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Fuzzy logic was developed by Lotfi A. Zadeh in the 1960s in order to provide mathematical rules and functions which permitted natural language queries. Fuzzy logic provides a means of calculating intermediate values between absolute true and absolute false with resulting values ranging between 0.0 and 1.0. With fuzzy logic, it is possible to calculate the degree to which an item is a member. For example, if a person is .83 of tallness, they are " rather tall. " Fuzzy logic calculates the shades of gray between black/white and true/false. Fuzzy logic is a super set of conventional (or Boolean) logic and contains similarities and differences with Boolean logic. Fuzzy logic is similar to Boolean logic, in that Boolean logic results are returned by fuzzy logic operations when all fuzzy memberships are restricted to 0 and 1. Fuzzy logic differs from Boolean logic in that it is permissive of natural language queries and is more like human thinking; it is based on degrees of truth.
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Fuzzy logic may appear similar to probability and statistics as well. Although, fuzzy logic is different then probability even though the results appear similar. The probability statement, " There is a 70% chance that Bill is tall" supposes that Bill is either tall or he is not. There is a 70% chance that we know which set Bill belongs. The fuzzy logic statement, " Bill's degree of membership in the set of tall people is .80 " supposes that Bill is rather tall. The fuzzy logic answer determines not only the set which Bill belongs, but also to what degree he is a member. There are no probability statements that pertain to fuzzy logic. Fuzzy logic deals with the degree of membership. Fuzzy logic has been applied in many areas; it is used in a variety of ways. Household appliances such as dishwashers and washing machines use fuzzy logic to determine the optimal amount of soap and the correct water pressure for dishes and clothes. Fuzzy logic is even used in self-focusing cameras. Expert systems, such as decision-support and meteorological systems, use fuzzy logic. Fuzzy logic has many varied applications
A fuzzy set is a set whose elements have degrees of membership. That is, a member of a set can be full member (100% membership status) or a partial member (eg. less than 100% membership and greater than 0% membership). To fully understand fuzzy sets, one must first understand traditional sets.
A traditional or crisp set can formally be defined as the following:
A subset U of a set S is a mapping from the elements of S to the elements of the set {0,1}. This is represented by the notation: U: S-> {0,1}
The mapping is represented by one ordered pair for each element S where the first element is from the set S and the second element is from the set {0,1}. The value zero represents non-membership, while the value one represents membership.
Essentially this says that an element of the set S is either a member or a non-member of the subset U. There are no partial members in traditional sets.
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Here is an example of a traditional set: Consider a set X that contains all the real numbers between 0 and 10 and a subset A of the set X that contains all the real numbers between 5 and 8. Subset A is represented in the figure below. In the figure, the interval on the x-axis between 5 and 8 has y-value of one. This indicates that any number in this interval is a member of the subset A. Any number that has a y-value of zero is considered to be a non-member of the subset A.
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Again a fuzzy set is a set whose elements have degrees of membership. These can formally be defined as the following: A fuzzy subset F of a set S can be defined as a set of ordered pairs. The first element of the ordered pair is from the set S, and the second element from the ordered pair is from the interval [0,1]. The value zero is used to represent non-membership; the value one is used to represent complete membership, and the values in between are used to represent degrees of membership.
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EXAMPLE 1 Here is an example describing a set of young people using fuzzy sets. In general, young people range from the age of 0 to 20. But, if we use this strict interval to define young people, then a person on his 20th birthday is still young (still a member of the set). But on the day after his 20th birthday, this person is now old (not a member of the young set). How can one remedy this? By RELAXING the boundary between the strict separation of young and old. This separation can easily be relaxed by considering the boundary between young and old as "fuzzy". The figure below graphically illustrates a fuzzy set of young and old people. Notice in the figure that people whose ages are >= zero and <= 20 are complete members of the young set (that is, they have a membership value of one). Also note that people whose ages are > 20 and < 30 are partial members of the young set. For example, a person who is 25 would be young to the degree of 0.5. Finally people whose ages are >= 30 are non-members of the young set.
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Membership Functions A membership function is a mathematical function which defines the degree of an element's membership in a fuzzy set. The best way to illustrate this concept is with an example. This example describes a fuzzy set for tallness. Below in the membership function for tallness. tall(x)= { 0, if height(x) < 5ft, (height(x)-5ft)/2, if 5ft <= height(x) <= 7ft, 1, if height(x) > 7ft } Essentially this function calculates the membership value of a certain height. For example, if a person is less 4'9", then this person has a membership value of 0.0 and thus is not a member of the set tall. If a person is 7'6", then this person has a membership value of 1.0 and thus is a member of the set tall. Finally, if a person is 5'5", then this person has a membership value of 0.21 and is a partial member of the set tall.
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Below is a graphical representation of the fuzzy set for tallness.
Now that we understand what fuzzy sets and membership functions are, we can discuss three basic operation on sets: negation, intersection, and union of fuzzy sets. In L.A. Zadeh first paper, he formally defined these operations in the following mann er:
Negation
membership_value(not x)= 1- membership_value(x) where x is the fuzzy set being negated
Intersection
membership_value(x and y) = minimum( membership_value(x), membership_value(y) ) where x and y are the fuzzy set being negated
Union
membership_value(x or y) = maximum (membership_value(x), membership_value(y) ) where x and y are the fuzzy set being negated
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Negation In this figure, the red line is a fuzzy set. To negate this fuzzy set, subtract the membership value in the fuzzy set from one. For example, the membership value at 5 is one. In the negation, the membership value at 5 would be zero (1-1=0). For exa mple, if the membership value is 0.4. In the negation, the membership value would be 0.6 (1-0.4=0.6). Put the mouse over the image to see the negation of the fuzzy set (blue curve).
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Intersection In this figure, the red and green lines are fuzzy sets. To find the intersection of these sets take the minimum of the two membership values at each point on the x-axis (see the formal definition above). For example, in the figure the red fuzzy set has a membership of ZERO when x = 4 and the green fuzzy set has a membership of ONE when x = 4. The intersection would have a membership value of ZERO when x = 4 because the minimum of zero and one is zero.
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Union In this figure, the red and green lines are fuzzy sets. To find the union of these sets take the maximum of the two membership values at each point on the x-axis (see the formal definition above). For example, in the figure the red fuzzy set has a membership of ZERO when x = 4 and the green fuzzy set has a membership of ONE when x = 4. The union would have a membership value of ONE when x = 4 because the maximum of zero and one is one.
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The Concept of Hedging Much has been made about the relationship of Fuzzy Logic to the human thought process and the ability to handle imprecise conditions that may arise. One of the terms frequently seen in the Fuzzy Logic literature is the concept of Hedging . Hedging can be described as the modifiers to a certain set, much like the way adjectives and adverbs modify statements in the English language. When referring to a fuzzy set, hedges are used to adjust the characteristics of that fuzzy set by either: Approximating Complementing Diluting Intensifying
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Some specific words and their effect on the fuzzy set include: In general, when a hedge is used to dilute a set, the set is expanded. When a set is intensified with a hedge, the set is compressed. Intensify the set
Automobile and other vehicle subsystems, such as automatic transmissions, ABS and cruise control (e.g. Tokyo monorail)
Air conditioners
The Massive engine used in the Lord of the Rings films, which helped huge scale armies create random, yet orderly movements
Cameras
Digital image processing, such as edge detection
Rice cookers
Dishwashers
Elevators
Washing machines and other home appliances
Video game artificial intelligence
Language filters on message boards and chat rooms for filtering out offensive text
Pattern recognition in Remote Sensing
Fuzzy logic has also been incorporated into some microcontrollers and microprocessors, for instance, the Freescale 68HC12.
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Bus Time Tab les How accurately do the schedules predict the actual travel time on the bus? Bus schedules are formulated on information that does not remain constant. They use fuzzy logic because it is impossible to give an exact answer to when the bus will be at a certain stop. Many unforseen incidents can occur. There can be accidents, abnormal traffic backups, or the bus could break down. An observant scheduler would take all these possibilities into account, and include them in a formula for figuring out the approximate schedule. It is that formula which imposes the fuzziness.
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Predicting genetic traits Genetic traits are a fuzzy situation for more than one reason. There is the fact that many traits can't be linked to a single gene. So only specific combinations of genes will create a given trait. Secondly, the dominant and recessive genes that are frequently illustrated with Punnet squares, are sets in fuzzy logic. The degree of membership in those sets is measured by the occurrence of a genetic trait. In clear cases of dominant and recessive genes, the possible degrees in the sets are pretty strict. Take, for instance, eye color. Two brown-eyed parents produce three blue-eyed children. Sounds impossible, right? Brown is dominant, so each parent must have the recessive gene within them. Their membership in the blue eye set must be small, but it is still there. So their children have the potential for high membership in the blue eye set, so that trait actually comes through. According to the Punnet square, 25% of their children should have blue eyes, with the other 75% having brown.
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Temperature control (heating/cooling) I don't think the university has figured this one out yet ;-) The trick in temperature control is to keep the room at the same temperature consistently. Well, that seems pretty easy, right? But how much does a room have to cool off before the heat kicks in again? There must be some standard, so the heat (or air conditioning) isn't in a constant state of turning on and off. Therein lies the fuzzy logic. The set is determined by what the temperature is actually set to. Membership in that set weakens as the room temperature varies from the set temperature. Once membership weakens to a certain point, temperature control kicks in to get the room back to the temperature it should be.
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Auto-Focus on a camera How does the camera even know what to focus on? Auto-focus cameras are a great revolution for those who spent years struggling with "old-fashioned" cameras. These cameras somehow figure out, based on multitudes of inputs, what is meant to be the main object of the photo. It takes fuzzy logic to make these assumptions. Perhaps the standard is to focus on the object closest to the center of the viewer. Maybe it focuses on the object closest to the camera. It is not a precise science, and cameras err periodically. This margin of error is acceptable for the average camera owner, whose main usage is for snapshots. However, the "old-fashioned" manual focus cameras are preferred by most professional photographers. For any errors in those photos cannot be attributed to a mechanical glitch. The decision making in focusing a manual camera is fuzzy as well, but it is not controlled by a machine.
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Medical diagnoses How many of what kinds of symptoms will yield a diagnosis? How often are doctors in error? Surely everyone has seen those lists of symptoms for a horrible disease that say "if you have at least 5 of these symptoms, you are at risk". It is a hypochondriac's haven. The question is, how do doctors go from that list of symptoms to a diagnosis? Fuzzy logic. There is no guaranteed system to reach a diagnosis. If there were, we wouldn't hear about cases of medical misdiagnosis. The diagnosis can only be some degree within the fuzzy set.
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Predicting travel time This is especially difficult for driving, since there are plenty of traffic situations that can occur to slow down travel. As with bus timetabling, predicting ETA's is a great exercise in fuzzy logic. That's why it is called an estimated time of arrival. A major player in predicting travel time is previous experience. It took me six hours to drive to Philadelphia last time, so it should take me about that amount of time when I make the trip again. Unfortunately, other factors are not typically considered. Weather, traffic, construction, accidents should all be added into the fuzzy equation to deliver a true estimate.
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Antilock Braking System It's probably something you hardly think about when you're slamming on the brakes in your car The point of an ABS is to monitor the braking system on the vehicle and release the brakes just before the wheels lock. A computer is involved in determining when the best time to do this is. Two main factors that go into determining this are the speed of the car when the brakes are applied, and how fast the brakes are depressed. Usually, the times you want the ABS to really work are when you're driving fast and slam on the brakes. There is, of course, a margin for error. It is the job of the ABS to be "smart" enough to never allow the error go past the point when the wheels will lock. (In other words, it doesn't allow the membership in the set to become too weak.)
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http://www.mathworks.com/access/helpdesk/help/toolbox/ fuzzy/fp351dup8.html http://en.wikipedia.org/wiki/Neural_network http://www.dementia.org Fuzzy/Neurofuzzy Logic [online] Neurosciences. Available from internet: < http://www.neurosciences.com/nn_fzy.htm >. Goebel, Greg. An Introduction to Fuzzy Control Systems [ online ] 23 December 1995.[ cited 24 October 1999 ]. Available from the World Wide Web: < http://www.isis.ecs.soton.ac.uk/research/nfinfo/fuzzycontrol.html >.
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