Solvable and Unsolvable Problems (1954)

In Chapter 1 Turing proves the existence of mathematical problems that cannot be solved by the universal Turing machine. There he also advances the thesis, now called the Church–Turing thesis, that any systematic method for solving mathematical problems can be carried out by the universal Turing machine. Combining these two propositions yields the result that there are mathematical problems which cannot be solved by any systematic method—cannot, in other words, be solved by any algorithm. In ‘Solvable and Unsolvable Problems’ Turing sets out to explain this result to a lay audience. The article first appeared in Science News, a popular science journal of the time. Starting from concrete examples of problems that do admit of algorithmic solution, Turing works his way towards an example of a problem that is not solvable by any systematic method. Loosely put, this is the problem of sorting puzzles into those that will ‘come out’ and those that will not. Turing gives an elegant argument showing that a sharpened form of this problem is not solvable by means of a systematic method (pp. 591–2). The sharpened form of the problem involves what Turing calls ‘the substitution type of puzzle’. An typical example of a substitution puzzle is this. Starting with the word BOB, is it possible to produce BOOOB by replacing selected occurrences of the pair OB by BOOB and selected occurences of the triple BOB by O? The answer is yes: . . . BOB → BBOOB → BBOBOOB → BOOOB . . .Turing suggests that any puzzle can be re-expressed as a substitution puzzle. Some row of letters can always be used to represent the ‘starting position’ envisaged in a particular puzzle, e.g. in the case of a chess problem, the pieces on the board and their positions. Desired outcomes, for example board positions that count as wins, can be described by further rows of letters, and the rules of the puzzle, whatever they are, are to be represented in terms of permissible substitutions of groups of letters for other groups of letters.

The Chemical Basis of Morphogenesis (1952)

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogenous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading. In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge. The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points.

On Computable Numbers: Corrections and Critiques

As is not uncommon in work of such complexity, there are a number of mistakes in ‘On Computable Numbers’. Turing corrected some of these in his short note 2.1, published in the Proceedings of the London Mathematical Society a few months after the original paper had appeared. The mathematician Emil L. Post’s critique of ‘On Computable Numbers’ was published in 1947 and formed part of Post’s paper ‘Recursive Unsolvability of a Problem of Thue’. Post is one of the major figures in the development of mathematical logic in the twentieth century, although his work did not gainwide recognition until after his death. (Born in 1897, Post died in the same year as Turing.) By 1936 Post had arrived independently at an analysis of computability substantially similar to Turing’s. Post’s ‘problem solver’ operated in a ‘symbol space’ consisting of ‘a two way infinite sequence of spaces or boxes’. A box admitted ‘of but two possible conditions, i.e., being empty or unmarked, and having a single mark in it, say a vertical stroke’. The problem solver worked in accordance with ‘a fixed unalterable set of directions’ and could perform the following ‘primitive acts’: determine whether the box at present occupied is marked or not; erase any mark in the box that is at present occupied; mark the box that is at present occupied if it is unmarked; move to the box to the right of the present position; move to the box to the left of the present position. Later, Post considerably extended certain of the ideas in Turing’s ‘Systems of Logic Based on Ordinals’, developing the important field now called degree theory. In his draft letter to Church, Turing responded to Post’s remarks concerning ‘Turing convention-machines’. It is doubtful whether Turing ever sent the letter. The approximate time of writing can be inferred from Turing’s opening remarks: Kleene’s review appeared in the issue of the Journal of Symbolic Logic dated September 1947 (12: 90–1) and Turing’s ‘Practical Forms of Type Theory’ appeared in the same journal in June 1948.

Systems of Logic Based on Ordinals (1938)

On 23 September 1936 Turing left England on a vessel bound for New York. His destination was Princeton University, where the Mathematics Department and the Institute for Advanced Study combined to make Princeton a leading centre for mathematics. Turing had applied unsuccessfully for a Visiting Fellowship to Princeton in the spring of 1935. When a year later he learned of Church’s work at Princeton on the Entscheidungsproblem, which paralleled his own (see ‘Computable Numbers: A Guide’), Turing ‘decided quite definitely’ to go there. He planned to stay for a year. In mid-1937 the offer of a Visiting Fellowship for the next academic year persuaded him to prolong his visit, and he embarked on a Ph.D. thesis. Already advanced in his academic career, Turing was an unusual graduate student (in the autumn of 1937, he himself was appointed by Cambridge University to examine a Ph.D. thesis). By October 1937 Turing was looking forward to his thesis being ‘done by about Christmas’. It took just a little longer: ‘Systems of Logic Based on Ordinals’ was accepted on 7 May 1938 and the degree was awarded a few weeks later. The following year the thesis was published in the Proceedings of the London Mathematical Society. ‘Systems of Logic Based on Ordinals’ was written under Church’s supervision. His relationship to Turing—whose formalization of the concept of an effective procedure and work on the Entscheidungsproblem was ‘possibly more convincing’ than Church’s own—was hardly the usual one of doctoral supervisor to graduate student. In an interview given in 1984, Church remarked that Turing ‘had the reputation of being a loner’ and said: ‘I forgot about him when I was speaking about my own graduate students—truth is, he was not really mine.’ Nevertheless Turing and Church had ‘a lot of contact’ and Church ‘discussed his dissertation with him rather carefully’. Church’s influence was not all for the good, however. In May 1938 Turing wrote: My Ph.D. thesis has been delayed a good deal more than I had expected. Church made a number of suggestions which resulted in the thesis being expanded to an appalling length. I hope the length of it won’t make it difficult to get it published.

Can Digital Computers Think? (1951)

The lecture ‘Can Digital Computers Think?’ was broadcast on BBC Radio on 15 May 1951, and was repeated on 3 July of that year. (Sara Turing relates that Turing did not listen to the Wrst broadcast but did ‘pluck up courage’ to listen to the repeat.) Turing’s was the second lecture in a series with the general title ‘Automatic Calculating Machines’. Other speakers in the series included Newman, D. R. Hartree, M. V. Wilkes, and F. C. Williams. Turing’s principal aim in this lecture is to defend his view that ‘it is not altogether unreasonable to describe digital computers as brains’, and he argues for the proposition that ‘If any machine can appropriately be described as a brain, then any digital computer can be so described’. The lecture casts light upon Turing’s attitude towards talk of machines thinking. In Chapter 11 he says that in his view the question ‘Can machines think?’ is ‘too meaningless to deserve discussion’ (p. 449). However, in the present chapter he makes liberal use of such phrases as ‘programm[ing] a machine . . . to think’ and ‘the attempt to make a thinking machine’. In one passage, Turing says (p. 485): ‘our main problem [is] how to programme a machine to imitate a brain, or as we might say more briefly, if less accurately, to think.’ He shows the same willingness to discuss the question ‘Can machines think?’ in Chapter 14. Turing’s view is that a machine which imitates the intellectual behaviour of a human brain can itself appropriately be described as a brain or as thinking. In Chapter 14, Turing emphasizes that it is only the intellectual behaviour of the brain that need be considered (pp. 494–5): ‘To take an extreme case, we are not interested in the fact that the brain has the consistency of cold porridge. We don’t want to say ‘‘This machine’s quite hard, so it isn’t a brain, and so it can’t think.’’ ’ It is, of course, the ability of the machine to imitate the intellectual behaviour of a human brain that is examined in the Turing test (Chapter 11).

Intelligent Machinery, A Heretical Theory (c.1951)

Turing gave the presentation ‘Intelligent Machinery, A Heretical Theory’ on a radio discussion programme called The ’51 Society. Named after the year in which the programme first went to air, The ’51 Society was produced by the BBC Home Service at their Manchester studio and ran for several years. A presentation by the week’s guest would be followed by a panel discussion. Regulars on the panel included Max Newman, Professor of Mathematics at Manchester, the philosopher Michael Polanyi, then Professor of Social Studies at Manchester, and the mathematician Peter Hilton, a younger member of Newman’s department at Manchester who had worked with Turing and Newman at Bletchley Park. Turing’s target in ‘Intelligent Machinery, A Heretical Theory’ is the claim that ‘You cannot make a machine to think for you’ (p. 472). A common theme in his writing is that if a machine is to be intelligent, then it will need to ‘learn by experience’ (probably with some pre-selection, by an external educator, of the experiences to which the machine will be subjected). The present article continues the discussion of machine learning begun in Chapters 10 and 11. Turing remarks that the ‘human analogy alone’ suggests that a process of education ‘would in practice be an essential to the production of a reasonably intelligent machine within a reasonably short space of time’ (p. 473). He emphasizes the point, also made in Chapter 11, that one might ‘start from a comparatively simple machine, and, by subjecting it to a suitable range of ‘‘experience’’ transform it into one which was more elaborate, and was able to deal with a far greater range of contingencies’ (p. 473). Turing goes on to give some indication of how learning might be accomplished, introducing the idea of a machine’s building up what he calls ‘indexes of experiences’ (p. 474). (This idea is not mentioned elsewhere in his writings.) An example of an index of experiences is a list (ordered in some way) of situations in which the machine has found itself, coupled with the action that was taken, and the outcome, good or bad. The situations are described in terms of features.

Computing Machinery and Intelligence (1950)

Together with ‘On Computable Numbers’, ‘Computing Machinery and Intelligence’ forms Turing’s best-known work. This elegant and sometimes amusing essay was originally published in 1950 in the leading philosophy journal Mind. Turing’s friend Robin Gandy (like Turing a mathematical logician) said that ‘Computing Machinery and Intelligence’. . . was intended not so much as a penetrating contribution to philosophy but as propaganda. Turing thought the time had come for philosophers and mathematicians and scientists to take seriously the fact that computers were not merely calculating engines but were capable of behaviour which must be accounted as intelligent; he sought to persuade people that this was so. He wrote this paper—unlike his mathematical papers—quickly and with enjoyment. I can remember him reading aloud to me some of the passages— always with a smile, sometimes with a giggle. The quality and originality of ‘Computing Machinery and Intelligence’ have earned it a place among the classics of philosophy of mind. ‘Computing Machinery and Intelligence’ contains Turing’s principal exposition of the famous ‘imitation game’ or Turing test. The test first appeared, in a restricted form, in the closing paragraphs of ‘Intelligent Machinery’ (Chapter 10). Chapters 13 and 14, dating from 1951 and 1952 respectively, contain further discussion and amplification; unpublished until 1999, this important additional material throws new light on how the Turing test is to be understood. The imitation game involves three participants: a computer, a human interrogator, and a human ‘foil’. The interrogator attempts to determine, by asking questions of the other two participants, which of them is the computer. All communication is via keyboard and screen, or an equivalent arrangement (Turing suggested a teleprinter link). The interrogator may ask questions as penetrating and wide-ranging as he or she likes, and the computer is permitted to do everything possible to force a wrong identification. (So the computer might answer ‘No’ in response to ‘Are you a computer?’ and might follow a request to multiply one large number by another with a long pause and a plausibly incorrect answer.) The foil must help the interrogator to make a correct identification.

History of Hut 8 to December 1941 (1945)

Patrick Mahon (A. P. Mahon) was born on 18 April 1921, the son of C. P. Mahon, Chief Cashier of the Bank of England from 1925 to 1930 and Comptroller from 1929 to 1932. From 1934 to 1939 he attended Marlborough College before going up to Clare College, Cambridge, in October 1939 to read Modern Languages. In July 1941, having achieved a First in both German and French in the Modern Languages Part II, he joined the Army, serving as a private (acting lancecorporal) in the Essex Regiment for several months before being sent to Bletchley. He joined Hut 8 in October 1941, and was its head from the autumn of 1944 until the end of the war. On his release from Bletchley in early 1946 he decided not to return to Cambridge to obtain his degree but instead joined the John Lewis Partnership group of department stores. John Spedan Lewis, founder of the company, was a friend of Hut 8 veteran Hugh Alexander, who effected the introduction. At John Lewis, where he spent his entire subsequent career, Mahon rapidly achieved promotion to director level, but his health deteriorated over a long period. He died on 13 April 1972. This chapter consists of approximately the first half of Mahon’s ‘The History of Hut Eight, 1939–1945’. Mahon’s typescript is dated June 1945 and was written at Hut 8. It remained secret until 1996, when a copy was released by the US government into the National Archives and Records Administration (NARA) in Washington, DC. Subsequently another copy was released by the British government into the Public Record Office at Kew. Mahon’s ‘History’ is published here for the first time. Mahon’s account is first-hand from October 1941. Mahon says, ‘for the early history I am indebted primarily to Turing, the first Head of Hut 8, and most of the early information is based on conversations I have had with him’.

Can Automatic Calculating Machines Be Said To Think? (1952)

This discussion between Turing, Newman, R. B. Braithwaite, and G. Jefferson was recorded by the BBC on 10 January 1952 and broadcast on BBC Radio on the 14th, and again on the 23rd, of that month. This is the earliest known recorded discussion of artificial intelligence. The anchor man of the discussion is Richard Braithwaite (1900–90). Braithwaite was at the time Sidgwick Lecturer in Moral Science at the University of Cambridge, where the following year he was appointed Knightsbridge Professor of Moral Philosophy. Like Turing, he was a Fellow of King’s College. Braithwaite’s main work lay in the philosophy of science and in decision and games theory (which he applied in moral philosophy). Geoffrey Jefferson (1886–1961) retired from the Chair of Neurosurgery at Manchester University in 1951. In his Lister Oration, delivered at the Royal College of Surgeons of England on 9 June 1949, he had declared: ‘When we hear it said that wireless valves think, we may despair of language.’ Turing gave a substantial discussion of Jefferson’s views in ‘Computing Machinery and Intelligence’ (pp. 451–2), rebutting the ‘argument from consciousness’ that he found in the Lister Oration. In the present chapter, Jefferson takes numerous pot shots at the notion of a machine thinking, which for the most part Turing and Newman are easily able to turn aside. Jefferson may have thought little of the idea of machine intelligence, but he held Turing in considerable regard, saying after Turing’s death that he ‘had real genius, it shone from him’. From the point of view of Turing scholarship, the most important parts of ‘Can Automatic Calculating Machines Be Said to Think’ are the passages containing Turing’s exposition of the imitation game or Turing test. The description of the test that Turing gave in ‘Computing Machinery and Intelligence’ is here modified in a number of significant ways. The lone interrogator of the original version is replaced by a ‘jury’ (p. 495). Each jury must judge ‘quite a number of times’ and ‘sometimes they really are dealing with a man and not a machine’. For a machine to pass the test, a ‘considerable proportion’ of the jury ‘must be taken in by the pretence’.

Lecture on the Automatic Computing Engine (1947)

On 8 December 1943 the world’s first large-scale special-purpose electronic digital computer—‘Colossus’, as it became known—went into operation at the Government Code and Cypher School (see ‘Computable Numbers: A Guide’, ‘Enigma’, and the introduction to Chapter 4). Colossus was built by Thomas H. Flowers and his team of engineers at the Post Office Research Station in Doll is Hill, London. Until relatively recently, few had any idea that electronic digital computation was used successfully during the Second World War, since those who built and worked with Colossus were prohibited by the Official Secrets Act from sharing their knowledge. Colossus contained approximately the same number of electronic valves (vacuum tubes) as von Neumann’s IAS computer, built at the Princeton Institute of Advanced Study and dedicated in 1952. The IAS computer was forerunner of the IBM 701, the company’s first mass-produced stored-programme electronic computer (1953). The first Colossus had 1,600 electronic valves and Colossus II, installed in mid-1944, 2,400, while the IAS computer had 2,600. Colossus lacked two important features of modern computers. First, it had no internally stored programmes (see ‘Computable Numbers: A Guide’). To set up Colossus for a new task, the operators had to alter the machine’s physical wiring, using plugs and switches. Second, Colossus was not a general-purpose machine, being designed for a specific cryptanalytic task (involving only logical operations and counting). Nevertheless, Flowers had established decisively and for the first time that large-scale electronic computing machinery was practicable. The implication of Flowers’s racks of electronic equipment would have been obvious to Turing. Once Turing had seen Colossus it was, Flowers said, just a matter of Turing’s waiting to see what opportunity might arise to put the idea of his universal computing machine into practice. Precisely such an opportunity fell into Turing’s lap in 1945, when John Womersley invited him to join the Mathematics Division of the National Physical Laboratory (NPL) at Teddington in London, in order to design and develop an electronic stored-programme digital computer—a concrete form of the universal Turing machine of 1936.