Mathematics

A Mathematical Theory Of Communication

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**Claude Elwood Shannon** (April 30, 1916 – February 24, 2001), an American electrical engineer and mathematician, has been called "the father of information theory",^{[1]} and was the founder of practical digital circuit design theory.

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Shannon was born in Petoskey, Michigan. The first sixteen years of Shannon's life were spent in Gaylord, Michigan, where he attended public school, graduating from Gaylord High School in 1932. While growing up, he worked as a messenger for Western Union. Shannon was a distant relative of Thomas Edison.

In 1932 he entered the University of Michigan, where he took a course that introduced him to the works of George Boole. He graduated in 1936 with two bachelor's degrees, one in electrical engineering and one in mathematics, then began graduate study at the Massachusetts Institute of Technology, where he worked on Vannevar Bush's differential analyzer, an analog computer.

While studying the complicated ad hoc circuits of the differential analyzer, Shannon saw that Boole's concepts could be used to great utility. A paper drawn from his 1937 master's thesis, *A Symbolic Analysis of Relay and Switching Circuits*, was published in the 1938 issue of the *Transactions of the American Institute of Electrical Engineers*. It also earned Shannon the Alfred Noble American Institute of American Engineers Award in 1940. Howard Gardner, of Harvard University, called Shannon's thesis "possibly the most important, and also the most famous, master's thesis of the century".

In this work, Shannon proved that Boolean algebra and binary arithmetic could be used to simplify the arrangement of the electromechanical relays then used in telephone routing switches, then turned the concept upside down and also proved that it should be possible to use arrangements of relays to solve Boolean algebra problems. Exploiting this property of electrical switches to do logic is the basic concept that underlies all electronic digital computers. Shannon's work became the foundation of practical digital circuit design when it became widely known among the electrical engineering community during and after World War II. The theoretical rigor of Shannon's work completely replaced the *ad hoc* methods that had previously prevailed.

Flush with this success, Vannevar Bush suggested that Shannon work on his dissertation at Cold Spring Harbor Laboratory, funded by the Carnegie Institution headed by Bush, to develop similar mathematical relationships for Mendelian genetics, which resulted in Shannon's 1940 PhD thesis at MIT, *An Algebra for Theoretical Genetics.*

Shannon then joined Bell Labs to work on fire-control systems and cryptography during World War II, under a contract with section D-2 (Control Systems section) of the National Defense Research Committee (NDRC).

In 1945, as the war was coming to an end, the NDRC was issuing a summary of technical reports as a last step prior to its eventual closing down. Inside the volume on Fire Control a special essay titled *Data Smoothing and Prediction in Fire-Control Systems*, coauthored by Richard B. Blackman, Hendrik Wade Bode, and Claude Shannon, formally introduced the problem of Fire Control as a special case of *transmission, manipulation and utilization of intelligence*, in other words it modeled the problem in terms of Data and Signal Processing and thus heralded the coming of the information age. Shannon was greatly influenced by this work. It is clear that the technological convergence of the information age was preceded by the synergy between these scientific minds and their collaborators.

In 1948 Shannon published *A Mathematical Theory of Communication* article in two parts in the July and October issues of the *Bell System Technical Journal*. This work focuses on the problem of how best to encode the information a sender wants to transmit. In this fundamental work he used tools in probability theory, developed by Norbert Wiener, which were in their nascent stages of being applied to communication theory at that time. Shannon developed information entropy as a measure for the uncertainty in a message while essentially inventing what became known as the dominant form of "information theory." The book, co-authored with Warren Weaver, *The Mathematical Theory of Communication*, reprints Shannon's 1948 article and Weaver's popularization of it, which is accessible to the non-specialist. Shannon's concepts were also popularized, subject to his own proofreading, in John Robinson Pierce's *Symbols, Signals, and Noise*.

Another notable paper published in 1949 is *Communication Theory of Secrecy Systems*, a major contribution to the development of a mathematical theory of cryptography where he also proved that all theoretically unbreakable ciphers must have the same requirements as the one-time pad. He is also credited with the introduction of Sampling Theory, which is concerned with representing a continuous-time signal from a (uniform) discrete set of samples.

He returned to MIT to hold an endowed chair in 1956.

Outside of his academic pursuits, Shannon was interested in juggling, unicycling, and chess. He also invented many devices, including rocket-powered flying discs, a motorized pogo stick, a wearable computer to predict the result of playing roulette [1], and a flame-throwing trumpet for a science exhibition. One of his more humorous devices was a box kept on his desk called the "Ultimate Machine", based on an idea by Marvin Minsky. Otherwise featureless, the box possessed a single switch on its side. When the switch was flipped, the lid of the box opened and a mechanical hand reached out, flipped off the switch, then retracted back inside the box.

Shannon came to the Massachusetts Institute of Technology (MIT) in 1956 to join its faculty and to conduct work in the Research Laboratory of Electronics (RLE). He continued to serve on the MIT faculty until 1978. To commemorate his achievements, there were celebrations of his work in 2001, and there are currently five statues of Shannon: one at the University of Michigan; one at MIT in the Laboratory for Information and Decision Systems; one in Gaylord, Michigan; one at the University of California at San Diego; and another at Bell Labs. After the breakup of the Bell system, the part of Bell Labs that remained with AT&T was named Shannon Labs in his honor.

Robert Gallager has called Shannon the greatest scientist of the 20th century. According to Neil Sloane, an AT&T Fellow who co-edited Shannon's large collection of papers in 1993, the perspective introduced by Shannon's communication theory (now called information theory) is the foundation of the digital revolution and every device containing a microprocessor or microcontroller is a conceptual descendant of Shannon's 1948 publication^{[2]}: "He's one of the great men of the century. Without him, none of the things we know today would exist. The whole digital revolution started with him."^{[3]}

However, Shannon was oblivious to the marvels of the digital revolution because his mind was ravaged by Alzheimer's disease. His wife mentioned in his obituary that "he would have been bemused" by it all.^{[3]}

Claude Shannon, among the other amusing machines and devices that he created, also built the maze-solving mouse Theseus.

Theseus was created in 1950 and it was a magnetic mouse controlled by a relay circuit that enabled it to move around a maze of 25 squares. Its dimensions were the same as an average mouse.^{[1]} The maze configuration was flexible and it could be modified at will.^{[1]} The mouse was designed to search through the corridors until it found the target. Having travelled through the maze, the mouse would then be placed anywhere it had been before and because of its prior *experience* it could go directly to the target. If placed in unfamiliar territory, it was programmed to search until it reached a known location and then it would proceed to the target, adding the new knowledge to its memory thus *learning*.^{[1]} Shannon's mouse appears to have been the first learning device of its kind.^{[1]}

In 1950 Shannon published a groundbreaking paper on computer chess entitled *Programming a Computer for Playing Chess*. It describes how a machine or computer could be made to play a reasonable game of chess. His process for having the computer decide on which move to make is a minimax procedure, based on an evaluation function of a given chess position. Shannon gave a rough example of an evaluation function in which the value of the black position was subtracted from that of the white position. *Material* was counted according to the usual relative chess piece point value (1 point for a pawn, 3 points for a knight or bishop, 5 points for a rook, and 9 points for a queen). He considered some positional factors, subtracting ½ point for each doubled pawn, backward pawn, and isolated pawn. Another positional factor in the evaluation function was *mobility*, adding 0.1 point for each legal move available. Finally, he considered checkmate to be the capture of the king, and gave it the artificial value of 200 points. Quoting from the paper:

*The coefficients .5 and .1 are merely the writer's rough estimate. Furthermore, there are many other terms that should be included. The formula is given only for illustrative purposes. Checkmate has been artificially included here by giving the king the large value 200 (anything greater than the maximum of all other terms would do).*

The evaluation function is clearly for illustrative purposes, as Shannon stated. For example, according to the function, pawns that are doubled as well as isolated would have no value at all, which is clearly unrealistic.

The reason for assigning checkmate a value higher than the maximum sum of all other terms is so that the minimax procedure will value checkmate above all else and thus it will sacrifice as much material as it has to in order to prevent itself from being checkmated, or to checkmate the opponent. The value is arbitrary — any number larger than the sum of all of the other terms would cause the minimax procedure to give the same result.

Shannon and his wife Betty also used to go on weekends to Las Vegas with M.I.T. mathematician Ed Thorp,^{[4]} and made very successful forays in roulette and blackjack using game theory type methods co-developed with fellow Bell Labs associate, physicist John L. Kelly Jr. based on principles of information theory.^{[5]} They made a fortune, as detailed in the book *Fortune's Formula* by William Poundstone and corroborated by the writings of Elwyn Berlekamp,^{[6]} Kelly's research assistant in 1960 and 1962.^{[7]} Shannon and Thorp also applied the same theory, later known as the *Kelly criterion*, to the stock market with even better results.^{[8]}

Shannon formulated a version of Kerckhoffs' principle as "the enemy knows the system". In this form it is known as **Shannon's Maxim**.

He met his wife Betty when she was a numerical analyst at Bell Labs.

- Alfred Noble Prize, 1940
- Morris Liebmann Memorial Award of the Institute of Radio Engineers, 1949
- Yale University (Master of Science), 1954
- Stuart Ballantine Medal of the Franklin Institute, 1955
- Research Corporation Award, 1956
- University of Michigan, honorary doctorate, 1961
- Rice University Medal of Honor, 1962
- Princeton University, honorary doctorate, 1962
- Marvin J. Kelly Award, 1962
- University of Edinburgh, honorary doctorate, 1964
- University of Pittsburgh, honorary doctorate, 1964
- Institute of Electrical and Electronics Engineers Medal of Honor, 1966
- National Medal of Science, 1966, presented by President Lyndon B. Johnson
- Golden Plate Award, 1967
- Northwestern University, honorary doctorate, 1970
- Harvey Prize, the Technion of Haifa, Israel, 1972
- Royal Netherlands Academy of Arts and Sciences (KNAW), foreign member, 1975
- University of Oxford, honorary doctorate, 1978
- Joseph Jacquard Award, 1978
- Harold Pender Award, 1978
- University of East Anglia, honorary doctorate, 1982
- Carnegie Mellon University, honorary doctorate, 1984
- Audio Engineering Society Gold Medal, 1985
- Kyoto Prize, 1985
- Tufts University, honorary doctorate, 1987
- University of Pennsylvania, honorary doctorate, 1991
- Eduard Rhein Prize, 1991
- National Inventors Hall of Fame inducted, 2004

- Shannon–Fano coding
- Shannon–Hartley law
- Nyquist–Shannon sampling theorem
- Shannon's theorem
- Rate distortion theory
- Information theory
- Confusion and diffusion
- One-time pad
- Shannon switching game
- Shannon number
- Claude E. Shannon Award
- Shannon index

- ^
^{a}^{b}^{c}^{d}^{e}Bell Labs website: "For example, Claude Shannon, the father of Information Theory, had a passion..." **^**C. E. Shannon:*A mathematical theory of communication.*Bell System Technical Journal, vol. 27, pp. 379–423 and 623–656, July and October, 1948- ^
^{a}^{b}Bell Labs digital guru dead at 84 -- Pioneer scientist led high-tech revolution (*The Star-Ledger*, obituary by Kevin Coughlin 27 February 2001) **^**American Scientist online: Bettor Math, article and book review by Elwyn Berlekamp**^**John Kelly by William Poundstone website**^**Elwyn Berlekamp (Kelly's Research Assistant) Bio details**^**Poundstone, William:*Fortune's Formula : The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street***^**William Pounstone website

- Claude E. Shannon:
*A Mathematical Theory of Communication*, Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, 1948. - Claude E. Shannon and Warren Weaver:
*The Mathematical Theory of Communication.*The University of Illinois Press, Urbana, Illinois, 1949. ISBN 0-252-72548-4 - Claude E. Shannon:
*Programming a Computer for Playing Chess*, Philosophical Magazine, Ser.7, Vol. 41, No. 314, March 1950. (Available online under*External links*below) - David Levy:
*Computer Gamesmanship: Elements of Intelligent Game Design*, Simon & Schuster, 1983. ISBN 0-671-49532-1 - Mindell, David A., "Automation's Finest Hour: Bell Labs and Automatic Control in World War II", IEEE Control Systems, December 1995, pp. 72-80.
- David Mindell, Jérôme Segal, Slava Gerovitch, "From Communications Engineering to Communications Science: Cybernetics and Information Theory in the United States, France, and the Soviet Union"
- Walker, Mark (Ed.),
*Science and Ideology: A Comparative History*, Routledge, London, 2003, pp. 66-95. - Poundstone, Willaim,
*Fortune's Formula*, Hill & Wang, 2005, ISNB-13 978-0-8090-4599-0

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