Alan Turing: The Enigma: The Book That Inspired the Film The Imitation Game

by Andrew Hodges

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  Alan’s first task was to write a report1, setting out a detailed design of an electronic universal machine, and an account of its operation. Surprisingly, the report that he submitted did not contain mention of Computable Numbers. Instead, it referred to the Draft Report on the EDVAC, with which his report was to be read ‘in conjunction’. However, the ACE proposal was effectively self-contained, and its roots lay not in the EDVAC, but in his own universal machine. Some fragmentary notes2, dating from this early period, made this clear:

  … In ‘Computable Numbers’ it was assumed that all the stored material was arranged linearly, so that in effect the accessibility time was directly proportional to the amount of material stored, being essentially the digit time multiplied by the number of digits stored. This was the essential reason why the arrangement in ‘Computable Numbers’ could not be taken over as it stood to give a practical form of machine.

  It was also implicit in an opening paragraph of the report he now wrote, which explained how new problems would be ‘virtually only a matter of paper work’, with examples,* and said:

  It may appear somewhat surprising that this can be done. How can one expect a machine to do all this multitudinous variety of things? The answer is that we should consider the machine as doing something quite simple, namely carrying out orders given to it in a standard form which it is able to understand.

  But he considerably amplified this idea in a talk given a year later in February 1947,3 in words which explained the origin of the ACE as he himself perceived it:

  Some years ago I was researching on what might now be described as an investigation of the theoretical possibilities and limitations of digital computing machines. I considered a type of machine which had a central mechanism and an infinite memory which was contained on an infinite tape. This type of machine appeared to be sufficiently general. One of my conclusions was that the idea of a ‘rule of thumb’ process and a ‘machine process’ were synonymous. The expression ‘machine process’ of course means one which could be carried out by the type of machine I was considering…. Machines such as the ACE* may be regarded as practical versions of this same type of machine. There is at least a very close analogy.

  Digital computing machines all have a central mechanism or control and some very extensive form of memory. The memory does not have to be infinite, but it certainly needs to be very large. In general the arrangement of the memory on an infinite tape is unsatisfactory in a practical machine, because of the large amount of time which is liable to be spent in shifting up and down the tape to reach the point at which a particular piece of information required at the moment is stored. Thus a problem might easily need a storage of three million entries, and if each entry was equally likely to be the next required the average journey up the tape would be through a million entries, and this would be intolerable. One needs some form of memory with which any required entry can be reached at short notice. This difficulty presumably used to worry the Egyptians when their books were written on papyrus scrolls. It must have been slow work looking up references in them, and the present arrangement of written matter in books which can be opened at any point is greatly to be preferred. We may say that storage on tape and papyrus scrolls is somewhat inaccessible. It takes a considerable time to find a given entry. Memory in book form is a good deal better, and is certainly highly suitable when it is to be read by the human eye. We could even imagine a computing machine that was made to work with a memory based on books. It would not be very easy but would be immensely preferable to the single long tape. Let us for the sake of argument suppose that the difficulties involved in using books as memory were overcome, that is to say that mechanical devices for finding the right book and opening it at the right page, etc. etc. had been developed, imitating the use of human hands and eyes. The information contained in the books would still be rather inaccessible because of the time occupied in the mechanical motions. One cannot turn a page over very quickly without tearing it, and if one were to do much book transportation, and do it fast, the energy involved would be very great. Thus if we moved one book every millisecond and each were moved ten metres and weighed 200 grams, and if the kinetic energy were wasted each time, we should consume 1010 watts, about half the country’s power consumption. If we are to have a really fast machine then we must have our information, or at any rate a part of it, in a more accessible form than can be obtained with books.

  After this flight of fancy, very typical of his conversation, he discussed various more serious proposals for storage, and commented that ‘the provision of proper storage is the key to the problem of the digital computer.’

  In my opinion this problem of making a large memory available at reasonably short notice is much more important than that of doing operations such as multiplication at high speed. Speed is necessary if the machine is to work fast enough for the machine to be commercially valuable, but a large storage is necessary if it is to be capable of anything more than rather trivial operations. The storage capacity is therefore the more fundamental requirement.

  He then continued to give a succinct definition of ‘building a brain’:

  Let us now return to the analogy of the theoretical computing machines with an infinite tape. It can be shown that a single special machine of that type can be made to do the work of all. It could in fact be made to work as a model of any other machine. The special machine may be called the universal machine; it works in the following quite simple manner. When we have decided what machine we wish to imitate, we punch a description of it on the tape of the universal machine. This description explains what the machine would do in every configuration in which it might find itself. The universal machine has only to keep looking at this description in order to find out what it should do at each stage. Thus the complexity of the machine to be imitated is concentrated in the tape and does not appear in the universal machine proper in any way.

  If we take the properties of the universal machine in combination with the fact that machine processes and rule of thumb processes are synonymous, we may say that the universal machine is one which, when supplied with the appropriate instructions, can be made to do any rule of thumb process. This feature is parallelled in digital computing machines such as the ACE. They are in fact practical versions of the universal machine. There is a certain central pool of electronic equipment, and a large memory. When any particular problem has to be handled the appropriate instructions for the computing process involved are stored in the memory of the ACE and it is then ‘set up’ for carrying out that process.

  His priorities were a large, fast memory, and then a hardware system that would be as simple as possible. The latter requirement was reminiscent of his ‘desert island’ mentality, doing everything with the least waste. But both features were to exploit the universality of the machine. His idea was always that anything in the way of refinement or convenience for the user could be performed by thought and not by machinery, by instructions and not by hardware.

  In his philosophy it was almost an extravagance to supply addition and multiplication facilities as hardware, since in principle they could be replaced by instructions applying only the more primitive logical operations of OR, AND, and NOT. Indeed, the Colossus, when it was ‘almost’ programmed to do multiplication, had done just that. Since these primitive logical operations (absent from the EDVAC draft design) were incorporated in his plan for the ACE, he could indeed have omitted adders and multipliers, and still have had a universal machine. In reality, he did include special hardware to perform arithmetical tasks, but even there he decomposed the arithmetical operations into small pieces so that he could economise on hardware at the cost of more stored instructions. The whole conception was very puzzling to his contemporaries, to whom a computer was a machine to do sums, and a multiplier the very essence of its function. To Alan Turing, the multiplier was a rather tiresome technicality; the heart lay in the logical control, which took the instructions from the memory, a
nd put them into operation.

  For similar reasons, his report placed no great emphasis upon the fact that the ACE would use binary arithmetic. He stated the advantage of the binary representation, namely that electronic switches could naturally represent ‘1’ and ‘0’ by ‘on’ and ‘off’. But that was all, apart from a terse statement that the input and output to the machine would be in ordinary decimal notation, and that the conversion process would have ‘virtually no outward and visible form’. In his 1947 talk he would elaborate on this briefest of all possible comments. The point was that the universality of the machine made it possible to encode numbers in any way one wished within the machine – in binary form, if that happened to suit the technology. It would be inappropriate to employ binary numbers in a cash register, because it would be more trouble than it was worth to make the conversions for input and output. On the universal ACE, however, no such conversion was required –

  This last statement sounds quite paradoxical, but it is a simple consequence of the fact that these machines can be made to do any rule of thumb process by remembering suitable instructions. In particular it can be made to do binary decimal conversion. For example in the case of the ACE the provision of the converter involves no more than adding two extra delay lines to the memory. This situation is very typical of what happens with the ACE. There are many fussy little details which have to be taken care of, and which according to normal engineering practice would require special circuits. We are able to deal with these points without modification of the machine itself, by pure paperwork, eventually resulting in feeding in appropriate instructions.

  Logical as this was, and certainly comprehensible to mathematicians, who had been familiar with binary numbers for three hundred years at least, the fact was that ‘fussy little details’ such as this were more of a headache for other people. To an engineer, in particular, it would come as a revelation that the concept of number could be separated from its representation in decimal form. Many people would see the ‘binary’ arithmetic of the ACE as itself a weird and wonderful innovation. Whilst he was perfectly correct in seeing this as a detail, it was a good example of his difficulties in communication with the kind of people who might fund, organise, and build his machine.

  But with such details disposed of, he concentrated his report upon the two really important things: the memory and the control.

  Considering the storage problem, he listed every form of discrete store that he and Don Bayley had thought of, including film, plugboards, wheels, relays, paper tape, punched cards, magnetic tape, and ‘cerebral cortex’, each with an estimate, in some cases obviously fanciful, of access time, and of the number of digits that could be stored per pound sterling. At one extreme, the storage could all be on electronic valves, giving access within a microsecond, but this would be prohibitively expensive. As he put it in his 1947 elaboration, ‘To store the content of an ordinary novel by such means would cost many millions of pounds.’ It was necessary to make a trade-off between cost and speed of access. He agreed with von Neumann, who in the EDVAC report had referred to the future possibility of developing a special ‘Iconoscope’ or television screen,* for storing digits in the form of a pattern of spots. This he described as ‘much the most hopeful scheme, for economy combined with speed.’ But in a prescient paragraph of the ACE report he also suggested an approach more on the home-made, ‘least waste of energy’ line:

  It seems probable that a suitable storage system can be developed without involving any new types of tube, using in fact an ordinary cathode ray tube with tin-foil over the screen to act as a signal plate. It will be necessary to furbish up the charge pattern from time to time, as it will tend to become dissipated …. It will be necessary to stop the beam from scanning in the refurbishing cycle, switch to the point from which the information required is to be taken, do some scanning there, replace the information removed by the scanning, and return to refurbishing from the point left off. Arrangements must also be made to make sure that refurbishing does not get neglected for too long because of more pressing duties. None of this involves any fundamental difficulty, but no doubt it will take time to develop.

  Lacking such cathode ray tube storage, he had to plump for the mercury delay lines, not with any great enthusiasm, but because they were already working. They held the obvious disadvantage, from the point of view of accessibility, of involving a delay. His plan was for a delay line to hold a sequence of 1024 pulses, so it was like chopping up the ‘tape’ of the Universal Turing Machine into segments each of 1024 squares in length. It would take an average of 512 units of time to reach a given entry. However, this was an improvement upon the ‘papyrus scroll’.

  As for the other most important aspect of the machine, this was the ‘Logical Control’. It corresponded to the ‘scanner’ of the Universal Turing Machine. The principle was simple: ‘The universal machine has only to keep looking at this description’ – that is, at the instructions on its tape – ‘to find out what it should do at each stage.’ So the Logical Control was a piece of electronic hardware which would contain two pieces of information: where it was on the ‘tape’, and what instruction it had read there. The instruction would take up thirty-two ‘squares’ or pulses in a delay line store, and might be of two kinds, in the design that he proposed. It might simply cause the ‘scanner’ to go on to another point of the ‘tape’ for its next instruction. Alternatively, it might prescribe an operation of adding, multiplying, shifting or copying, of numbers stored elsewhere on the ‘tape’. In the latter case, the ‘scanner’ was to move to the next point on the ‘tape’ for its next instruction. None of this involved anything but the reading, writing, erasing, changing of state, and moving to left and right, that was to be done by the theoretical Universal Turing Machine working on the description numbers on its tape – except that there were special facilities added so that addition and multiplication could be achieved in only a few steps, rather than with thousands of more elementary operations.

  Of course, there was to be no physical motion when the ‘scanner’ went to fetch an instruction, or operate upon numbers stored on the ‘tape’; no motion but that of electrons. Instead, the control of the ACE would work by a process rather like that of dialling a telephone number, to reach the right place. Most of the complexity of the electronic circuits arose from the demands of this ‘tree’ system. There was also a complexity in the way that thirty-two half-way houses, ‘temporary storage’ locations consisting of special short delay lines, were provided for the shunting around of pulses. This was quite different from the EDVAC conception, in which all the arithmetic would be done by shunting numbers in and out of a central ‘accumulator’. In the ACE design the arithmetical operations were ‘distributed’ around the thirty-two ‘temporary storage’ delay lines in an ingenious way.

  The point of this complexity was that it increased the speed of operation. Speed took a slightly higher priority than simplicity. This was reflected also in the fact that Alan planned the pulse rate of the ACE to be a million a second, straining electronic technology to the full.* His emphasis on speed was natural enough, granted his Bletchley experience, in which speed was of the very essence, a few hours marking the difference between usefulness and pointlessness. It was also related to the universality of the electronic computer. In 1942 they had tried to get faster Bombes to cope with the fourth rotor; they had been saved by the German slip-up with the weather-reporting signal system, but without this stroke of fortune they would have had to wait over a year for the machinery to match the problem. One virtue of a universal machine would lie in its ability to take on any new problem immediately – but this meant that it must be as fast as possible from the start. It would hardly be desirable to re-engineer a universal machine for the sake of a special problem. The whole point was to get the engineering out of the way once and for all, so that all the work thereafter could go into the devising of instruction tables.

  Although the ACE was based upon the idea of the
Universal Turing Machine, in one way it departed from it. The departure lay in the feature, at first sight an extraordinary one, that it had no facility for conditional branching. In this respect it might seem to lack the crucial idea that Babbage had introduced a hundred years earlier. For the ‘scanner’, or Logical Control, of the ACE could only hold one ‘address’, or position on the tape, at a time. It had no way of holding more than one, and then of selecting a next destination according to some criterion.

  The omission, however, was only on the surface. The reason for it was that it was a case where the hardware could be simplified, at the cost of more stored instructions. Alan set out a way in which conditional branching could be done without needing the logical control to hold more than one ‘address’ at a time. It was not the best technical solution, but it had the merit of a brutal simplicity. Suppose that it were wished to execute instruction 50 if some digit D were 1, and to execute instruction 33 if it were 0. His idea was to ‘pretend that the instructions were really numbers and calculate D × (Instruction 50) + (1 – D) × (Instruction 33).’ The result of this calculation would be an instruction which would have the effect that was required. The ‘IF’ was effected not by hardware, but by extra programming. It was a device which had led him to mix up data (the digit D) with instructions. This in itself was of great significance, for he had allowed himself to modify the stored program. But this was only the beginning.