Turing machine

The Man Who Invented the Computer

by Jane Smiley  · 18 Oct 2010  · 253pp  · 80,074 words

whether every answer was “true” or not. The lambda calculus “represented an elegant and powerful symbolism for mathematical processes of abstraction and variation,” but the Turing machine was a thought experiment that posited a mechanical operation, to be done by either a mechanism or by a human mind. Andrew Hodges, Turing’s

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mathematics, not only a play of symbols, for it involved thinking about what people did in the physical world … His machines—soon to be called Turing Machines—offered a bridge, a connection between abstract symbols and the physical world. Indeed, his imagery was, for Cambridge, almost shockingly industrial.” In May 1936, Alan

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, unlike Mauchly and Eckert, he happened to be quite familiar with “On Computable Numbers” and had even toyed with designing a mechanical version of a Turing machine before the war (his partner, like Mauchly and Eckert’s original partner, was in the horse-racing pari-mutuel totalizer business), he offered Turing £800

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clear that what he had in mind building was something very like the theoretical model in his Computability paper, the model we now call a Turing machine. It worked on one bit at a time, but used a huge amount of memory to do anything of consequence. Since he had proved that

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about it and building it simply occupied his mind almost completely and drove almost every other consideration, including mortality, out of his consciousness. 2. A Turing machine has been constructed. It can be seen on YouTube: http://www.youtube.com/watch?v=E3keLeMwfHY. 3. One computer that conformed to Turing’s ideas

The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant From Two Centuries of Controversy

by Sharon Bertsch McGrayne  · 16 May 2011  · 561pp  · 120,899 words

Oxford and Cambridge universities. Among that handful of men was Alan Mathison Turing, who would father the modern computer, computer science, software, artificial intelligence, the Turing machine, the Turing test—and the modern Bayesian revival. Turing had studied pure mathematics at Cambridge and Princeton, but his passion was bridging the gap between

The Music of the Primes

by Marcus Du Sautoy  · 26 Apr 2004  · 434pp  · 135,226 words

special machines that could effectively be made to behave like any person or machine that was doing arithmetic computations. They would later be known as Turing machines. Hilbert had been rather vague about what he meant by a machine that could tell whether statements could be proved. Now, thanks to Turing, Hilbert

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out one day, running along the banks of the River Cam, Turing experienced the second flash of enlightenment that told him why none of these Turing machines could be made to distinguish between statements that had proofs and those that didn’t. As he paused for a breather, lying on his back

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the fractions were matched with the real numbers. Turing took this technique and used it to produce a ‘left-over’ true statement for which the Turing machine could not possibly decide whether a proof existed. The beauty of Cantor’s argument was that if you tried to adapt the machine to include

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, he turned it over in his mind, probing for any weaknesses. There was one point that worried him. He had shown that none of his Turing machines could answer Hilbert’s Decision Problem. But how could he convince people that there wasn’t some other machine that could answer Hilbert’s problem

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: the idea of a universal machine. He drew up a blueprint for a single machine that could be taught to behave like all of his Turing machines or like any other machine that might answer Hilbert’s problem. He was already beginning to understand the power of a program that could teach

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felt sure that there should be a way to exploit the groundwork laid by Turing. She understood that each Turing machine gives rise to a sequence of numbers. For example, one of the Turing machines could be made to produce a list of the square numbers 1, 4, 9, 16, …, whilst another generated the

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Problem had been to prove that no program existed that could decide whether, given a Turing machine and a number, that number is the output of the machine. Robinson was looking for a connection between equations and Turing machines. Each Turing machine, she believed, would correspond to a particular equation. If there were such a connection

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, Robinson hoped that asking whether a number was the output of a particular Turing machine would translate into asking whether the equation corresponding to that machine had a solution. So if she could establish such a connection, she would be

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program could be used, via Robinson’s as yet hypothetical connection between equations and Turing machines, to check which numbers were outputs of Turing machines. But Turing had shown that no such program – one that could decide about the outputs of Turing machines – existed. Therefore there could be no program that could decide whether equations had

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solutions. The answer to Hilbert’s tenth problem would be ‘no’. Robinson set about understanding why each Turing machine might have its own equation. She wanted an equation whose solutions were connected to the sequence of numbers output by the

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Turing machine. She was rather amused by the question she had set herself. ‘Usually in mathematics you have an equation and you want to find a solution.

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. By the end of the 1960s they had reduced the problem to something simpler. Rather than having to find all equations for all outputs of Turing machines, they discovered that if they found an equation for one particular sequence of numbers, they would have proved Robinson’s hunch. It was a remarkable

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solutions – there was a silver lining. Robinson had been proved right in her belief that lists of numbers produced by Turing machines could be described by equations. Mathematicians knew that there was one Turing machine that could reproduce the list of prime numbers. So, thanks to the work of Robinson and Matijasevich, in theory

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negative. Even theoretically it is somewhat lacking in significance. Robinson and Matijasevich had shown that any list of numbers that can be produced by a Turing machine will have such an equation, so in that sense there is nothing special about the primes as opposed to any other list of numbers. This

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and the Enigma code 175, 190–91, 205, 206 and Hardy 187, 188 death 192 homosexuality 192 and the Riemann Hypothesis 175, 188, 191, 212 Turing machines 182–93, 197, 198, 199, 202–3, 204, 207, 213, 215 twin autistic-savants 8–9, 39 Twin Primes Conjecture 39, 181, 257, 258 uranium

The Secret Life of Bletchley Park: The WWII Codebreaking Centre and the Men and Women Who Worked There

by Sinclair McKay  · 24 May 2010  · 351pp  · 107,966 words

for the United States, to Princeton. Building bridges between the two disciplines of mathematics and applied physics, he threw himself into the construction of a ‘Turing machine’, a machine that could carry out logical binary calculations. Having seen a tide-predicting machine some years back in Liverpool, it occurred to him that

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death in 1954, his home in Manchester was filled with extraordinary and sometimes pungent chemical experiments. Turing had fixed upon the idea of a ‘Universal Turing Machine’ in the 1930s; the inspiration had been provided by a mathematical problem posed in Cambridge, concerning the provability of any given mathematical assertion. Turing had

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a human operator. His argument was that any calculation that a human could perform, a machine could perform as well. The bombes were not Universal Turing Machines. Far from it. Nor were they an extension of the Polish ‘bomba’ machines, from which their name was taken. The British bombe was quite a

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who in the 1930s, with his lectures on ‘mechanical approaches’ to solving mathematical problems, first led Alan Turing to start pondering on the idea of ‘Turing Machines’. Indeed, Newman had lectured Turing directly. Newman, born in 1897, was a very popular figure at the Park; many veterans recall his openness, and his

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, still only thirty-four at the end of the war, technology was drawing closer and closer to enable him to realise his concept of a ‘Turing Machine’; equally, though, his homosexuality, and the British establishment’s attitude towards it, were to contribute to his tragic – and wholly pointless – death. The transition from

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wanted to return to the question that had been haunting him since the 1930s, that of constructing a thinking machine – an electronic brain. A Universal Turing Machine. A machine of such complexity that it could not only speedily handle any kind of mathematical calculation, but also store a memory of the process

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by this stage, he was widely admired within the British scientific community. At conferences, mathematicians would vie for his attention. Work on the Mark II Turing machine – an even larger computer than the first – was under way. He even managed a reunion with his old colleague Don Bayley, who now lived in

Turing's Cathedral

by George Dyson  · 6 Mar 2012

lives of those who brought them into existence, and how code took over the world.” eISBN: 978-0-307-90706-6 1. Computers—History. 2. Turing machines. 3. Computable functions. 4. Random access memory. 5. Von Neumann, John, 1903–1957. 6. Turing, Alan Mathison, 1912–1954. I. Title. QA76.17.D97 2012

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(1972), Nicholas Metropolis’s History of Computing in the Twentieth Century (1980), Andrew Hodges’s Alan Turing: The Enigma (1983), Rolf Herken’s The Universal Turing Machine: A Half-Century Survey (1988), and William Aspray’s John von Neumann and the Origins of Modern Computing (1990). Julian Bigelow and his colleagues designed

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of a class of devices (including an obedient human being) that could read, write, remember, and erase marks on an unbounded supply of tape. These “Turing machines” were able to translate, in both directions, between bits embodied as structure (in space) and bits encoded as sequences (in time). Turing then demonstrated the

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on paper tape. “Being digital should be of more interest than being electronic,” Turing pointed out.6 Von Neumann set out to build a Universal Turing Machine that would operate at electronic speeds. At its core was a 32-by-32-by-40-bit matrix of high-speed random-access memory—the

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were demonstrations, if not the direct offspring, of his, Gödel’s, ideas. “What von Neumann perhaps had in mind appears more clearly from the universal Turing machine,” he later explained to Arthur Burks. “There it might be said that the complete description of its behavior is infinite because, in view of the

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non existence of a decision procedure predicting its behavior, the complete description could be given only by an enumeration of all instances. The universal Turing machine, where the ratio of the two complexities is infinity, might then be considered to be a limiting case.”53 Leibniz’s belief in a universal

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computation is the transformation of a bit between its two possible forms of existence: as structure (memory) or as sequence (code). This is what a Turing Machine does when reading a mark (or the absence of a mark) on a square of tape, changing its state of mind accordingly, and making (or

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. You can’t rely upon chance.”72 According to Bigelow, the forty-fold parallel architecture, despite its deviations, was descended directly from the pure-serial Turing Machine. “Turing’s machine does not sound much like a modern computer today, but nevertheless it was,” Bigelow explains. “It was the germinal idea. If you

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these operations the m-configuration may be changed.12 Turing introduced two fundamental assumptions: discreteness of time and discreteness of state of mind. To a Turing machine, time exists not as a continuum, but as a sequence of changes of state. Turing assumed a finite number of possible states at any given

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which seriously affects computation, since the use of more complicated states of mind can be avoided by writing more symbols on the tape.”13 The Turing machine thus embodies the relationship between an array of symbols in space and a sequence of events in time. All traces of intelligence were removed. The

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infinite, but if more tape is needed, the supply can be counted on never to run out. Each step in the relationship between tape and Turing machine is determined by an instruction table listing all possible internal states, all possible external symbols, and, for every possible combination, what to do (write or

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erase a symbol, move right or left, change the internal state) in the event that combination comes up. The Turing machine follows instructions and never makes mistakes. Complicated behavior does not require complicated states of mind. By taking copious notes, the

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Turing machine can function with as few as two internal states. Behavioral complexity is equivalent whether embodied in complex states of mind (m-configurations) or complex symbols (

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,” he announced.14 The Universal Machine, when provided with a suitably encoded description of some other machine, executes this description to produce equivalent results. All Turing machines, and therefore all computable functions, can be encoded by strings of finite length. Since the number of possible machines is countable but the number of

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be given a finite description but could not be computed by finite means. One of these was the halting function: given the number of a Turing machine and the number of an input tape, it returns either the value 0 or the value 1 depending on whether the computation will ever come

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for seventy-six years. In March of 1937, Alonzo Church reviewed “On Computable Numbers” in the Journal of Symbolic Logic, and coined the term Turing machine. “Computability by a Turing machine,” wrote Church, “has the advantage of making the identification with effectiveness in the ordinary (not explicitly defined) sense evident immediately.”18 Church’s

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Turing showed that undecidable statements, resistant to the assistance of an external oracle, could still be constructed, and the Entscheidungsproblem would remain unsolved. The Universal Turing Machine of 1936 gets all the attention, but Turing’s O-machines of 1939 may be closer to the way intelligence (real and artificial) works: logical

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cryptography, where a little ingenuity in encoding a message can resist a large amount of ingenuity if the message is intercepted along the way. A Turing machine can be instructed to conceal meaningful statements in what appears to be meaningless noise—unless you know the key. A

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Turing machine can also be instructed to search for meaningful statements, but since there will always be uncountably more meaningless statements than meaningful ones, concealment would appear

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Newman, who had set all these wheels in motion when he sparked Turing’s original interest in the Entscheidungsproblem in 1935. Colossus was an electronic Turing machine, and if not yet universal, it had all the elements in place. Colossus was so successful (and subspecies of Fish so prolific) that by the

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the ENIAC, and was both running and replicated while the one-of-a-kind ENIAC was still being built. Each Fish was a form of Turing machine, and the process by which the Colossi were used to break the various species of Fish demonstrated how the function (or partial function) of one

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Turing machine could be encoded for execution by another Turing machine. Since the British did not know the constantly changing state of the Fish, they had to guess. Colossus, trained to sense

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mathematical fit. J. R. Womersley, superintendent of the Mathematics Division of the National Physical Laboratory, who had read “On Computable Numbers” and become interested in Turing machines before the war, had been sent to the United States in the spring of 1945 to survey the latest (and still-secret) computer developments, including

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the cost of a cerebral cortex at £300 per annum—his King’s College fellowship for the year. Viewed as part of a finite-state Turing machine, the delay line represented a continuous loop of tape, 1,000 squares in length and making 1,000 complete passes per second under the read

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pseudomorphosis resumed its course. The von Neumann address matrix is becoming the basis of a non–von Neumann address matrix, and Turing machines are being assembled into systems that are not Turing machines. Codes—we now call them apps—are breaking free from the intolerance of the numerical address matrix and central clock cycle

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their ability to compute. Sending an e-mail, or transferring a file, does not physically move anything; it creates a new copy somewhere else. A Turing machine is, by definition, able to make exact copies of any readable sequence—including its own state of mind and the sequence stored on its own

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tape. A Turing machine can, therefore, make copies of itself. This caught the attention of von Neumann at the same time as the Institute for Advanced Study computer project

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ones on it. What is needed … is an automaton whose output is other automata.”7 Using the same method of logical substitution by which a Turing machine can be instructed to interpret successively higher-level languages—or by which Gödel was able to encode metamathematical statements within ordinary arithmetic—it was possible

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to design Turing machines whose coded instructions addressed physical components, not memory locations, and whose output could be translated into physical objects, not just zeros and ones. “Small variations

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-reproduction is fundamentally a problem of communication, over a noisy channel, from one generation to the next. “Turing!” refers to the powers of the Universal Turing Machine, and “Not Turing!” refers to the limitations of those powers—and how they might be transcended by living and nonliving things. “Pitts-McCulloch!” refers to

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a memory of 25 kilobytes, or 2 × 105 bits. The present scale of the digital universe has been estimated at 1022 bits. The number of Turing machines populating this universe is unknown, and increasingly these machines are virtual machines that do not necessarily map to any particular physical hardware at any particular

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the local conditions, might be the development of the Universal Machine. Over long distances, it is expensive to transport structures, and inexpensive to transmit sequences. Turing machines, which by definition are structures that can be encoded as sequences, are already propagating themselves, locally, at the speed of light. The notion that one

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migrate from one place to another as an electromagnetic signal, as long as there’s a digital world—a civilization that has discovered the Universal Turing Machine—for it to colonize when it gets there. And that’s why von Neumann and you other Martians got us to build all these computers

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of the greatest significance,” but never received an answer, von Neumann having given up his correspondence by that time. “It is easy to construct a Turing machine that allows us to decide, for each formula F of the restricted functional calculus and every natural number n, whether F has a proof of

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. 10. I. J. Good to Sara Turing, December 9, 1956, AMT; Robin Gandy, “The Confluence of Ideas in 1936,” in Rolf Herken, ed., The Universal Turing Machine: A Half-Century Survey (Oxford: Oxford University Press, 1988), p. 85. 11. Alan Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,” Proceedings of

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, AMT. 17. Freeman Dyson, interview with author, May 5, 2004, GBD; Martin Davis, “Influences of Mathematical Logic on Computer Science,” in Herken, ed., The Universal Turing Machine, p. 315. 18. Alonzo Church, “Review of A.M. Turing, ‘On Computable Numbers, with an Application to the Entscheidungsproblem,’ ” Journal of Symbolic Logic 2, no

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, 8.1, 8.2 and termination of the ECP, 14.1, 18.1, 18.2 on time (vs. sequence), 16.1, 16.2 on Universal Turing Machine, 8.1, 14.1 visits Manchester (1948) on von Neumann, 5.1, 5.2, 7.1, 8.1, 8.2, 14.1, 14.2 on

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, 8.2, 14.1 reliability of, 7.1, 7.2, 8.1, 8.2, 11.1, 12.1 speed (and asynchronous arithmetic) of as Universal Turing Machine, 1.1, 8.1 see also ECP; Williams (memory) tubes IBM (International Business Machines), 4.1, 7.1, 7.2, 8.1, 8.2, 10

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, 17.2 and nondeterministic machines on “oracle machines” on unorganized machines and von Neumann, 5.1, 13.1, 13.2, 13.3 see also Universal Turing Machine Turing, Ethel Sara Turing, Julius Mathison Turing, Sara, 13.1, 13.2, 13.3 Turing’s cathedral, Google as turtles Ujelang (Marshall Islands) Ulam, Adam

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also incompleteness United States Atomic Energy Commission, see Atomic Energy Commission (AEC) UNIVAC (Universal Automatic Computer), 5.1, 18.1 Universal Constructor (von Neumann) Universal Turing Machine, prf.1, 1.1, 5.1, 6.1, 11.1, 13.1, 13.2, 13.3, 14.1, 15.1, 15.2, 17.1 and

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.1, 8.1, 10.1, 16.1 superstitions of and theory of self-reproducing automata, 1.1, 15.1, 15.2 and Turing (and Universal Turing Machine), 1.1, 3.1, 5.1, 6.1, 8.1, 13.1, 13.2, 13.3, 13.4, 13.5, 15.1 and Stan Ulam

From Bacteria to Bach and Back: The Evolution of Minds

by Daniel C. Dennett  · 7 Feb 2017  · 573pp  · 157,767 words

determined by the instructions, which gave them a remarkable competence: they could do anything computational. In other words, a programmable digital computer is a Universal Turing Machine, capable of mimicking any special-purpose digital computer by following a set of instructions that implement that special-purpose computer in software.13 (You don

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beaver dams are, so there is no radical discontinuity, no need for a skyhook, to get us from spiders and beaver dams to Turing and Turing machines. Still, there is a large gap to be filled, because Turing’s way of making things was strikingly different from the spider’s way and

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illustrious example. For the sake of argument let’s concede that evolution by natural selection could not directly evolve a living digital computer (a Turing machine tree or a Turing machine turtle, for example). But there is an indirect way: let natural selection first evolve human minds, and then they can intelligently design Hamlet

Human Compatible: Artificial Intelligence and the Problem of Control

by Stuart Russell  · 7 Oct 2019  · 416pp  · 112,268 words

the second device on its input, produce the same output that the second device would have produced. We now call this first device a universal Turing machine. To prove its universality, Turing introduced precise definitions for two new kinds of mathematical objects: machines and programs. Together, the machine and program define a

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even if we had tried to build intelligent machines from chemical reactions or biological cells, those assemblages would have turned out to be implementations of Turing machines in nontraditional materials. Whether an object is a general-purpose computer has nothing to do with what it’s made of. 31. Turing’s breakthrough

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paper defined what is now known as the Turing machine, the basis for modern computer science. The Entscheidungsproblem, or decision problem, in the title is the problem of deciding entailment in first-order logic: Alan

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probability theory and, 273–84 United Nations (UN), 250 universal basic income (UBI), 121 Universal Declaration of Human Rights (1948), 107 universality, 32–33 universal Turing machine, 33, 40–41 unpredictability, 29 utilitarian AI, 217–27 Utilitarianism ((Mill), 217–18 utilitarianism/utilitarian AI, 214 challenges to, 221–27 consequentialist AI, 217–19

The Dream Machine: J.C.R. Licklider and the Revolution That Made Computing Personal

by M. Mitchell Waldrop  · 14 Apr 2001

Activity," was essentially a demonstration that their idealized neural networks were functionally equivalent to Turing machines. That is, any problem that a Turing machine could solve, an appropriately designed network could also solve. And conversely, anything that was beyond a Turing machine's power-such as the decidability problem-was likewise beyond a network's power

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. As the science historian William Aspray has written, "With the Turing machines providing an abstract characterization of thinking in the machine world and McCulloch and Pitts's neuron nets providing one in the biological world, the equivalence

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."14 Or, as McCulloch himself would put it in his 1965 autobiography Embodiments of Mind, he and Pitts had proved the equivalence of all general Turing machines, whether "man-made or begotten." John von Neumann was deeply impressed by McCulloch and Pitts's neural- network ideas from the moment he saw their

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bit of leeway in his answer, thanks to Alan Turing. As he was undoubtedly well aware, his abstract architecture was logically equiva- lent to a Turing machine, with the memory, input, and output units collectively THE LAST TRANSITION 61 corresponding to the tape, and the central arithmetic and central control units collectively

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instead of hard-wiring it into the read/write head. And it had precisely the same implications for universality: a stored-program computer, like a Turing machine, could treat its program instructions as just another kind of data. In practical engineering terms, meanwhile, the stored-program approach had the obvious advantage of

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Neumann observed, you have an automaton that can turn out very complex parts but not another machine tool. Likewise, a universal Turing machine can output an arbitrarily complex tape but not another Turing machine. However, in almost any biological organ- ism, you have an automaton that can not only reproduce identical copies of it

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and brain in terms of information processing (and in some cases, in terms of neural networks) or to hear computer scientists talking about information flow, Turing machines, and chip-level logic gates. So, yes: as a set of concepts, the cybernetics movement is still very much alive, and we are all members

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abstract, you can think of them as collectively forming a "machine"-an automaton that generates word strings in much the same way that an abstract Turing machine computes numbers. And once you do that, he said, you can ask essentially the same question that Alan Turing asked of his imaginary machine: What

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question (Whom did John kiss?)- you need a grammar from the most powerful class of all, the one that is mathe- matically equivalent to a Turing machine. To put it another way, the very fact that we human beings use language in the way we do is proof that, in some sense

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, our brains have the computational power of a Turing machine. Or to express it still another way, the pinnacle of all possi- ble mathematical machines-the Turing machine-is also the baseline, the mini- mum needed for human cognition. Anybody who seriously wants to understand

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, a title that McCarthy hated. But with contributions from von Neumann and many others it added up to present a greatly enriched understanding of what Turing machine-like automata-that is, com- puters-can and cannot do. The book is now considered a landmark in computer science. 162 THE DREAM MACHINE was

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moored to the physical operation of the com- puter. That Lisp kernel, known as apply, thus provided a particularly elegant example of a uni- versal Turing machine: it was the universal function that took the definition of any other function as input and then executed that function. By no coincidence, McCarthy imple

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of Newton's laws of motion, say- "and I couldn't find them. There were certainly some deep ideas- for example, information theory, or the Turing machine. But those ideas had not led to a large body of theory as there was in physics." The upshot was that his plan for the

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, 100 Truong, Thl T., 430 Truscott, Tom, 427 Tukey, John, 81 Tunng, Alan, 47-53, 58, 80, 83, 122-25,131,161 Tunng Award, 240n Turing machine, 50-51, 58, 60-61,62,88, 131-32, 172n TUflng test, 122-24 Turkle, Sherry, 64, 437 "turtle" robot, 328, 360 TX-O, 144

The Man From the Future: The Visionary Life of John Von Neumann

by Ananyo Bhattacharya  · 6 Oct 2021  · 476pp  · 121,460 words

of a particular type. Later in the paper, he uses these to help build a ‘universal computing machine’ that is capable of simulating any other Turing machine. Computer programmers today would recognize Turing’s strategy: modern programmes make use of libraries of simpler programmes known as ‘subroutines’. Subroutines simplify the structure of

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easier to understand, improve and troubleshoot. Though Turing describes his computing machine in purely abstract terms, it is quite easy to imagine building one. A Turing machine that can execute a single task might comprise a scanner, print head (erasing characters is admittedly a little trickier) and a motor that moves a

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another (clean laundry, punched cards with calculated artillery trajectories). The universal machine Turing described, however, is quite different. When fed the instruction table of another Turing machine, the universal machine can simulate it exactly. Turing first describes a system to convert an instruction table into a form that his machines can digest

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his machines are capable of carrying out any algorithmic process that can be performed by their human counterparts. Conversely, a human may compute anything a Turing machine can – assuming the human in question does not die of boredom first – but nothing that the machine cannot. The universal

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Turing machine is now considered an abstract prototype of a general-purpose ‘stored program’ computer – one that can, like any laptop or smart phone today, execute an

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could learn, calculate, store data and execute logical functions – they could, in short, compute. Whether they had ‘proved, in substance, the equivalence of all general Turing machines – man-made or begotten’, as McCulloch later claimed, is a point of contention even today. Von Neumann adopted the terminology and notation of McCulloch and

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the question ‘can machines reproduce?’ but he would be the first to answer it. At the heart of von Neumann’s theory is the Universal Turing machine. Furnished with a description of any other Turing machine and a list of instructions, the universal machine can imitate it. Von Neumann begins by considering what a

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Turing-machine-like automaton would need to make copies of itself, rather than just compute. He argues that three things are necessary and sufficient. First, the machine

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-up, entitled ‘Man Viewed as a Machine’, is barely less sensational than the sci-fi story. ‘Are we, as rational beings, basically different from universal Turing machines?’ he asks. ‘The usual answer is that whatever else machines can do, it still takes a man to build the machine. Who would dare to

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cells (representing ‘1’).19 Adding a control unit that can read from and write in tape cells, von Neumann is able to reproduce a universal Turing machine in two dimensions. He then designs a constructing arm which snakes out to any cell on the grid, stimulates it into the desired state then

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had smuggled a trick question into Gardner’s column to tantalize his readers. The glider was the first piece he needed to build a Universal Turing machine within Life. Like von Neumann’s immensely more complicated cellular automaton, Conway wanted to prove that Life would have the power to compute anything. The

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designed Life organisms capable of executing the basic logical operations and storing data.29 Conway did not bother to finish the job – he knew a Turing machine could be built from the components he had assembled. He had done enough to prove that his automaton could carry out any and all computations

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myself to write you about a mathematical problem, of which your opinion would very much interest me …’ He then launches into the description of a Turing machine, which if ever shown to actually exist, ‘would have consequences of the greatest importance’. Namely, despite Turing’s negative answer to the Entscheidungsproblem, Gödel said

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Economic Behavior, Princeton University Press, Princeton. Petzold, Charles, 2008, The Annotated Turing: A Guided Tour Through Alan Turing’s Historic Paper on Computability and the Turing Machine, Wiley, Hoboken. Poundstone, William, 1992, Prisoner’s Dilemma: John von Neumann, Game Theory and the Puzzle of the Bomb, Doubleday, New York. Reid, Constance, 1986

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Turing’s paper is abridged from Charles Petzold, 2008, The Annotated Turing: A Guided Tour Through Alan Turing’s Historic Paper on Computability and the Turing Machine, Wiley, Hoboken, and ‘Computable Numbers: A Guide’, in Jack B. Copeland (ed.), 2004, The Essential Turing: Seminal Writings in Computing, Logic, Philosophy, Artificial Intelligence, and

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something like this: where P0 means ‘print a 0’, P1 means ‘print a 1’ and R means move right one square. 34. Charles Petzold describes Turing machines that can, for example, add and multiply two binary numbers together. See Petzold, The Annotated Turing. 35. Supplied with a copy of the standard description

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, all the universal machine needs to start work is the initial state of the Turing machine it is imitating and the first symbol that machine is supposed to read. It begins by scanning the standard description on its tape for the

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this information to hand, the universal machine winds its tape back to where it started, prints the next symbol and the new state of the Turing machine it is simulating. Now the universal machine has a new symbol and state to look for in the standard description on its tape. This cyclical

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of Automata and Numerical Analysis, Pergamon Press, Oxford. 37. Briefly, Turing’s strategy is this: he begins by considering whether it is possible for some Turing machine to tell from another machine’s standard description whether it will print digits for ever or halt. A little confusingly, he calls machines that print

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. 28. Quoted in Levy, Artificial Life. 29. These are the AND, OR and NOT operations of Boolean algebra. 30. In 2001, Paul Rendell implemented a Turing machine in Life and, later, a universal version: http://www.rendell-attic.org/gol/tm.htm 31. Quoted in Fred Hapgood, 1987, ‘Let There Be Life

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123 stored-program 120, 121, 122 see also First Draft of a Report on the EDVAC subroutines 119 Turing’s contribution 118–21 the universal Turing machine 118–21, 306–7n35, 307n37 virus, first 236 VNs contribution 122, 125–76, 129–130, 131, 139–140, 308n48 VNs early interests in 79–80

…

237–41, 243 hexagonal packing of circles 237, 238 Life 239–41, 239, 240, 242, 243, 244, 245, 257 Universal Turing Machine within 241, 243 survey of life forms 240 Universal Turing machine 241, 243 cooperative game theory 172–3, 176, 178, 196–7 Copeland, B. Jack 121–2, 307n37 Copenhagen 58, 76 Copenhagen

…

test 91–2 Turing’s influence 121–2, 301n22 unification of quantum mechanics 30, 36, 37–9, 43–9 unique gift 12–13 on universal Turing machine 120 at University of Budapest 11–13 at University of Berlin 12, 39–41 view of the Cold War 208–9 vision of future technical

…

27, 70, 71–2, 77, 85, 85, 98, 100, 104, 132, 133–4, 136, 234, 241, 272, 279, 282 uncertainty principle 33, 69, 294n11 Universal Turing machine 119–20, 121, 229–30, 235, 241, 243, 306–7n35, 307n37 universal wave function, the 58 University of California in Berkeley 78 uranium, instability 77

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a tune into a Pianola. The answer to the multiplication would be output via another tape. Turing imagined a whole series of these so-called Turing machines, each specially designed to tackle a particular task, such as dividing, squaring or factoring. Then Turing took a more radical step. He imagined a machine

…

whose internal workings could be altered so that it could perform all the functions of all conceivable Turing machines. The alterations would be made by inserting carefully selected tapes, which transformed the single flexible machine into a dividing machine, a multiplying machine, or any

…

other type of machine. Turing called this hypothetical device a universal Turing machine because it would be capable of answering any question that could logically be answered. Unfortunately, it turned out that it is not always logically possible

…

to answer a question about the undecidability of another question, and so even the universal Turing machine was unable to identify every undecidable question. Mathematicians who read Turing’s paper were disappointed that Gödel’s monster had not been subdued but, as

…

a consolation prize, Turing had given them the blueprint for the modern programmable computer. Turing knew of Babbage’s work, and the universal Turing machine can be seen as a reincarnation of Difference Engine No. 2. In fact, Turing had gone much further, and provided computing with a solid theoretical

…

basis, imbuing the computer with a hitherto unimaginable potential. It was still the 1930s though, and the technology did not exist to turn the universal Turing machine into a reality. However, Turing was not at all dismayed that his theories were ahead of what was technically feasible. He merely wanted recognition from

…

piece of cryptanalysis, and only Turing, with his unique background in mathematical machines, could ever have come up with it. His musings on the imaginary Turing machines were intended to answer esoteric questions about mathematical undecidability, but this purely academic research had put him in the right frame of mind for designing

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