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Fuzzy Logic Yeni Herdiyeni http://www.cs.ipb.ac.id/~yeni Departemen Ilmu Komputer IPB 1 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 2 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 3 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. 4 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. 5 Introduction (cont.) UNCERTAINTY (Uncertainty-based information) COMPLEXITY CREDIBILITY (Description-algorithmic infor.) (relevance) USEFULNESS 6 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. 7 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. 8 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. 9 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. 10 Introduction (cont.) Fuzzy Logic was motivated by two objectives: First, it aims to minimize 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. 11 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,... 12 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‟ 13 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. 14 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. 15 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. 16 Introduction Most of the phenomena we encounter everyday are imprecise - the imprecision may be associated with their shapes, position, color, texture, semantics that describe what they are Fuzziness primarily describes uncertainty (partial truth) and imprecision The key idea of fuzziness comes from the multivalued logic: Everything is a matter of degree Imprecision raises in several faces, e.g. as a semantic ambiguity 17 Introduction By fuzzifying crisp data obtained from measurements, FL enhances the robustness of a system Imprecision raises in several faces - for example, as a semantic ambiguity the statement “the soup is HOT” is ambiguous, but not fuzzy Definition of The temperature of the soup the domain Hot of The amount of spices used discourse e.g. [20º,80º] Transaction to Fuzziness 18 The word “fuzzy” can be defined as “imprecisely defined, confused, vague” Humans represent and manage natural language terms (data) which are vague. Almost all answers to questions raised in everyday life are within some proximity of the absolute truth 19 Probability vs uncertainty vs Fuzziness Probability theory is one of the most traditional theories for representing uncertainty in mathematical models Nature of uncertainty in a problem is a point which should be clearly recognized by engineer - there is uncertainty that arises from chance, from imprecision, from a lack of knowledge, from vagueness, from randomness… probability theory deals with the expectation of an event (future event, its outcome is not known yet), i.e. it is a theory of random events 20 Fuzziness deals with the impression of meaning of concepts expressed in natural language - it is not concerned with events at all Fuzzy theory handles nonrandom uncertainty Random Uncertain Certain Fuzzy, imprecise, vague 21 a Fuzzy System (FS) is defined as a system with operating principles based on fuzzy information processing and decision making There are several ways to represent knowledge, but the most commonly used has a form of rules: IF (premise)A THEN (conclusion)B 22 From a knowledge representation viewpoint, a fuzzy IF- THEN rule is a scheme for capturing knowledge that involves imprecision - if we know a premise (fact), then we can infer another fact (conclusion) A fuzzy system (FS) is constructed from a collection of fuzzy IF-THEN rules Acquisition of knowledge captured in IF-THEN rules is NOT a trivial task (expert knowledge, systems measurements, etc.) The building blocks for fuzzy IF-THEN rules are FUZZY SETS 23 The rule “IF the air is cool THEN set the motor speed to slow” has a form: IF x is A THEN y is B, where fuzzy sets “cool” and “slow” are labeled by A and B, correspondingly A and B characterize fuzzy propositions about variables x and y Most of the information involved in human communication uses natural language terms that are often vague, imprecise, ambiguous by their nature, and fuzzy sets can serve as the mathematical foundation of natural language 24 A Fuzzy Set is a set with a smooth boundaries Fuzzy Set Theory generalizes classical set theory to allow partial membership Fuzzy Set A is a universal set U is determined by a membership function A(x) that assigns to each element xU a number A(x) in the unit interval [0,1] Universal set U (Universe of Discourse) contains all possible elements of concern for a particular application Fuzzy set has a one-to-one correspondence with its membership function 25 Fuzzy set A is defined as A = { (x, A(x)) }, xU, A(x)[0,1] A(x) = Degree(xA) is a grade of membership of element xU in set A X1 X2 X3 . 0 1/2 1 . . unit interval xN . . U (universe of discourse) 26 The membership functions themselves are NOT fuzzy - they are precise mathematical functions; once a fuzzy property is represented by a membership function, nothing is fuzzy anymore Suppose U is the interval [0,85] representing the age of ordinary human beings, and the linguistic term “young” as a function of age (value of the variable age) can be defined as 25 85 -1 x - 25 2 A " young" 0 1 1 x 5 25 x [see the graphical representation on the next slide] [ !! pay attention to the usage of the symbol “ / “ ] 27 1 (x) Universe of A discourse U is 0.41 continuos 31 25 0 U (universe of discourse) 85 If U is a set of integers from 1 to 10 ( U={1,2,…,10} ), then “small” is a fuzzy subset of U, and it can be defined using enumeration (summation notation): A = “small” = 1/1+1/2+0.85/3+0.75/4+0.5/5+0.3/6+0.1/7 28 In the previous example elements of U (universal set) with zero membership degrees are not included into enumeration A notion of a fuzzy set provides a convenient way of defining abstraction - a process which plays a basic role in human thinking and communication All theories that use the basic concept of fuzzy set can be called in a whole Fuzzy Theory Rough classification of Fuzzy Theory can be depicted as follows [note that dependencies between the branches are not shown] : 29 Fuzzy Theory Fuzzy Fuzzy Uncertainty & Mathematics Decision-Making Information Fuzzy Systems Fuzzy Logic & AI 30 31 32 33 34 35 crisp (classical) set A A = set of TALL people fuzzy set A 1.0 1.0 0.65 0.0 0.0 1.75m height 1.75m 36 37 Membership Functions Main types of membership functions (MF): (a) Triangular MF is specified by 3 parameters {a,b,c}: 0, if x a (x - a) (b - a), if a x b trn(x : a, b, c) (c - x) (c - b), if b x c 0, if x c (b) Trapezoidal MF is specified by 4 parameters {a,b,c,d}: 0, if x a (x - a) (b - a), if a x b trp(x : a,b,c, d) 1, if b x c (d - x) (d - c), if c x d 0, if x d 38 (c) Gaussian MF is specified by 2 parameters {a,}: - (x - a)2 gsn(x : a, ) e xp 2 (d) Bell-shaped MF is specified by 3 parameters {a,b,}: 1 bll(x : a,b,) 2b x- 1 a (e) Sigmoidal MF is specified by 2 parameters {a,b}: 1 sgm(x : a,b) 1 e- a(x -b) 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Terima Kasih 57