Overview: The works of Alan Turing ( )

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Overview: The works of Alan Turing (1912-1954) Dan Hallin <danhallin@yahoo.

1 Overview: The works of Alan Turing ( ) Dan Hallin Introduction Course in Computer Science (CD5600) The methodology of Science in Technology (CT3620) Mälardalen University 1

2 Abstract The main objective of the report is to give a general and chronological overview of the more important scientific contributions of Alan M. Turing ( ). Turing did major works in a wide array of areas: logic, mathematics, biology, philosophy, cryptanalysis, and in the then undefined areas which later became known as computer science, cognitive science, artificial intelligence and artificial life. Each of the works included is set in a background and explained in brief. The works include the classic On computable numbers where the so called universal Turing machine was first defined. Next is his PhD thesis where he defined ordinal logics and then his secret work for the British government breaking the German Enigma codes during World War II. After the war is his work of constructing the first nontheoretical universal Turing machines, stored-programme computers. At the end comes his works in the areas of artificial intelligence and artificial life. 2

3 Contents Overview: The works of Alan Turing ( )... 1 Abstract... 2 Contents... 3 Works of Alan Turing (chronologically) On computable numbers, with an application to the Entscheidungsproblem4 Background... 4 The systematic method... 4 The Turing machine... 4 The Church-Turing thesis Systems of logic based on ordinals... 5 Background... 5 The thesis Work at Bletchley Park... 6 Background... 6 Turing s contributions The first computers Artificial Intelligence Artificial Life... 8 Conclusions... 8 References

4 Works of Alan Turing (chronologically) 1936 On computable numbers, with an application to the Entscheidungsproblem Background In 1900 David Hilbert proposed the making of a complete, consistent, decidable formal system [Copeland04] for mathematics, which could express the whole thought content of mathematics in a uniform way [Copeland04]. In other words, it would be possible to decide the true or false status (this requires consistency) of any (completeness) statement by determining whether it is provable in the system (decidability) [Copeland04] and this would have to be possible to do with a systematic method. [Copeland04] When Kurt Gödel published his first incompleteness theorem he proved that such a system, if consistent, must be incomplete. This means that there will be true statements which cannot be proven within the system. The theorem was a devastating blow to the work of realizing Hilbert s system. It did however leave the so called Entscheidungsproblem (decidability problem). It is the problem of finding a systematic method which can decide whether any statement is provable in the system or not. In a lecture course held by M. H. A. Newman in 1935 Turing learned about this problem. [Copeland04] [Hodges95] [Wiki05d] The systematic method A systematic method... is any mathematical method of which all the following are true: The method can, in practice or in principle, be carried out by a human computer working with paper and pencil; The method can be given to the human computer in the form of a finite number of instructions; The method demands neither insight nor ingenuity on the part of the human being carrying it out; The method will definitely work if carried out without error; The method produces the desired result in a finite number of steps; or if the desired result is some infinite sequence of symbols..., then the method produces each individual symbol in the sequence in some finite number of steps. [Copeland04] In the definition above the terms insight and ingenuity are not explicitly defined. To tackle the problem of decidability Turing had to create a rigorously defined expression of a systematic method. He did this using an abstract machine with a basic set of simple rules. Turing referred to it as a computing machine, now called the Turing machine. [Copeland04] The Turing machine The machine Turing proposed consisted of a scanner and an unlimited tape divided into squares. The squares could contain either a blank, or any one symbol from a finite alphabet. The scanner could move one square at a time to the right or left along the tape, and read the symbol in its current square. The scanner could also erase the symbol in the square, or replace it with another. Finally the scanner had the ability to change into a finite number of different states at any time, the states defining the behaviour of the machine, moving, printing, erasing and changing state, in accordance to the scanned squares. Turing argued that this theoretical machine would be capable of carrying out any systematic method. [Copeland04] [Hodges04] [Turing36] 4

5 A special machine was a machine with a list of states such that it could carry out one certain method. Turing was able to show that a special machine capable of reading its states from the tape would be able to simulate any other special machine, this was called the universal computing machine. [Copeland04] [Turing36] [Wiki05b] The Church-Turing thesis In April 1936 Alonzo Church published A note on the Entsheidungsproblem where he stated that to each [systematic] method there corresponds a lambda-definable function [Copeland04]. Turing who had not yet published his results by this time had to reference to Church s work, but was able to prove that the universal computing machine was equivalent to his work. The universal computing machine was however a more intuitive solution to the problem and it has become the generally accepted notion, even if the thesis is called the Church-Turing thesis. [Copeland04] Both Church and Turing could with these systems independently prove that no consistent formal system of arithmetic is decidable [Copeland04], nor was any formal system based on functional calculus (now called first order predicate calculus ), which includes many important mathematical systems. [Copeland04] [Turing36] 1938 Systems of logic based on ordinals Background At this time Princeton University in USA was an intellectual centre, much due to the scientists who had been exiled from Nazi Germany. This was where Alonzo Church was active, and Turing had unsuccessfully applied for a Visiting Fellowship there in When Alonzo Church s parallel work was published in 1936 Turing decided yet again to go there and arrived in September 1936, intending to stay for one year. In the mid of 1937 he was offered a Visiting Fellowship, and came to stay another year. During this period he wrote his PhD thesis, being supervised by Church. [Copeland04] [Hodges04] The supervision of Church was not all for the better, Church suggested additions to the thesis which made it expand[...] to an appalling length [Turing38]. Moreover Turing also choose to use Church s more abstruse lambda-calculus notation making the thesis less accessible than if he had chosen to continue using Turing machines. [Copeland04] [Hodges04] The thesis In this work Turing tries to work around the implications of his own and Gödels work. Gödel had shown that there would always be statements which could be intuitively realized as true, but were not provable within the system. Turing wanted to extend logical systems by including each unprovable statement as a new rule and thus create new systems recursively. Each such new system would infinitely also include unprovable statements, but each system would also be more complete than the previous, capable of proving more statements. Turing called this ordinal logic. [Copeland04] Gödel had shown there was no systematic method of finding the unprovable true statements. As the Turing machine was equivalent to the systematic method, Turing had to introduce a new concept in the machine, the oracle, the new machine being called the o-machine. The oracle had the capability of answering questions which no Turing-machine could simulate, that is, the ones requiring intuition. Turing let the oracles function remain undefined - We 5

6 shall not go any further into the nature of this oracle apart from saying it cannot be a machine. [Turing39]. [Copeland04] Work at Bletchley Park Background After returning from America in September 1938 Turing went back to work at King s college. He was contacted by the Government Code and Cypher School (GC&CS) in London and secretly worked part-time on decrypting the Enigma cipher used by the German forces. The Enigma was a machine, looking much like an old-fashioned type-writer, which could be used to encrypt text messages before being sent over radio. The most important part of the Enigma were three wheels, which for each keystroke rotated and thus changed the internal wiring of the machine and the letter produced. In addition somewhere between 3 and 10 pairs of keys could be swapped onto each other. To decode a message, one needed not only the machine, but also the settings of the wheels and the swapped keys. The settings changed at regular intervals according to monthly tables. [Copeland04] Turing s contributions The German Army s routine of unnecessarily sending the current days wheel-setup as an uncoded preamble to the encrypted message, together with a twicely encoded message wheelsetup (called indicator) had made it relatively easy to decrypt messages. The British knew this routine was likely to change sooner or later and set out to find methods of decrypting messages without having this advantage. The day after the declaration of war on 3 September 1939, Turing and several other scientists moved to work at GS&CS, now in Bletchley Park outside London. [Copeland04] In late 1939 Turing did the logical design of an improved version of the Polish Bomba, a machine used in finding the correct indicators. The new machine was named Bombe, it was a logic machine consisting of 36 replicas of an Enigma. It could do the same work as the Bomba, but also had the ability of searching for something Turing called cribs. Cribs allowed the personnel to guess at a longer sentence within an encoded message and the machines would try to find a correct wheel-position given this. Guessing was possible by looking at stereotypical messages, for example reports from weather stations. Turing also worked out a method of finding which letters had been swapped with each other, by finding logical loops within the system and coupling the Bombes to mimic those. [Copeland04] In 1940 a lot of the traffic sent by the German Air Force could be decrypted. The German navy however used a more advanced technique for encrypting the indicator, and as it was thought of as impossible to break, no one was assigned to breaking to it. Turing quite enjoyed the idea of having a problem no one else was working on. By looking at old traffic where all the other message settings were known, he was soon able to figure the technique behind the indicators. They were hand-encrypted an extra time according to given tables and the tables were to always be destroyed before abandoning a ship. [Copeland04] In March and May 1941, captures of German ships brought in material which made it possible to reconstruct the tables. In June and July they were for the first time able to decrypt messages within hours of reception. This came at a critical time; the German submarines had sunk so many convoy ships headed for Britain that there was not enough food or oil. The convoys could now have their routes changed, and during the first 23 days of June not a single convoy ship sank from submarine attack. In July 1941 Turing and two of his co-workers were 6

7 summoned to the Foreign Office in London to be awarded for their services with 200 each, more than half a years wage for Turing. At the end of the war he received the Order of the British Empire. [Copeland04] [Mohan45] When operations were set up and running in Bletchley park, there was no longer enough work to occupy Turing, and instead he worked as a general scientific consultant. He only returned shortly in 1942 to help in breaking Tunny, a new system the Germans used in high-level army communications. In November 1942 he went on a journey to the USA to oversee the production of Bombes and the set up of an enciphered speech system for Churchill- Roosevelt messages. After returning in March the next year he spent the rest of the war on designing a secure voice communication device. [Copeland04] [Hodges95] [Wiki05a] The first computers At the end of the war Turing had seen the capabilities of Colossus, the special-purpose electronic digital computer built at Bletchley Park and had himself studied electronics. In the autumn of 1945 Turing was recruited to the Mathematics Division at the National Physics Laboratory (NPL) in London to design a stored-programme digital computer. He spent the end of 1945 writing a detailed report of the Automatic Computing Engine (ACE) and it was accepted in March [Copeland04] [Hodges04] His complex ACE design was cut down severely for a smaller pilot-version. The building of the ACE was not done in NPL, but by the engineers at the Post Office Research Station, where it was not given a high priority. The different sites also meant there was less communication between the designers and the implementers. [Copeland04] [Hodges04] Disheartened by the slow progress, Turing left NPL in autumn 1947 to go to Cambridge for a sabbatical year, from which he was never to return. In May 1948 Newman offered him a job as Deputy Director of the Computing Machine Laboratory at Manchester University where he came to design input and programming systems for an expanded version of the Manchester Baby, which had been the world s first stored-programme computer. [Copeland04] [Hodges04] Artificial Intelligence The first records of Turing mentioning intelligence in machines date as early as 1941, while he was working on the Enigma codes. He mentioned such things as problem-solving machines and learning machines. He designed chess-playing programs, and in lack of computers to run them on he drew out the operations by means of paper-and-pencil. [Copeland04] In 1948 Turing finished a report he called Intelligent Machinery. It was a far-sighted work, introducing many of the concepts which later became central to Artificial Intelligence (AI). His director, Sir Charles Darwin however thought of it as a school boy s essay [Copeland04] and Turing never published his ideas. Some of them were not discovered until reinvented by others. [Copeland04] Influenced by his experiences during the war, where machines searched for correct keys, in his work he hypothesized that intellectual activity consists mainly of various kinds of search [Copeland04]. He further defined logic-based problem-solving, and an early version of the imitation game, now known as the Turing test. He also proposed a system with a network of neurons, capable of learning. He claimed to have a proof (which has never been found) that such a network could be setup to learn to become a universal computing 7

8 machine (with a finite amount of memory), and proposed this to be similar to the functioning of an infants brain. [Copeland04] [Turing48] His next major contribution came with the publishing of his essay Computing machinery and Intelligence in In this work he looks at the definitions of thinking and intelligence, and how intelligence can be achieved by learning systems. This was also where he defined the complete setup of the conversational imitation game, where a computer taking the place of a person will be able to succeed equally well as a person. [Copeland04] [Hodges04] [Turing50] Artificial Life When Turing got his hands on the Ferranti Mark I, the first commercial electronic generalpurpose computer he set out into a new field. He wanted to study the generation of organization and pattern in biological structures. He was convinced that those were results of physical and mathematical laws and that the processes behind them could be simulated on the computer. In his 1952 publication The Chemical Basis of Morphogenesis he proposed the reaction-diffusion model, a mechanism capable of explaining the creation of animal patterns such as stripes, flecks and spots. Turing s ideas were so complex the computer power to execute them was not available until the early 80 s. When Turing died in 1954 he left behind material in this area which is yet to be understood. [Copeland04] [Hodges04] Conclusions The report shows Alan Turing to be a multifaceted man. While educated as a mathematician, he was able to commence and succeed in any area of logics interesting him. During his short life he made great contributions in the areas of logic, mathematics, philosophy, biology, cryptanalysis and he more or less came to define the areas of computer science, cognitive science, artificial intelligence and artificial life. 8

9 References [Copeland04] B. Jack Copeland, The essential Turing: seminal writings in computing, logic, philosophy, artifical intelligence, and artificial life plus the secrets of Enigma, Oxford University Press, ISBN , [Hodges95] Andrew Hodges, Alan Turing: a short biography, Alan Turing, at Short Turing Biography < (7 Oct 2005) [Hodges04] Andrew Hodges, The Alan Turing Internet Scrapbook, Alan Turing, at Alan Turing Internet Scrapbook < (7 Oct 2005) [Mohan45] Patrick Mohan, History of Hut 8 to December 1941, as reprinted in [Copeland04], [Turing36] A. M. Turing, On computable numbers, with an application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, Series 2, 42, [Turing38] A. M. Turing, Turing s letters home, as reprinted in [Copeland04], May [Turing39] A. M. Turing, Systems of logic based on ordinals, Proceedings of the London Mathematical Society, Series 2, 45, As reprinted in [Copeland04]. [Turing48] A. M. Turing, Intelligent Machinery, as reprinted in [Copeland04], [Turing50] A. M. Turing, Computing machinery and Intelligence, Mind, vol. LIX, no. 236, p , As reprinted in [Copeland04]. [Wiki05a] Alan Turing Wikipedia, the free encyclopedia, Alan Turing, at Wikipedia, the free Encyclopedia < (7 Oct 2005) [Wiki05b] Church Turing thesis Wikipedia, the free encyclopedia, Church Turing thesis, at Wikipedia, the free Encyclopedia < (7 Oct 2005) [Wiki05c] Turing machine Wikipedia, the free encyclopedia, Turing Machine, at Wikipedia, the free Encyclopedia < (7 Oct 2005) [Wiki05d] Entscheidungsproblem Wikipedia, the free encyclopedia, Entscheidungsproblem, at Wikipedia, the free Encyclopedia < (14 Oct 2005) 9

Alan Turing and the Enigma of Computability

Alan Turing and the Enigma of Computability http://kosmoi.com/technology//computer/turing/ Alan Matheson Turing, b. June 23, 1912, d. June 7, 1954, was a British mathematician who conceived of a machine