Category: fuzzy Logic

Fuzzy Logic – Useful Resources The following resources contain additional information on Fuzzy Logic. Please use them to get more in-depth knowledge on this. Useful Links on Fuzzy Logic − Wikipedia Reference for Fuzzy Logic Useful Books on Fuzzy Logic To enlist your site on this page, please drop an email to [email protected] Learning working make money

Adaptive Fuzzy Controller In this chapter, we will discuss what is an Adaptive Fuzzy Controller and how it works. Adaptive Fuzzy Controller is designed with some adjustable parameters along with an embedded mechanism for adjusting them. Adaptive controller has been used for improving the performance of controller. Basic Steps for Implementing Adaptive Algorithm Let us now discuss the basic steps for implementing adaptive algorithm. Collection of observable data − The observable data is collected to calculate the performance of controller. Adjustment of controller parameters − Now with the help of controller performance, calculation of adjustment of controller parameters would be done. Improvement in performance of controller − In this step, the controller parameters are adjusted to improve the performance of controller. Operational Concepts Design of a controller is based on an assumed mathematical model that resembles a real system. The error between actual system and its mathematical representation is calculated and if it is relatively insignificant than the model is assumed to work effectively. A threshold constant that sets a boundary for the effectiveness of a controller, also exists. The control input is fed into both the real system and mathematical model. Here, assume $xleft ( t right )$ is the output of the real system and $yleft ( t right )$ is the output of the mathematical model. Then the error $epsilon left ( t right )$ can be calculated as follows − $$epsilon left ( t right ) = xleft ( t right ) – yleft ( t right )$$ Here, $x$ desired is the output we want from the system and $mu left ( t right )$ is the output coming from controller and going to both real as well as mathematical model. The following diagram shows how the error function is tracked between output of a real system and Mathematical model − Parameterization of System A fuzzy controller the design of which is based on the fuzzy mathematical model will have the following form of fuzzy rules − Rule 1 − IF $x_1left ( t_n right )in X_{11} : AND…AND: x_ileft ( t_n right )in X_{1i}$ THEN $mu _1left ( t_n right ) = K_{11}x_1left ( t_n right ) + K_{12}x_2left ( t_n right ) : +…+ : K_{1i}x_ileft ( t_n right )$ Rule 2 − IF $x_1left ( t_n right )in X_{21} : AND…AND : x_ileft ( t_n right )in X_{2i}$ THEN $mu _2left ( t_n right ) = K_{21}x_1left ( t_n right ) + K_{22}x_2left ( t_n right ) : +…+ : K_{2i}x_ileft ( t_n right ) $ . . . Rule j − IF $x_1left ( t_n right )in X_{k1} : AND…AND : x_ileft ( t_n right )in X_{ki}$ THEN $mu _jleft ( t_n right ) = K_{j1}x_1left ( t_n right ) + K_{j2}x_2left ( t_n right ) : +…+ : K_{ji}x_ileft ( t_n right ) $ The above set of parameters characterizes the controller. Mechanism Adjustment The controller parameters are adjusted to improve the performance of controller. The process of calculating the adjustment to the parameters is the adjusting mechanism. Mathematically, let $theta ^left ( n right )$ be a set of parameters to be adjusted at time $t = t_n$. The adjustment can be the recalculation of the parameters, $$theta ^left ( n right ) = Theta left ( D_0,: D_1, : …, :D_n right )$$ Here $D_n$ is the data collected at time $t = t_n$. Now this formulation is reformulated by the update of the parameter set based on its previous value as, $$theta ^left ( n right ) = phi ( theta ^{n-1}, : D_n)$$ Parameters for selecting an Adaptive Fuzzy Controller The following parameters need to be considered for selecting an adaptive fuzzy controller − Can the system be approximated entirely by a fuzzy model? If a system can be approximated entirely by a fuzzy model, are the parameters of this fuzzy model readily available or must they be determined online? If a system cannot be approximated entirely by a fuzzy model, can it be approximated piecewise by a set of fuzzy model? If a system can be approximated by a set of fuzzy models, are these models having the same format with different parameters or are they having different formats? If a system can be approximated by a set of fuzzy models having the same format, each with a different set of parameters, are these parameter sets readily available or must they be determined online? Learning working make money

Fuzzy Logic – Control System Fuzzy logic is applied with great success in various control application. Almost all the consumer products have fuzzy control. Some of the examples include controlling your room temperature with the help of air-conditioner, anti-braking system used in vehicles, control on traffic lights, washing machines, large economic systems, etc. Why Use Fuzzy Logic in Control Systems A control system is an arrangement of physical components designed to alter another physical system so that this system exhibits certain desired characteristics. Following are some reasons of using Fuzzy Logic in Control Systems − While applying traditional control, one needs to know about the model and the objective function formulated in precise terms. This makes it very difficult to apply in many cases. By applying fuzzy logic for control we can utilize the human expertise and experience for designing a controller. The fuzzy control rules, basically the IF-THEN rules, can be best utilized in designing a controller. Assumptions in Fuzzy Logic Control (FLC) Design While designing fuzzy control system, the following six basic assumptions should be made − The plant is observable and controllable − It must be assumed that the input, output as well as state variables are available for observation and controlling purpose. Existence of a knowledge body − It must be assumed that there exist a knowledge body having linguistic rules and a set of input-output data set from which rules can be extracted. Existence of solution − It must be assumed that there exists a solution. ‘Good enough’ solution is enough − The control engineering must look for ‘good enough’ solution rather than an optimum one. Range of precision − Fuzzy logic controller must be designed within an acceptable range of precision. Issues regarding stability and optimality − The issues of stability and optimality must be open in designing Fuzzy logic controller rather than addressed explicitly. Architecture of Fuzzy Logic Control The following diagram shows the architecture of Fuzzy Logic Control (FLC). Major Components of FLC Followings are the major components of the FLC as shown in the above figure − Fuzzifier − The role of fuzzifier is to convert the crisp input values into fuzzy values. Fuzzy Knowledge Base − It stores the knowledge about all the input-output fuzzy relationships. It also has the membership function which defines the input variables to the fuzzy rule base and the output variables to the plant under control. Fuzzy Rule Base − It stores the knowledge about the operation of the process of domain. Inference Engine − It acts as a kernel of any FLC. Basically it simulates human decisions by performing approximate reasoning. Defuzzifier − The role of defuzzifier is to convert the fuzzy values into crisp values getting from fuzzy inference engine. Steps in Designing FLC Following are the steps involved in designing FLC − Identification of variables − Here, the input, output and state variables must be identified of the plant which is under consideration. Fuzzy subset configuration − The universe of information is divided into number of fuzzy subsets and each subset is assigned a linguistic label. Always make sure that these fuzzy subsets include all the elements of universe. Obtaining membership function − Now obtain the membership function for each fuzzy subset that we get in the above step. Fuzzy rule base configuration − Now formulate the fuzzy rule base by assigning relationship between fuzzy input and output. Fuzzification − The fuzzification process is initiated in this step. Combining fuzzy outputs − By applying fuzzy approximate reasoning, locate the fuzzy output and merge them. Defuzzification − Finally, initiate defuzzification process to form a crisp output. Advantages of Fuzzy Logic Control Let us now discuss the advantages of Fuzzy Logic Control. Cheaper − Developing a FLC is comparatively cheaper than developing model based or other controller in terms of performance. Robust − FLCs are more robust than PID controllers because of their capability to cover a huge range of operating conditions. Customizable − FLCs are customizable. Emulate human deductive thinking − Basically FLC is designed to emulate human deductive thinking, the process people use to infer conclusion from what they know. Reliability − FLC is more reliable than conventional control system. Efficiency − Fuzzy logic provides more efficiency when applied in control system. Disadvantages of Fuzzy Logic Control We will now discuss what are the disadvantages of Fuzzy Logic Control. Requires lots of data − FLC needs lots of data to be applied. Useful in case of moderate historical data − FLC is not useful for programs much smaller or larger than historical data. Needs high human expertise − This is one drawback as the accuracy of the system depends on the knowledge and expertise of human beings. Needs regular updating of rules − The rules must be updated with time. Learning working make money

Fuzzy Logic – Set Theory Fuzzy sets can be considered as an extension and gross oversimplification of classical sets. It can be best understood in the context of set membership. Basically it allows partial membership which means that it contain elements that have varying degrees of membership in the set. From this, we can understand the difference between classical set and fuzzy set. Classical set contains elements that satisfy precise properties of membership while fuzzy set contains elements that satisfy imprecise properties of membership. Mathematical Concept A fuzzy set $widetilde{A}$ in the universe of information $U$ can be defined as a set of ordered pairs and it can be represented mathematically as − $$widetilde{A} = left { left ( y,mu _{widetilde{A}} left ( y right ) right ) | yin Uright }$$ Here $mu _{widetilde{A}}left ( y right )$ = degree of membership of $y$ in widetilde{A}, assumes values in the range from 0 to 1, i.e., $mu _{widetilde{A}}(y)in left [ 0,1 right ]$. Representation of fuzzy set Let us now consider two cases of universe of information and understand how a fuzzy set can be represented. Case 1 When universe of information $U$ is discrete and finite − $$widetilde{A} = left { frac{mu _{widetilde{A}}left ( y_1 right )}{y_1} +frac{mu _{widetilde{A}}left ( y_2 right )}{y_2} +frac{mu _{widetilde{A}}left ( y_3 right )}{y_3} +…right }$$ $= left { sum_{i=1}^{n}frac{mu _{widetilde{A}}left ( y_i right )}{y_i} right }$ Case 2 When universe of information $U$ is continuous and infinite − $$widetilde{A} = left { int frac{mu _{widetilde{A}}left ( y right )}{y} right }$$ In the above representation, the summation symbol represents the collection of each element. Operations on Fuzzy Sets Having two fuzzy sets $widetilde{A}$ and $widetilde{B}$, the universe of information $U$ and an element 𝑦 of the universe, the following relations express the union, intersection and complement operation on fuzzy sets. Union/Fuzzy ‘OR’ Let us consider the following representation to understand how the Union/Fuzzy ‘OR’ relation works − $$mu _{{widetilde{A}cup widetilde{B} }}left ( y right ) = mu _{widetilde{A}}vee mu _widetilde{B} quad forall y in U$$ Here ∨ represents the ‘max’ operation. Intersection/Fuzzy ‘AND’ Let us consider the following representation to understand how the Intersection/Fuzzy ‘AND’ relation works − $$mu _{{widetilde{A}cap widetilde{B} }}left ( y right ) = mu _{widetilde{A}}wedge mu _widetilde{B} quad forall y in U$$ Here ∧ represents the ‘min’ operation. Complement/Fuzzy ‘NOT’ Let us consider the following representation to understand how the Complement/Fuzzy ‘NOT’ relation works − $$mu _{widetilde{A}} = 1-mu _{widetilde{A}}left ( y right )quad y in U$$ Properties of Fuzzy Sets Let us discuss the different properties of fuzzy sets. Commutative Property Having two fuzzy sets $widetilde{A}$ and $widetilde{B}$, this property states − $$widetilde{A}cup widetilde{B} = widetilde{B}cup widetilde{A}$$ $$widetilde{A}cap widetilde{B} = widetilde{B}cap widetilde{A}$$ Associative Property Having three fuzzy sets $widetilde{A}$, $widetilde{B}$ and $widetilde{C}$, this property states − $$(widetilde{A}cup left widetilde{B}) cup widetilde{C} right = left widetilde{A} cup (widetilde{B}right )cup widetilde{C})$$ $$(widetilde{A}cap left widetilde{B}) cap widetilde{C} right = left widetilde{A} cup (widetilde{B}right cap widetilde{C})$$ Distributive Property Having three fuzzy sets $widetilde{A}$, $widetilde{B}$ and $widetilde{C}$, this property states − $$widetilde{A}cup left ( widetilde{B} cap widetilde{C}right ) = left ( widetilde{A} cup widetilde{B}right )cap left ( widetilde{A}cup widetilde{C} right )$$ $$widetilde{A}cap left ( widetilde{B}cup widetilde{C} right ) = left ( widetilde{A} cap widetilde{B} right )cup left ( widetilde{A}cap widetilde{C} right )$$ Idempotency Property For any fuzzy set $widetilde{A}$, this property states − $$widetilde{A}cup widetilde{A} = widetilde{A}$$ $$widetilde{A}cap widetilde{A} = widetilde{A}$$ Identity Property For fuzzy set $widetilde{A}$ and universal set $U$, this property states − $$widetilde{A}cup varphi = widetilde{A}$$ $$widetilde{A}cap U = widetilde{A}$$ $$widetilde{A}cap varphi = varphi$$ $$widetilde{A}cup U = U$$ Transitive Property Having three fuzzy sets $widetilde{A}$, $widetilde{B}$ and $widetilde{C}$, this property states − $$If : widetilde{A}subseteq widetilde{B}subseteq widetilde{C},:then:widetilde{A}subseteq widetilde{C}$$ Involution Property For any fuzzy set $widetilde{A}$, this property states − $$overline{overline{widetilde{A}}} = widetilde{A}$$ De Morgan’s Law This law plays a crucial role in proving tautologies and contradiction. This law states − $$overline{{widetilde{A}cap widetilde{B}}} = overline{widetilde{A}}cup overline{widetilde{B}}$$ $$overline{{widetilde{A}cup widetilde{B}}} = overline{widetilde{A}}cap overline{widetilde{B}}$$ Learning working make money

Fuzzy Logic – Traditional Fuzzy Refresher Logic, which was originally just the study of what distinguishes sound argument from unsound argument, has now developed into a powerful and rigorous system whereby true statements can be discovered, given other statements that are already known to be true. Predicate Logic This logic deals with predicates, which are propositions containing variables. A predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable. Following are a few examples of predicates − Let E(x, y) denote “x = y” Let X(a, b, c) denote “a + b + c = 0” Let M(x, y) denote “x is married to y” Propositional Logic A proposition is a collection of declarative statements that have either a truth value “true” or a truth value “false”. A propositional consists of propositional variables and connectives. The propositional variables are dented by capital letters (A, B, etc). The connectives connect the propositional variables. A few examples of Propositions are given below − “Man is Mortal”, it returns truth value “TRUE” “12 + 9 = 3 – 2”, it returns truth value “FALSE” The following is not a Proposition − “A is less than 2” − It is because unless we give a specific value of A, we cannot say whether the statement is true or false. Connectives In propositional logic, we use the following five connectives − OR (∨∨) AND (∧∧) Negation/ NOT (¬¬) Implication / if-then (→→) If and only if (⇔⇔) OR (∨∨) The OR operation of two propositions A and B (written as A∨BA∨B) is true if at least any of the propositional variable A or B is true. The truth table is as follows − A B A ∨ B True True True True False True False True True False False False AND (∧∧) The AND operation of two propositions A and B (written as A∧BA∧B) is true if both the propositional variable A and B is true. The truth table is as follows − A B A ∧ B True True True True False False False True False False False False Negation (¬¬) The negation of a proposition A (written as ¬A¬A) is false when A is true and is true when A is false. The truth table is as follows − A ¬A True False False True Implication / if-then (→→) An implication A→BA→B is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true. The truth table is as follows − A B A→B True True True True False False False True True False False True If and only if (⇔⇔) A⇔BA⇔B is a bi-conditional logical connective which is true when p and q are same, i.e., both are false or both are true. The truth table is as follows − A B A⇔B True True True True False False False True False False False True Well Formed Formula Well Formed Formula (wff) is a predicate holding one of the following − All propositional constants and propositional variables are wffs. If x is a variable and Y is a wff, ∀xY and ∃xY are also wff. Truth value and false values are wffs. Each atomic formula is a wff. All connectives connecting wffs are wffs. Quantifiers The variable of predicates is quantified by quantifiers. There are two types of quantifier in predicate logic − Universal Quantifier Existential Quantifier Universal Quantifier Universal quantifier states that the statements within its scope are true for every value of the specific variable. It is denoted by the symbol ∀. ∀xP(x) is read as for every value of x, P(x) is true. Example − “Man is mortal” can be transformed into the propositional form ∀xP(x). Here, P(x) is the predicate which denotes that x is mortal and the universe of discourse is all men. Existential Quantifier Existential quantifier states that the statements within its scope are true for some values of the specific variable. It is denoted by the symbol ∃. ∃xP(x) for some values of x is read as, P(x) is true. Example − “Some people are dishonest” can be transformed into the propositional form ∃x P(x) where P(x) is the predicate which denotes x is dishonest and the universe of discourse is some people. Nested Quantifiers If we use a quantifier that appears within the scope of another quantifier, it is called a nested quantifier. Example ∀ a∃bP(x,y) where P(a,b) denotes a+b = 0 ∀ a∀b∀cP(a,b,c) where P(a,b) denotes a+(b+c) = (a+b)+c Note − ∀a∃bP(x,y) ≠ ∃a∀bP(x,y) Learning working make money

Fuzzy Logic Tutorial Job Search Fuzzy Logic resembles the human decision-making methodology and deals with vague and imprecise information. This is a very small tutorial that touches upon the very basic concepts of Fuzzy Logic. Audience This tutorial will be useful for graduates, post-graduates, and research students who either have an interest in this subject or have this subject as a part of their curriculum. The reader can be a beginner or an advanced learner. Prerequisites Fuzzy Logic is an advanced topic, so we assume that the readers of this tutorial have preliminary knowledge of Set Theory, Logic, and Engineering Mathematics. Learning working make money

Fuzzy Logic – Introduction The word fuzzy refers to things which are not clear or are vague. Any event, process, or function that is changing continuously cannot always be defined as either true or false, which means that we need to define such activities in a Fuzzy manner. What is Fuzzy Logic? Fuzzy Logic resembles the human decision-making methodology. It deals with vague and imprecise information. This is gross oversimplification of the real-world problems and based on degrees of truth rather than usual true/false or 1/0 like Boolean logic. Take a look at the following diagram. It shows that in fuzzy systems, the values are indicated by a number in the range from 0 to 1. Here 1.0 represents absolute truth and 0.0 represents absolute falseness. The number which indicates the value in fuzzy systems is called the truth value. In other words, we can say that fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness. There can be numerous other examples like this with the help of which we can understand the concept of fuzzy logic. Fuzzy Logic was introduced in 1965 by Lofti A. Zadeh in his research paper “Fuzzy Sets”. He is considered as the father of Fuzzy Logic. Learning working make money

Fuzzy Logic – Quantification In modeling natural language statements, quantified statements play an important role. It means that NL heavily depends on quantifying construction which often includes fuzzy concepts like “almost all”, “many”, etc. Following are a few examples of quantifying propositions − Every student passed the exam. Every sport car is expensive. Many students passed the exam. Many sports cars are expensive. In the above examples, the quantifiers “Every” and “Many” are applied to the crisp restrictions “students” as well as crisp scope “(person who)passed the exam” and “cars” as well as crisp scope ”sports”. Fuzzy Events, Fuzzy Means and Fuzzy Variances With the help of an example, we can understand the above concepts. Let us assume that we are a shareholder of a company named ABC. And at present the company is selling each of its share for ₹40. There are three different companies whose business is similar to ABC but these are offering their shares at different rates – ₹100 a share, ₹85 a share and ₹60 a share respectively. Now the probability distribution of this price takeover is as follows − Price ₹100 ₹85 ₹60 Probability 0.3 0.5 0.2 Now, from the standard probability theory, the above distribution gives a mean of expected price as below − $100 × 0.3 + 85 × 0.5 + 60 × 0.2 = 84.5$ And, from the standard probability theory, the above distribution gives a variance of expected price as below − $(100 − 84.5)2 × 0.3 + (85 − 84.5)2 × 0.5 + (60 − 84.5)2 × 0.2 = 124.825$ Suppose the degree of membership of 100 in this set is 0.7, that of 85 is 1, and the degree of membership is 0.5 for the value 60. These can be reflected in the following fuzzy set − $$left { frac{0.7}{100}, : frac{1}{85}, : frac{0.5}{60}, right }$$ The fuzzy set obtained in this manner is called a fuzzy event. We want the probability of the fuzzy event for which our calculation gives − $0.7 × 0.3 + 1 × 0.5 + 0.5 × 0.2 = 0.21 + 0.5 + 0.1 = 0.81$ Now, we need to calculate the fuzzy mean and the fuzzy variance, the calculation is as follows − Fuzzy_mean $= left ( frac{1}{0.81} right ) × (100 × 0.7 × 0.3 + 85 × 1 × 0.5 + 60 × 0.5 × 0.2)$ $= 85.8$ Fuzzy_Variance $= 7496.91 − 7361.91 = 135.27$ Learning working make money

Fuzzy Logic – Approximate Reasoning Following are the different modes of approximate reasoning − Categorical Reasoning In this mode of approximate reasoning, the antecedents, containing no fuzzy quantifiers and fuzzy probabilities, are assumed to be in canonical form. Qualitative Reasoning In this mode of approximate reasoning, the antecedents and consequents have fuzzy linguistic variables; the input-output relationship of a system is expressed as a collection of fuzzy IF-THEN rules. This reasoning is mainly used in control system analysis. Syllogistic Reasoning In this mode of approximation reasoning, antecedents with fuzzy quantifiers are related to inference rules. This is expressed as − x = S1A′s are B′s y = S2C′s are D′s ———————— z = S3E′s are F′s Here A,B,C,D,E,F are fuzzy predicates. S1 and S2 are given fuzzy quantifiers. S3 is the fuzzy quantifier which has to be decided. Dispositional Reasoning In this mode of approximation reasoning, the antecedents are dispositions that may contain the fuzzy quantifier “usually”. The quantifier Usually links together the dispositional and syllogistic reasoning; hence it pays an important role. For example, the projection rule of inference in dispositional reasoning can be given as follows − usually( (L,M) is R ) ⇒ usually (L is [R ↓ L]) Here [R ↓ L] is the projection of fuzzy relation R on L Fuzzy Logic Rule Base It is a known fact that a human being is always comfortable making conversations in natural language. The representation of human knowledge can be done with the help of following natural language expression − IF antecedent THEN consequent The expression as stated above is referred to as the Fuzzy IF-THEN rule base. Canonical Form Following is the canonical form of Fuzzy Logic Rule Base − Rule 1 − If condition C1, then restriction R1 Rule 2 − If condition C1, then restriction R2 . . . Rule n − If condition C1, then restriction Rn Interpretations of Fuzzy IF-THEN Rules Fuzzy IF-THEN Rules can be interpreted in the following four forms − Assignment Statements These kinds of statements use “=” (equal to sign) for the purpose of assignment. They are of the following form − a = hello climate = summer Conditional Statements These kinds of statements use the “IF-THEN” rule base form for the purpose of condition. They are of the following form − IF temperature is high THEN Climate is hot IF food is fresh THEN eat. Unconditional Statements They are of the following form − GOTO 10 turn the Fan off Linguistic Variable We have studied that fuzzy logic uses linguistic variables which are the words or sentences in a natural language. For example, if we say temperature, it is a linguistic variable; the values of which are very hot or cold, slightly hot or cold, very warm, slightly warm, etc. The words very, slightly are the linguistic hedges. Characterization of Linguistic Variable Following four terms characterize the linguistic variable − Name of the variable, generally represented by x. Term set of the variable, generally represented by t(x). Syntactic rules for generating the values of the variable x. Semantic rules for linking every value of x and its significance. Propositions in Fuzzy Logic As we know that propositions are sentences expressed in any language which are generally expressed in the following canonical form − s as P Here, s is the Subject and P is Predicate. For example, “Delhi is the capital of India”, this is a proposition where “Delhi” is the subject and “is the capital of India” is the predicate which shows the property of subject. We know that logic is the basis of reasoning and fuzzy logic extends the capability of reasoning by using fuzzy predicates, fuzzy-predicate modifiers, fuzzy quantifiers and fuzzy qualifiers in fuzzy propositions which creates the difference from classical logic. Propositions in fuzzy logic include the following − Fuzzy Predicate Almost every predicate in natural language is fuzzy in nature hence, fuzzy logic has the predicates like tall, short, warm, hot, fast, etc. Fuzzy-predicate Modifiers We discussed linguistic hedges above; we also have many fuzzy-predicate modifiers which act as hedges. They are very essential for producing the values of a linguistic variable. For example, the words very, slightly are modifiers and the propositions can be like “water is slightly hot.” Fuzzy Quantifiers It can be defined as a fuzzy number which gives a vague classification of the cardinality of one or more fuzzy or non-fuzzy sets. It can be used to influence probability within fuzzy logic. For example, the words many, most, frequently are used as fuzzy quantifiers and the propositions can be like “most people are allergic to it.” Fuzzy Qualifiers Let us now understand Fuzzy Qualifiers. A Fuzzy Qualifier is also a proposition of Fuzzy Logic. Fuzzy qualification has the following forms − Fuzzy Qualification Based on Truth It claims the degree of truth of a fuzzy proposition. Expression − It is expressed as x is t. Here, t is a fuzzy truth value. Example − (Car is black) is NOT VERY True. Fuzzy Qualification Based on Probability It claims the probability, either numerical or an interval, of fuzzy proposition. Expression − It is expressed as x is λ. Here, λ is a fuzzy probability. Example − (Car is black) is Likely. Fuzzy Qualification Based on Possibility It claims the possibility of fuzzy proposition. Expression − It is expressed as x is π. Here, π is a fuzzy possibility. Example − (Car is black) is Almost Impossible. Learning working make money

Fuzzy Logic – Database and Queries We have studied in our previous chapters that Fuzzy Logic is an approach to computing based on “degrees of truth” rather than the usual “true or false” logic. It deals with reasoning that is approximate rather than precise to solve problems in a way that more resembles human logic, hence database querying process by the two valued realization of Boolean algebra is not adequate. Fuzzy Scenario of Relations on Databases The Fuzzy Scenario of Relations on Databases can be understood with the help of the following example − Example Suppose we have a database having the records of persons who visited India. In simple database, we will have the entries made in the following way − Name Age Citizen Visited Country Days Spent Year of Visit John Smith 35 U.S. India 41 1999 John Smith 35 U.S. Italy 72 1999 John Smith 35 U.S. Japan 31 1999 Now, if anyone queries about the person who visited India and Japan in the year 99 and is the citizen of US, then the output will show two entries having the name of John Smith. This is simple query generating simple output. But what if we want to know whether the person in the above query is young or not. According to the above result, the age of the person is 35 years. But can we assume the person to be young or not? Similarly, same thing can be applied on the other fields like days spent, year of visit, etc. The solution of the above issues can be found with the help of Fuzzy Value sets as follows − FV(Age){ very young, young, somewhat old, old } FV(Days Spent){ barely few days, few days, quite a few days, many days } FV(Year of Visit){distant past, recent past, recent } Now if any query will have the fuzzy value then the result will also be fuzzy in nature. Fuzzy Query System A fuzzy query system is an interface to users to get information from the database using (quasi) natural language sentences. Many fuzzy query implementations have been proposed, resulting in slightly different languages. Although there are some variations according to the particularities of different implementations, the answer to a fuzzy query sentence is generally a list of records, ranked by the degree of matching. Learning working make money