MATHEMATICS, LOGIC AND INFORMATION THEORY

Mathematics

 

Logic

 

Analog and Digital

 

Boolean Logic

 

Probability Theory

 

Information Theory

 

Game Theory

 

Possibility Theory

 

Fuzzy Logic

 

Nash equilibrium graph (http://www.math.ku.dk/"brunke/game.html)

 

 

"Fuzzy sets... bring the reasoning used by computers closer to that used by people." - Lofti Zadeh

 

Mathematics

"What makes society turn is science, and the language of science is math, and the structure of math is logic, and the bedrock of logic is Aristotle, and that's what goes out with fuzzy." (Fuzzy Logic, The Revolutionary Computer Technology That Is Changing Our World, McNeill, Daniel and Freiberger, Paul, Simon & Schuster, 1993, p 45)

 

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Digital computing devices handles numbers by counting discrete units while analog devices represent numbers by measuring continuous quantities. These devices need not be computers. Thermometers, older clocks, rulers, astrolabes and slide rules are all analog devices. During WWII, digital computers were not as fast as analog machines but they did give more precise results and were more easily adapted to different problems. Postwar research focused on digital devices. "...Most electronic computing devices now use digital design. Curiously, the results of digital computations may be displayed in what appears to be an analog form. For example, some electronic calculators plot graphs, and the readings of a scanning tunneling microscope are translated into computer images of whatever surface is studied...In all of these instances, however, fundamental calculations are carried out digitally and the images only appear to be continuous." (Landmarks in Digital Computing, Ceruzzi, Paul and Kidwell, Peggy, Smithsonian Institute Press, 1994)

 

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The basis of western logic comes to us from the Greeks. It started with Aristotle (384-322 B.C.) who has dominated western thought longer than any other secular figure. Some consider him to be the inventor of logic. He created logical classes and formal logic itself from the step-by-step methods of mathematics.

The basis of Aristotle's thought are two axioms. The Law of Contradiction was the first: "A cannot be both B and not-B." "The second was the Law of the Excluded Middle- more properly, the Law of Bivalence. It states, 'A must be either B or not-B'. In the Metaphysics, Arisotle expressed it thus: 'Of any subject, one thing must be either asserted or denied'." (Fuzzy Logic, p 52)

"Fuzzy sets flout both laws. They let a person be jolly in degrees, so she can be partly jolly and not-jolly at once. Hence the intersection of a fuzzy set and its complement is not a void. Overlaps abound. Likewise, the fuzzy union of a set and its complement is the higher of each person's jolliness and not jolliness, and it does not fill the universe of discourse." (Fuzzy Logic, p 53)

"...Silhouette definitions are crucial to Aristotle's logic. Definitions are the hairline. They must cleanly separate items into chair and not-chair. Yet at the outset of De Anima he admits the difficulty of finding any method of definition. It is a remarkable concession, for if we can't specify the hairline, classes blur at the edges and the whole two-valued system becomes an artifice." (Fuzzy Logic, p 56)

 

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George Boole (1815-1864) was a self taught mathematician who made significant contributions to the fields of mathematics and logic. He the was the founder of modern symbolic logic. He reduced logic to boolean algebra in his papers : "The Mathematical Analysis of Logic, Being an Essay Towards a Calculus of Deductive Reasoning" (1847) and "An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probability" (1854).

Boolean algebra is important in logic, the theory of probability, information theory and has led to the design of electronic computers by interpreting boolean combinations as sets of switching circuits. Charles Babbage and his successors were able to design mechanical computing devices that could perform logical tasks because Boole reduced logic to very simple algebraic systems.

Boolean logic is a way to use operators to tell the computer how certain words or data sets relate to one another.

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Claude Shannon in 1948 presented the conception of information theory regarding the design of information processing and transmission systems. "Shannon laid the foundation for the design and analysis of modern communications systems. He proved that nearly error-free information transmission over a noisy communications link is possible by encoding signals prior to transmission over the link and by decoding the received signals."

"Shannon suggested that the amount of information conveyed by the occurrence of an event is related to its uncertainity and was defined to be inversely related to the probability of occurrence of that event. Information theory also provides fundamental limits on the transmission of information, on the extraction of information from the environment, and on the representation of information."

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Fuzzy Logic "resembles human decision making with an ability to generate precise solutions with certain or approximate information."

In 1965, Lofti Zadeh, an Iranian engineer, introduced the term in a paper entitles "Fuzzy Sets." The importance is the simplicity of the idea. In thinking and in life we deal with things in matters of degrees rather than mutually exclusive dichotomies. "How big does a pond have to be to qualify as a lake? How much of an apple do you have to eat for what is left no longer to count as an apple? What amount of hair loss categorizes you as bald? The world is more a place of grey areas than clear choices of black or white, and so is better coped with by systems sensitive to grades rather than discontinuities." (The Quarterly Review of Biology, June 1995, Colin Beer, p 210)

In 1987, a large electronic combine in Japan began using fuzzy logic principles in the design of appliances. Other Asian companies soon followed along developing "smart" refrigerators, washing machines, vacuum cleaners and automatic transmissions.

When General Motors wanted to develop an automatic transmission for the Saturn automobile that would save gas by shifting less and operating less arbitrarily, they turned to fuzzy logic. General Electric, Motorola, and Rockwell are incorporating it into their products. Whirlpool recently won a $30 million dollar prize "for the most energy efficient and environmentally friendly refrigerator." (Financial World, October 10, 1995,Srikumar S. Rao, p 96)

"Fuzzy logic enables programmers to incorporate qualities that frequently accompany human descriptions such as: 'slightly', 'many', 'few', 'most', and 'about' into decision making. It is easier, faster, and less expensive to design a system because it is more intuitive and takes fewer, simpler rules. As a result, it is easier to operate, maintain, and modify. Fuzzy logic also reduces the hardware costs because the code can be smaller, requires less memory, and the programs run faster." (Journal of Systems Management, March/April 1996, Mohsen Attaran, p 11)

Fuzzy logic operates on a scale of 1 to 0. Therefore the answer is a measure of degree rather than an absolute of 0 or 1 as in a typical binary computing system application. "Take short men. Suppose Bob is 5 feet 4 inches. Is he short? Conventional logic would answer that by deciding arbitrarily that any man shorter than, say, 5 feet 5 inches is short. In contrast, fuzzy logic asks, 'How short is Bob'? The answer it comes up with is a partial membershiup in the fuzzy set short men, such as 0.7. So within a scale of 0 to 1 for shortness, Bob ranks 0.7." (Financial World, p 92)

"A typical fuzzy logic system consists of three components - the fuzzifier, the rule base (fuzzy associative memory) and the defuzzifier. The fuzzifier takes physical input values or signals and maps them to fuzzy variables. All of the fuzzy input variables are interpreted by the rule base to generate fuzzy output variables. Finally, the defuzzifier translates the fuzzy outputs into physical values that can be used to drive a process. The fuzzy rule base can be tuned using operator experience to achieve the desired performance." (http://www.controladv.com/fuzzy.html)

Fuzzy logic has proven very successful in the area of practical applications including cement kiln control which was the first major commercial application. It required monitoring four internal kiln states, controlling four sets of operations and managing 40 to 50 "rules of thumb" about their interrelationships. "Other applications which have benefited through the use of fuzzy systems theory have been information retrieval systems, a navigation system for automatic cars, a predictive fuzzy-logic controller for automatic operation of trains, laboratory water level controllers, controllers for robot arc-welders, feature-definition controllers for robot vision, graphics controllers for automated police sketchers and more." (http://www.ortech-engr.com/fuzzy/tutor.txt) Fuzzy logic controllers are essential to expert systems in the areas of decision support systems, financial planners, soybean pathogen diagnostic systems and meterological systems.

Fuzzy systems are also currently used in decision support systems in the areas of bid preparation, investment portfolio, stock market analysis, database management, medical diagnostic systems, pattern recognition, production scheduling, cost-volume-profit analysis and counterfeit bank note detectors. "Worldwide fuzzy-logic revenues were $2 billion in 1994, a number that will reach $8 billion by 1998." (Byte, June 1995, Randy Cronk, p 34)

"Fuzzy logic is not restricted to control applications but is also applicable to any system that requires logical reasoning, thus making such techniques extremely versatile." (www.controladv.com)

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