Introduction to Fuzzy Logic Lecture 2 Introduction “Mathematics that refers to reality is not certain and mathematics that is certain does not refer to reality” Albert Einstein “While the mathematician constructs a theory in terms of ´perfect´objects, the experimental observes objects of which the properties demanded by theory are and can, in the very nature of measurement, be only approximately true” Max Black “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 is what goes out with fuzzy logic” Bart Kosko Introduction (cont.) Uncertainty is produced when a lack of information exists. The complexity also involves the degree of uncertainty. It is possible to have a great deal of data (facts collected from observations or measurements) and at the same time lack of information (meaningful interpretation and correlation of data that allows one to make decisions.) Data Information Knowledge & Intelligence Database Intelligent information systems Knowledge base & AI Introduction (cont.) Knowledge is information at a higher level of abstraction. Ex: Ali is 10 years old (fact) Ali is not old (knowledge) Our problems are: Decision Management Prediction Solutions are: Faster access to more information and of increased aid in analysis Understanding – utilizing information available Managing with information not avaliable Large amount of information with large amount of uncertainty lead to complexity. Avareness of knowledge (what we know and what we do not know) and complexity goes together. Ex: Driving a car is complex, driving in an iced road is more compex, since more knowledge is needed for driving in an iced road. Introduction (cont.) Fuzzy logic provides a systematic basis for representation of uncertainty, imprecision, vagueness, and/or incompletenes. Uncertain information: Information for which it is not possible to determine whether it is true or false. Ex: a person is “possibly 30 years old” Imprecise information: Information which is not available as precise as it should be. Ex: A person is “around 30 years old.” Vague information: Information which is inherently vague. Ex: A person is “young.” Inconsistent information: Information which contains two or more assertions that cannot be true at the same time. Ex: Two assertions are given: “Ali is 16” and “Ali is older than 20” Incomplete information: information for which data is missing or data is partially available. Ex: A person’s age is “not known” or a person is “between 25 and 32 years old” Combination of the various types of such information may also exist. Ex: “possibly young”, “possibly around 30”, etc. Introduction (cont.) UNCERTAINTY (Uncertainty-based information) COMPLEXITY CREDIBILITY (Description-algorithmic infor.) (relevance) USEFULNESS Introduction (cont.) Example: When uncertainties like heavy traffic, unfamiliar roads, unstable wheather conditions, etc. increase, the complexity of driving a car increases. How do we go with the complexity? We try to simplify the complexity by making a satisfactory trade-off between information available to us and the amount of uncertainty we allow. We increase the amount of uncertainty by replacing some of the precise information with vague but more useful information. Introduction (cont.) Examples: Travel directions: try to do it in mm terms (or turn the wheel % 23 left, etc.), which is very precise and complex but not very useful. So replace mm information with city blocks, which is not as precise but more meaningful (and/or useful) information. Parking a car: doing it in mm terms, which is very precise and complex but difficult and very costly and not very useful. So replace mm information with approximate terms (between two lines), which is not as precise but more meaningful (or useful) information and can be done in less cost. Describing wheather of a day: try to do it in % cloud cover, which is very precise and complex but not very useful. So replace % cloud information with vague terms (very cloudy, sunny etc.), which is not as precise but more meaningful (or useful) information. Introduction (cont.) Fuzzy logic has been used for two different senses: In a narrow sense: refers to logical system generalizing crisp logic for reasoning uncertainty. In a broad sense: refers to all of the theories and technologies that employ fuzzy sets, which are classes with imprecise boundaries. The broad sense of fuzzy logic includes the narrow sense of fuzzy logic as a branch. Other areas include fuzzy control, fuzzy pattern recongnition, fuzzy arithmetic, fuzzy probability theory, fuzzy decision analysis, fuzzy databases, fuzzy expert systems, fuzzy computer SW and HW, etc. Introduction (cont.) With Fuzzy Logic, one can accomplish two things: Ease of describing human knowledge involving vague concepts Enhanced ability to develop a cost-effective solution to real- world In another word, fuzzy logic not only provides a cost effective way to model complex systems involving numeric variables but also offers a quantitative description of the system that is easy to comprehend. Introduction (cont.) Fuzzy Logic was motivated by two objectives: First, it aims to alleviate difficulties in developing and analyzing complex systems encountered by conventional mathematical tools. This motivation requires fuzzy logic to work in quantitative and numeric domains. Second, it is motivated by observing that human reasoning can utilize concepts and knowledge that do not have well defined, sharp boundaries (i.e., vague concepts). This motivation enables fuzzy logic to have a descriptive and qualitative form. This is related to AI. Introduction (cont.) Components of Fuzzy Logic Fuzzy Predicates: tall, small, kind, expensive,... Predicates modifiers (hedges): very, quite, more or less, extremely,.. Fuzzy truth values: true, very true, fairly false,... Fuzzy quantifiers: most, few, almost, usually, .. Fuzzy probabilities: likely, very likely, highly likely,... Introduction (cont.) Applications Control: “If the temperature is very high and the presure is decreasing rapidly, then reduce the heat significantly.” Database: “Retrieve the names of all candidates that are fairly young, have a strong background in algorithms, and a modest administrative experience.” Medicine: Hepatitis is characterized by the statement, ‘Total proteins are usually normal, albumin is decreased, - globulins are slightly decreased, -globulins are slightly decreased, -globulins are increased’ Introduction (cont.) Probability theory vs fuzzy set theory: Probability measures the likelihood of a future event, based on something known now. Probability is the theory of random events and is not capable of capturing uncertainty resulting from vagueness of linguistic terms. Fuzziness is not the uncertainty of expectation. It is the uncertainty resulting from imprecision of meaning of a concept expressed by a linguistic term in NL, such as “tall” or “warm” etc. Introduction (cont.) Probability theory vs fuzzy set theory (cont): Fuzzy set theory makes statements about one concrete object; therefore, modeling local vagueness, whereas probability theory makes statements about a collection of objects from which one is selected; therefore, modeling global uncertainty. Fuzzy logic and probability complement each other. Example: “highly probable” is a concept that involves both randomness and fuziness. The behaviour of a fuzzy system is completely deterministic. Fuzzy logic differs from multivalued logic by introducing concepts such as linguistic variables and hedges to capture human linguistic reasoning. Introduction (cont.) Even though the broad sense of fuzzy logic covers a wide range of theories and techniques, its core technique is based on four basic concepts: Fuzzy sets: sets with smooth boundaries; Linguistic variables: variables whose values are both qualitatively and quantitatively described by a fuzzy set; Possibility distribution: constraints on the value of a linguistic variable imposed by assigning it a fuzzy set; and Fuzzy if-then rules: a knowledge representation scheme for describing a functional mapping (fuzzy mapping rules) or a logical formula that generalizes an implication in two-valued logic (fuzzy implication rules). The first three concepts are fundamental for all subareas in fuzzy logic, but the fourth one is also important. • Fuzzy set represent fuzzy logic means to model the uncertainty associated with vagueness, imprecision and lack of information regarding a problem or system. • EG: Consider “short person”. For individual X, short person may be 4’25. For individual Y, short person may be 3’90. • Word “short” is linguistic descriptor & it represent imprecision existing in the system. • Term short provides same meaning to individual X & Y, but it can be seen that they both do not provide unique definition. • The basis of theory lies in making the membership function over a range of real number 0.0 to 1.0. The fuzzy set is characterized by (0.0,0,1.0). • Dr Zadeh proposed set membership idea to make suitable decision when uncertainty occurs. • If we take “short” as a height equal to or less than 4 feet than 3’90 would easily become member of the set short and 4’25 will not become the member of set “short”. • Membership value is “1” it belongs to the set and “0” if it does not belong to set. • Thus membership in a s set is found to be binary i.e. either the element of a set or not. It can be indicated as follow. • is the membership of element