setting deciphers, and no letter ever enciphers to itself. Marian Rejewski had used the cycle structure of such products to break the Enigma in 1932, and the Polish Cipher Bureau built its own bomba in 1938. The Poles gave their methods to the British and French in July 1939.

Cribs and menus

A crib is a guessed piece of plaintext, such as a weather report’s standard opening. Because no letter enciphers to itself, a crib can only sit at offsets where it never agrees letter-for-letter with the ciphertext. Once placed, each pair \((p_i,c_i)\) gives \(c_i=P S_i P\, p_i\), that is

Draw a graph, the menu, with a vertex for each letter and an edge labelled \(i\) joining \(p_i\) and \(c_i\).

Proof. \(P(B)=S_i P(A)\), \(P(C)=S_j P(B)\), \(P(A)=S_k P(C)\). Compose. ∎

A random permutation has on average one fixed point, so each loop cuts the candidates for \(P(A)\) to about one letter. Turing’s bombe tested a rotor position by hypothesizing \(P(A)=x\) and propagating every consequence along the menu electrically. If the hypothesis forced some letter to be plugged to two different letters, it was false. The machine stepped through all \(26^3=17{,}576\) rotor positions for a given wheel order, and stopped only where some hypothesis survived. Gordon Welchman’s diagonal board of 1940 added the symmetry \(P(x)=y\iff P(y)=x\), greatly increasing the deductions from a short menu. The first bombe, “Victory”, was installed on 18 March 1940; the second, “Agnus”, with the diagonal board, on 8 August 1940. Harold Keen of the British Tabulating Machine Company in Letchworth engineered them.

Weight of evidence

For the general reader. Codebreaking is a stream of weak clues. Turing measured each clue by how much more likely it is if a hypothesis is true than if it is false, took logarithms so that clues could be added, and called the unit a “ban”. Clerks at Bletchley added up scores in decibans to decide which hypotheses deserved scarce machine time. This was Bayesian statistics, done at industrial scale, years before the subject was respectable.

The name comes from Banbury, the town where the long punched sheets used in Banburismus were printed. Turing’s colleague I. J. Good later wrote that a deciban is about the smallest change in odds that a person can perceive.

Banburismus

Naval Enigma messages sent on the same day began from related positions. If two messages overlap “in depth” (enciphered at the same rotor positions), then where their plaintexts agree, their ciphertexts agree too. German naval plaintext has a letter-repeat rate of about \(1/17\), against \(1/26\) for unrelated text. Analysts slid one punched sheet over another on a light box, counted repeats at each offset, and scored them.

Each repeat is worth about +1.8 decibans and each non-repeat about −0.09. In practice Turing and his colleagues refined the scoring for repeated bigrams and longer fragments, and used the results to infer the right-hand and middle wheels, which reduced the number of bombe runs needed. Hut 8 used Banburismus from 1940 until mid-1943, when there were enough bombes to do without it.

The Good–Turing estimator

In a sample of \(N\) observations, let \(n_r\) be the number of distinct species seen exactly \(r\) times. Turing suggested, and Good published in 1953, that the total probability of all species not yet seen is estimated by \(n_1/N\), and more generally that a species seen \(r\) times has probability about \((r+1)\,n_{r+1}/(n_r N)\). The estimator is used today in language modelling and ecology.

The ACE, rounding errors, zeta zeros

For the general reader. After the war Turing set out to build a real universal machine. His design for the Automatic Computing Engine was ambitious and detailed, and while it was being built he asked a new question: when a machine does millions of arithmetic steps, each slightly rounded, how wrong is the answer? He also turned the first computers on a question he had pursued since 1939, the zeros of the Riemann zeta function.

The design

Turing joined the National Physical Laboratory on 1 October 1945. His report “Proposed Electronic Calculator”, written that winter and presented to the NPL Executive Committee in February 1946, gave a complete design: a stored-program machine with mercury delay-line memory, accessed at about 1 MHz, a minimal instruction set, and programs built from subroutines, including tables of instructions that would let the machine handle abbreviated codes. The NPL was slow to build it. The reduced Pilot ACE ran its first program on 10 May 1950, and its successors, the English Electric DEUCE and the Bendix G-15, were sold commercially.

Rounding-off errors (1948)

In “Rounding-off errors in matrix processes” Turing analysed Gaussian elimination as the factorization \(A=LU\) into lower and upper triangular matrices, and introduced the term condition number for the quantity that controls how errors are magnified.

Proof. \(\delta x=A^{-1}\delta b\), so \(\|\delta x\|\le\|A^{-1}\|\|\delta b\|\), and \(\|b\|\le\|A\|\|x\|\). Divide. ∎ Turing’s own “N-condition” and “M-condition” numbers used the Frobenius norm and the largest entry, with normalizing factors; the modern \(\kappa\) is the same idea.

Zeros of the zeta function

Let \(N(T)\) be the number of zeros \(\rho\) of \(\zeta(s)\) with \(0<\operatorname{Im}\rho\le T\). The Riemann–von Mangoldt formula says

Zeros on the critical line can be located by sign changes of a real function \(Z(t)\). To verify the Riemann hypothesis up to height \(T\), one must also show that no zeros were missed, which needs the exact value of \(N(T)\). Turing’s method, in his 1953 paper, avoids computing \(\arg\zeta\) directly. He proved an explicit bound of the form \(\big|\int_{t_1}^{t_2}S(t)\,dt\big|\le a+b\log t_2\), so that \(S\) has small average. If the count of sign changes found in a range were too low, \(S\) would be pushed away from zero on average, contradicting the bound. In June 1950 he ran the computation on the Manchester computer for \(2\pi\cdot63^2<t<2\pi\cdot64^2\) and found no counterexample.

Morphogenesis

For the general reader. How does a ball of nearly identical cells develop spots, stripes, or a regular arrangement of tentacles? In 1952 Turing showed that two chemicals reacting and spreading at different speeds can turn a uniform state into a regular pattern, all by themselves. Diffusion, which normally smooths things out, here creates structure. Such “Turing patterns” have since been found in chemical reactors, animal skins and developing limbs.

“The Chemical Basis of Morphogenesis” appeared in the Philosophical Transactions of the Royal Society B 237 (1952). Turing called the chemicals morphogens and worked with a ring of cells and with continuous tissue. Consider

with a spatially uniform steady state \((u_0,v_0)\), \(f=g=0\) there. Write \(J=\begin{pmatrix}f_u&f_v\\ g_u&g_v\end{pmatrix}\) at that state and \(D=\operatorname{diag}(D_u,D_v)\). A perturbation \(e^{\lambda t}e^{i\mathbf k\cdot\mathbf x}\) grows according to the eigenvalues \(\lambda\) of \(J-k^2D\).

Proof. \(\operatorname{tr}(J-k^2D)=\operatorname{tr}J-k^2(D_u+D_v)<0\) for all \(k\), so instability needs \(\det(J-k^2D)<0\) for some \(k^2>0\). This determinant is \(h(q)=D_uD_vq^2-(D_vf_u+D_ug_v)q+\det J\) with \(q=k^2\). Since \(h(0)>0\), \(h\) takes negative values on \(q>0\) iff its minimum, at \(q=(D_vf_u+D_ug_v)/(2D_uD_v)\), lies at positive \(q\) and is negative, which gives the two conditions. ∎

Since \(f_u+g_v<0\) but \(D_vf_u+D_ug_v>0\), one of \(f_u,g_v\) is positive (say \(f_u>0\): \(u\) is self-activating), and then \(D_v\) must exceed \(D_u\): the inhibitor must spread faster than the activator. This is “local activation, long-range inhibition”, the principle Alfred Gierer and Hans Meinhardt made central in 1972. The first unambiguous chemical Turing patterns were produced in 1990 in the chlorite–iodide–malonic acid reaction by Castets, Dulos, Boissonade and De Kepper.

Bletchley, the NPL and Manchester, 1939–1951

At Bletchley Park Turing took on the problem most thought hopeless, the naval Enigma, whose indicator system was more secure than the army and air force versions. He worked out the indicator system by the end of 1939, specified the bombe, and devised Banburismus. Captured key material, notably from German trawlers in 1940 and 1941, combined with these methods to let Hut 8, which Turing led, read naval Enigma from mid-1941. In October 1941 Turing, Gordon Welchman, Hugh Alexander and Stuart Milner-Barry wrote directly to Winston Churchill about shortages of staff; Churchill ordered that they be given what they needed. Turing’s colleagues remembered him as kind, absent-minded and entirely without pretence. In 1941 he became engaged to Joan Clarke, a fellow cryptanalyst in Hut 8; he told her he was homosexual, and they ended the engagement, remaining friends.

In July 1942 he devised “Turingery”, a hand method for recovering wheel patterns of the German Lorenz teleprinter cipher. From November 1942 to March 1943 he was in the United States, advising on naval Enigma and the US Navy’s bombes, and on speech encipherment at Bell Laboratories, where he met Claude Shannon. Back in England he moved to Hanslope Park and designed and built, with the engineer Donald Bayley, a portable speech-encipherment device called Delilah. He was appointed OBE for his wartime work; its nature stayed secret until the 1970s.

Computers

At the NPL from October 1945 he designed the ACE (Lesson 09). Frustrated by slow progress, he spent 1947–48 on sabbatical in Cambridge. He was a serious runner: he came fifth in the 1947 Amateur Athletic Association marathon, and an injury ended his hopes of the 1948 Olympic trials. In 1948 he moved to Manchester as a reader, where Frederic Williams and Tom Kilburn had built the first working stored-program computer, and he became deputy director of the Computing Machine Laboratory. He wrote the programming manual for the Ferranti Mark 1.

In 1950 Mind published “Computing Machinery and Intelligence”. It opens: “I propose to consider the question, ‘Can machines think?’” Turing replaced the question by the imitation game, now the Turing test, and answered objections one by one. He was elected a Fellow of the Royal Society in 1951.

Prosecution, death and legacy, 1952–present

Prosecution

In January 1952 Turing’s house in Wilmslow was burgled, and he reported it to the police. In the investigation he acknowledged a sexual relationship with Arnold Murray, a young man he had met in Manchester, whose acquaintance had committed the burglary. Sexual acts between men were then criminal in Britain. Turing and Murray were both charged with gross indecency under section 11 of the Criminal Law Amendment Act 1885. At his trial on 31 March 1952 Turing pleaded guilty. He was sentenced to twelve months’ probation on condition that he undergo hormone treatment with stilboestrol, a synthetic oestrogen, which he accepted in preference to prison. His security clearance was withdrawn, ending his consultancy with GCHQ. He kept his post at Manchester and continued to work on morphogenesis and on the zeta function.

Death

On 8 June 1954 his housekeeper found him dead at his home in Wilmslow. He had died on 7 June, aged 41. The post-mortem found cyanide poisoning. A half-eaten apple lay beside the bed; it was never tested for cyanide. The inquest returned a verdict of suicide. His mother believed his death was an accident, caused by careless handling of chemicals in the electrolysis experiments he ran at home. In 2012 the philosopher Jack Copeland argued that the evidence presented at the inquest did not establish suicide and that accidental inhalation of cyanide fumes was possible. Andrew Hodges, author of the standard biography, treats suicide as the most likely explanation. The evidence does not settle the question. He was cremated at Woking on 12 June 1954.

Recognition

• Apology and pardon: on 10 September 2009 Prime Minister Gordon Brown issued a public apology for the way Turing was treated. On 24 December 2013 he received a posthumous royal pardon. The Policing and Crime Act 2017, known as the Alan Turing law, extended pardons to men convicted under the historical laws against homosexual acts.
• Honours: the Association for Computing Machinery has given the A. M. Turing Award, the field’s highest honour, since 1966. On 23 June 2021, his birthday, the Bank of England issued a £50 note bearing his portrait. The Alan Turing Institute, the UK national institute for data science and artificial intelligence, bears his name.
• Mathematics: computability theory and degree theory, the theory of algorithms and complexity, Bayesian sequential analysis, numerical linear algebra, and mathematical biology all draw on his papers. His student Robin Gandy, who received his PhD under Turing in 1953, became a leading logician and editor of his collected works.

Connections

People linked to Alan Turing across this collection. A name without a link has no page here yet. See Alan Turing in the network of all 337 lives →

Teachers and advisors

• Alonzo Church 1936-1938

Students

• Robin Gandy 1953

Influenced by

• Ada Lovelace
• Bernhard Riemann
• Charles Babbage
• David Hilbert
• Kurt Gödel
• Marian Rejewski

Influenced

• Emil Post

Colleagues and collaborators

• Claude Shannon 1943
• John von Neumann 1936-1938
• Stephen Cole Kleene
• Gordon Welchman 1939-1945
• Hugh Alexander 1940-1945
• I. J. Good
• Ludwig Wittgenstein 1939
• Max Newman 1935-1954

Friends

• Joan Clarke 1941

Also mentioned in this life

• Thomas Bayes
• Albert Einstein
• Georg Frobenius
• Henri Lebesgue
• Carl Ludwig Siegel

Fields

• mathematical logic 10 lives
• cryptanalysis 3 lives
• numerical analysis 6 lives
• mathematical biology 3 lives

On CoursesHub

• Gödel & Turing · History of Mathematics
• Numerical Methods & Computational Physics

Sources and further reading

Biography

1. A. Hodges, Alan Turing: The Enigma, Burnett Books, 1983; centenary edition, Princeton University Press, 2012. Companion site: turing.org.uk.
2. J. J. O’Connor and E. F. Robertson, Alan Turing, MacTutor History of Mathematics.
3. Alan Turing, Wikipedia (with extensive references on the prosecution, inquest and pardon).
4. M. H. A. Newman, Alan Mathison Turing 1912–1954, Biographical Memoirs of Fellows of the Royal Society 1 (1955).
5. B. J. Copeland, Turing: Pioneer of the Information Age, Oxford University Press, 2012.
6. Mathematics Genealogy Project, Alan Mathison Turing.

Mathematics

1. A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proc. London Math. Soc. (2) 42 (1937) 230–265; correction 43 (1938) 544–546.
2. A. M. Turing, Systems of logic based on ordinals, Proc. London Math. Soc. (2) 45 (1939) 161–228.
3. A. M. Turing, Rounding-off errors in matrix processes, Quart. J. Mech. Appl. Math. 1 (1948) 287–308.
4. A. M. Turing, Computing machinery and intelligence, Mind 59 (1950) 433–460.
5. A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. R. Soc. Lond. B 237 (1952) 37–72.
6. A. M. Turing, Some calculations of the Riemann zeta-function, Proc. London Math. Soc. (3) 3 (1953) 99–117.
7. B. J. Copeland (ed.), The Essential Turing, Oxford University Press, 2004.
8. S. Feferman, Turing in the land of O(z), in R. Herken (ed.), The Universal Turing Machine: A Half-Century Survey, 1988.
9. I. J. Good, Studies in the history of probability and statistics XXXVII: A. M. Turing’s statistical work in World War II, Biometrika 66 (1979).
10. Bombe and Banburismus, Wikipedia.
11. M. Rubinstein, Turing and the primes.
12. A. M. Odlyzko, Turing and the Riemann zeta function.
13. J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, 3rd ed. 2003, ch. 2.
14. V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett. 64 (1990) 2953.