The Turing Machine

By 1933 Turing had already introduced himself to Russell and Whitehead's Principia Mathematica and so to the then arcane area of mathematical logic. Bertrand Russell had thought of logic as a solid foundation for mathematical truth, but many questions had since been raised about how truth could be captured by any formalism. In particular, in 1931 Goedel had shattered Russell's picture by showing the incompleteness of mathematics: the existence of true statements about numbers which could not be proved by the formal application of set rules of deduction. In 1935, Turing learnt from the lecture course of the Cambridge topologist M. H. A. Newman that a further question, posed by Hilbert, still lay open. It was the question of Decidability, the Entscheidungsproblem. Could there exist, at least in principle, any definite method or process by which all mathematical questions could be decided?

Alan Turing in 1934

To answer such a question needed a definition of 'method' which would be not only precise but compelling. This is what Turing supplied. Seizing on the idea of a process applied mechanically, he formulated the concept of a machine working on a paper tape on which symbols were printed, and asked what was the most general type of process such a machine could perform. He showed that this was equivalent to what could ever be achieved by a person working on a set of logical instructions. More daringly, he argued that it was also equivalent to what could be achieved by the human brain, assuming a finite number of possible states of mind.

This triple correspondence between logical instructions, the action of the mind, and a machine which could in principle be embodied in a practical physical form, was Turing's definitive contribution. Having made this novel definition of what should count as a 'definite method' --- in modern language, an algorithm --- it was not too hard to answer Hilbert's question in the negative: no such decision procedure exists.

In April 1936 he showed his result to Newman; but at the same moment the parallel conclusion of the American logician Alonzo Church became known, and Turing was robbed of the full reward for his originality. His paper, On Computable Numbers with an application to the Entscheidungsproblem, had to refer to Church's work, and was delayed until August 1936. However it was seen at the time that Turing's approach was original and different; Church relied upon an assumption internal to mathematics, rather than appealing to operations that could actually be done by real things or people in the physical world. Subsequently, the concept of the Turing machine has become the foundation of the modern theory of computation and computability.

Turing worked in isolation from the powerful school of logical theory centred on Church at Princeton University, and his work emerged as that of a complete outsider. One can only speculate, but it looks as if Turing found in the concept of the Turing machine something that would satisfy the fascination with the problem of Mind that Christopher Morcom had sparked; his total originality lay in seeing the application of mathematical logic to a problem originally seen as one of physics. In this paper, as in so many aspects of his life, Turing made a bridge between the logical and the physical worlds, thought and action, which crossed conventional boundaries.

His work introduced a concept of immense practical significance: the idea of the Universal Turing Machine. The concept of 'the Turing machine' is like that of 'the formula' or 'the equation'; there is an infinity of possible Turing machines, each corresponding to a different 'definite method' or algorithm. But imagine, as Turing did, each particular algorithm written out as a set of instructions in a standard form. Then the work of interpreting the instructions and carrying them out is itself a mechanical process, and so can itself be embodied in a particular Turing machine, namely the Universal Turing machine. A Universal Turing machine can be made do what any other particular Turing machine would do, by supplying it with the standard form describing that Turing machine. One machine, for all possible tasks.

It is hard now not to think of a Turing machine as a computer program, and the mechanical task of interpreting and obeying the program as what the computer itself does. Thus, the Universal Turing Machine embodies the essential principle of the computer: a single machine which can be turned to any well-defined task by being supplied with the appropriate program.

Additionally, the abstract Universal Turing Machine naturally exploits what was later seen as the 'stored program' concept essential to the modern computer: it embodies the crucial twentieth-century insight that symbols representing instructions are no different in kind from symbols representing numbers. But computers, in this modern sense, did not exist in 1936. Turing created these concepts out of his mathematical imagination. Only nine years later would electronic technology be tried and tested sufficiently to make it practical to transfer the logic of his ideas into actual engineering. In the meanwhile the idea lived only in his mind.

In common with other outstanding young scientists, Turing spent two years at Princeton University enrolled as a graduate student. He arrived in September 1936. On Computable Numbers... was published at the very end of 1936 and attracted some attention; by the time he left, the idea had come to the attention of the leading Hungarian-American mathematician John von Neumann. But Turing certainly did not shoot to fame. He worked on showing that his definition of computability coincided with that of Church; on an extension of his ideas (Ordinal Logics) which provided a Ph.D. thesis; also on algebra and number theory. True to the concreteness of the Turing machine, however, he also spent time making a cipher machine based on using electromagnetic relays to multiplying binary numbers. Even then he saw a link from 'useless' logic to practical computation. Although not one of the political intellectuals of the 1930s, Turing followed current events and was influenced in studying ciphers by the prospect of war with Germany.