From natural or artificial language. For example, the linguistic variable "speed" can have the values "high", "medium", "very low", etc. Phrases, the value of which takes on a variable, in turn, are the names of fuzzy variables and are described by a fuzzy set.
Mathematical definition
The linguistic variable is called five , where - variable name; - some set of values of a linguistic variable , each of which is a fuzzy variable on the set ; there is a syntactic rule for the formation of names of new values ; there is a semantic procedure that allows you to transform a new name generated by the procedure , into a fuzzy variable (set the type of the membership function), associates the name with its meaning, concept.
also called the base term set, since it specifies the minimum number of values based on which, using the rules and you can form the rest allowable values linguistic variable. A bunch of and newly educated with the help and the values of the linguistic variable form an extended term set.
Example: fuzzy age
Consider a linguistic variable describing a person's age, then:
- : "age";
- : a set of integers from the interval;
- : meanings "young", "mature", "old". a bunch of - a lot of fuzzy variables, for each value: "young", "mature", "old", it is necessary to set a membership function that specifies information about what age people should be considered young, mature, old;
- : "Very", "not very". Such additives allow the formation of new meanings: "very young", "not very old", etc.
- : mathematical rule, which determines the form of the membership function for each value formed using the rule .
Write a review on the article "Linguistic Variable"
Excerpt characterizing linguistic variable
The count went behind the partition again and lay down. The countess went up to Natasha, touched her head with an inverted hand, as she did when her daughter was sick, then touched her forehead with her lips, as if to find out if there was a fever, and kissed her.- You're cold. You're trembling all over. You should go to bed, ”she said.
- Go to bed? Yes, okay, I'll go to bed. I'll go to bed now, ”Natasha said.
Since Natasha had been told this morning that Prince Andrey had been seriously wounded and was traveling with them, only at the first minute had she asked a lot about where? as? is he dangerously injured? and can she see him? But after she was told that she could not see him, that he was seriously wounded, but that his life was not in danger, she obviously did not believe what she was told, but making sure that no matter how much she said, she would be answer the same thing, stopped asking and talking. All the way from big eyes whom the Countess knew so much and of which expression was so afraid, Natasha sat motionless in the corner of the carriage and was now sitting in the same way on the bench on which she sat. Something she was planning, something she was deciding, or had already decided in her mind now — the countess knew that, but what it was, she did not know, and it frightened and tormented her.
- Natasha, undress, my dear, lie down on my bed. (Only one countess had a bed on the bed; m me Schoss and both young ladies were to sleep on the floor in the hay.)
“No, Mom, I'll lie here on the floor,” Natasha said angrily, went to the window and opened it. The adjutant groan from open window was heard more clearly. She stuck her head out into the damp air of the night, and the Countess saw her slender shoulders shaking with sobs and hitting the frame. Natasha knew that it was not Prince Andrew who was moaning. She knew that Prince Andrew was lying in the same connection where they were, in another hut through the passage; but this terrible incessant groan made her sob. The Countess exchanged glances with Sonya.
“Lie down, my dear, lie down, my friend,” said the countess, lightly touching Natasha's shoulder with her hand. - Well, lie down.
“Oh, yes ... I’ll go to bed now, now,” said Natasha, hastily undressing and breaking off the ties of her skirts. Throwing off her dress and putting on a jacket, she twisted her legs, sat down on the bed prepared on the floor and, throwing her short thin braid over her shoulder in front, began to intertwine it. Thin long familiar fingers quickly, deftly disassembled, weaved, tied a braid. Natasha's head, with a habitual gesture, turned in one direction or the other, but her eyes, feverishly open, stared straight ahead. When the night suit was finished, Natasha quietly sank down on a sheet, laid on the hay at the edge of the door.
In an informal discussion of the concept of a linguistic variable in §1, we formulated that a linguistic variable differs from a numerical variable in that its values are not numbers, but words or sentences in a natural or formal language. Since words are generally less precise than numbers, the concept of a linguistic variable makes it possible to roughly describe phenomena that are so complex that they cannot be described in generally accepted quantitative terms. In particular, a fuzzy set, which is a constraint associated with the values of a linguistic variable, can be considered as an aggregate characteristic of various subclasses of elements of a universal set. In this sense, the role fuzzy sets is analogous to the role that words and sentences play in natural language. For example, the adjective beautiful reflects a complex of characteristics of an individual's appearance. This adjective can also be thought of as the name of a fuzzy set, which is a limitation due to a fuzzy variable. beautiful... From this point of view, the terms very beautiful, ugly, extremely beautiful, quite handsome etc. - the names of fuzzy sets formed by the action of modifiers very, not, extremely, quite etc. to a fuzzy set beautiful... In essence, these fuzzy sets, together with the beautiful fuzzy set, play the role of the values of the linguistic variable Appearance.
An important aspect of the concept of a linguistic variable is that this variable is of a higher order than a fuzzy variable, in the sense that the values of a linguistic variable are fuzzy variables. For example, the values of the linguistic variable Age may be: young, middle-aged, old, very old, middle-aged and not old, quite old etc. Each of these values is the name of a fuzzy variable. If is the name of a fuzzy variable, then the restriction due to this name can be interpreted as the meaning of the fuzzy variable. So, if the limitation due to a fuzzy variable old, is a fuzzy subset of a set of the form
, , (5.1)
Another important aspect of the concept of a linguistic variable is that two rules correspond to a linguistic variable: (1) a syntactic rule, which can be given in the form of a grammar that generates the names of the values of the variable; (2) a semantic rule that defines an algorithmic procedure for computing the meaning of each value. These rules form an essential part of describing the structure of a linguistic variable.
Rice. 5.1. Compatibility functions for values and 
.
Since a linguistic variable is a variable of a higher order than a fuzzy variable, its description should also be more complicated than the description of a fuzzy variable given in Definition 4.1.
Definition 5.1. A linguistic variable is characterized by the set 
, in which is the name of the variable; (or simply) denotes a term-set of a variable, that is, a set of names of linguistic values of a variable, each of such values being a fuzzy variable with values from a universal set with a base variable; - a syntactic rule (usually in the form of a grammar) that generates the names of the values of a variable, and - a semantic rule that associates each fuzzy variable with its meaning, that is, a fuzzy subset of the universal set. Specific title generated by a syntactic rule is called a term. A term consisting of one word or several words that always appear together with each other is called an atomic term. A term consisting of one or more atomic terms is called a compound term. The concatenation of some of the components of a compound term is a subterm. If - terms in, then can be represented as a union
(5.2)
If it is necessary to explicitly indicate what was generated by the grammar, we will write.
The meaning of a term is defined as a constraint on the base variable due to a fuzzy variable:
, (5.3)
meaning that and, therefore, can be considered as a fuzzy subset of a set that has a name. The relationship between its linguistic meaning and the base variable is illustrated in Fig. 1.3.
Remark 5.2. To avoid a large number symbols, it is prudent to assign multiple meanings to some of the symbols found in Definition 5.1, relying on context to resolve possible ambiguities. In particular:
a) We will often use the symbol to denote both the name of the variable itself and the general name of its values. Likewise, will denote both the general name of the variable values and the name of the variable itself.
b) We will use the same symbol to denote a set and its name. So, symbols, and will be interchangeable, although, strictly speaking, as a name (or) is not the same as a fuzzy set. In other words, when we say that a term (for example, young) is the value of the variable (for example, Age), then we mean that actual value is, but is just the name of this value.
Example 5.3. Age, i.e. , let it go . The linguistic value of the variable Age maybe for example old, and the value old is an atomic term. Another meaning could be very old, i.e., a compound term in which old - atomic term, and very and old- subterms.
Meaning more or less young variable Age - compound term in which the term young - atomic, and More or less- subterm. Variable term set Age can be written as follows:
(5.4)
Here each term is the name of a fuzzy variable in the universal set. A limitation due to a term, say, has a linguistic meaning old... Thus, if it is determined according to (5.1), then the meaning of the linguistic meaning old defined by the expression
, (5.5)
or simpler (see Remark 5.2)
. (5.6)
Likewise, the meaning of such linguistic meaning as very old, can be expressed as follows (see Fig.5.1):
The assignment equation in the case of a linguistic variable takes the form
whence it follows that the meaning assigned to a term is expressed by the equality
In other words, the meaning of a term is obtained by applying a semantic rule to the value of the term assigned according to the right side of equation (5.8). Moreover, it follows from definition (5.3), which is identical to the constraint caused by the term.
Remark 5.4. In accordance with Remark 5.2 (a), the assignment equation will usually be written as
, (5.10)
understanding this so that old- the constraint on the values of the base variable, defined by (5.1), is assigned to the linguistic variable Age... It is important to note that the equal sign in (5.10) does not denote a symmetric relationship, as in the case of arithmetic equality. So, it makes no sense to write (5.11) in the form
To illustrate the concept of a linguistic variable, we first consider a very simple example, which contains only small number terms, and the syntactic and semantic rules are trivial.
Example 5.5. Consider the linguistic variable Number, the finite term set of which has the form
where each term is a constraint on the values of the base variable in the universal set
It is assumed that these constraints are fuzzy subsets of the set and are defined as follows:
, (5.15) with a binary constraint are approximately equal.
To assign a value, say approximately equal linguistic variable, we write
where, as in (5.18), we mean that a binary fuzzy relation is assigned as the value of the variable approximately equal, which is a binary constraint on the values of the base variable in the universal set (5.20).
Rice. 5.2. Travel bag analogy for a linguistic variable
Remark 5.7. Using the analogy with the bag (see Remark 4.3), the linguistic variable in the sense of Definition 5.1 can be likened to a rigid bag in which you can put soft bags, as shown in Fig. 5.2. The soft bag corresponds to a fuzzy variable, which is the linguistic value of the variable, but plays the role of a label on the soft bag.
Fuzzy sets. Linguistic variable. Fuzzy logic. Fuzzy inference. Compositional inference rule.
(Abstract)
The concept of a fuzzy set (NI) is based on the idea that elements of a set possessing a common property can have different degrees of degeneracy of this property and, therefore, different degrees of belonging to this property.
Let U be some set. A fuzzy set à in U is a collection of pairs of the form ((µ à (u), u)), where u U, µ Ã.
The value µ Ã is called the degree of belonging of the object to the fuzzy set U.
µ Ã: U
µ Ã - is called a membership function.
An example of fuzzy sets is the age of people (Figure 19.1).
By analogy with traditional set theory, the following operations are defined in the IS theory:
An association:
, where 
Enumeration:
,
Addition:
Algebraic product:
, where
An n-ary fuzzy relation defined on sets is a fuzzy subset of Cartesian products 
Since a fuzzy relation is a set, all operations defined for fuzzy sets are valid for it. In practical applications of fuzzy set theory important role plays the operation of composition of fuzzy relations.
Fuzzy relationship composition
Let 2 two-place fuzzy relations be given:
The composition of fuzzy relations is defined by the following expression:
Degrees of membership of specific expressions
The linguistic variable is a five X is the name of the variable (age), U is the base set (0 ... 150), T (x) is the term of the set. Lots of linguistic meanings (young, middle-aged, elderly, old). Each linguistic meaning is a label of a fuzzy set defined in U. G is a syntactic rule that generates the linguistic meaning of the variable X (very young, very old). M is a semantic rule that associates each linguistic meaning with a fuzzy subset of the base set, that is, a membership function.
A fuzzy statement is a statement about which in this moment time can be judged on the degree of its truth or falsity. Truth takes on a value in the interval. A fuzzy statement that does not allow division into simpler ones is called elementary.
A fuzzy statement built on elementary ones using logical connectives is called a compound fuzzy statement. Operations on the truth of fuzzy statements correspond to logical connectives. - the degree of truthfulness of specific statements.
1)
2)
Thus, the algebra of fuzzy sets is isomorphic to the algebra of fuzzy statements.
4) operation of implication
Several definitions have been proposed for the implication operation in fuzzy logic. Basic:
1)
2)
3)
5) Equivalence
An n-ary fuzzy predicate defined on the sets U 1, U 2, ..., U n is an expression containing subject variables of these sets and turning into fuzzy statements when the subject variables are replaced by elements of the sets U 1, U 2, ..., U n.
Let U 1, U 2,…, U n be the basic sets of linguistic variables, and the yen of linguistic variables act as symbols of the subject variables. Then examples of fuzzy predicates are:
"Cylinder pressure low" - single predicate
"The temperature in the boiler is significantly higher than the temperature in the heat exchanger" - a two-place predicate.
If U k = 1.5, therefore "pressure in the boiler is low" = 0.7
In the construction and implementation of fuzzy algorithms, the compositional inference rule plays an important role.
Let be a fuzzy mapping
A fuzzy subset of the universe U, then generates in V a fuzzy subset
the compositional rule of inference is the basis for constructing inference in fuzzy logic.
Let a fuzzy statement be given, where and are fuzzy sets. Let also some statement be given (close to A, but not identical to it).
In classical logic, the Modus Ponens inference rule is widely used
This rule is generalized to the case of fuzzy logic as follows:
Let the set and be defined on the base set X, and on the base set Y. It is natural to assume that the statement if defines some fuzzy mapping from the set X to Y
Then, in accordance with the compositional inference rule, we have:
The relationship is based on the definition of the implication operation in fuzzy logic.
1)
If the temperature in the boiler is low (), then the heating is increased ()
Real fuzzy logic algorithms contain not one but many production rules
If S 1, then R 1, otherwise
If S n, then R n, otherwise
Therefore, fuzzy relationships must be built for each individual rule and then aggregated by superimposing
Either min or max is chosen as the aggregating operation, depending on the type of implication.
When fuzzy inference is used in a real object control loop, a clear control action must be given to the object. Therefore, it is necessary to transform the fuzzy set formed on the basis of the compositional inference rule into a crisp value. This procedure is called defuzzification procedure. Most often, 2 defuzzification methods are used:
1) The middle of the "plateau"
2) The center of gravity, a point that divides the area of a fuzzy set in half is determined.
The concept of fuzzy and linguistic variables is used to describe objects and phenomena using fuzzy sets.
Fuzzy variable characterized by three (α, X, A), where
α - the name of the variable;
X- universal set (domain of definition α);
A- fuzzy set on X, describing the constraints (i.e. μ A(x) ) to the values of the fuzzy variable α.
Linguistic variable (LP) is called a set ( β , T, X, G, M), where
β - the name of the linguistic variable;
T- the set of its values (term-set), which are the names of fuzzy variables, the domain of definition of each of which is the set X. A bunch of T called basic term-set linguistic variable;
G is a syntactic procedure that allows you to operate on the elements of the term-set T, in particular, to generate new terms (values). The set T∪G (T), where G (T) is the set of generated terms, is called the extended term-set of the linguistic variable;
M- a semantic procedure that allows you to turn each new value of a linguistic variable, formed by the procedure G, into a fuzzy variable, i.e. form the corresponding fuzzy set.
Comment. To avoid a lot of characters:
1) symbol β used both for the name of the variable itself and for all its values;
2) use the same symbol to denote a fuzzy set and its name, for example, the term "Young", which is the value of a linguistic variable β = "Age", at the same time there is a fuzzy set M("Young").
Assigning multiple meanings to symbols assumes that the context allows for potential ambiguities.
Example. Let the expert determine the thickness of the manufactured product using the concepts "Small thickness", "Average thickness" and "Large thickness", while minimum thickness is 10 mm, and the maximum is 80 mm.
Formalization of such a description can be carried out using the following linguistic variable ( β , T, X, G, M ), where
β - product thickness;
T- ("Small thickness", "Medium thickness", "Large thickness");
X— ;
G - the procedure for the formation of new terms using the connectives "and", "or" and modifiers such as "very", "not", "slightly", etc. For example: "Small or medium thickness", "Very small thickness", etc .;
M- the procedure for assigning X = fuzzy subsets A 1 = "Small thickness", A 2 = "Average thickness", A 3 = "Large thickness", as well as fuzzy sets for terms from G (T) in accordance with the rules of translation of fuzzy connectives and modifiers "and", "or", "not", "very", "slightly" and other operations on fuzzy sets of the form: A⋂V,A∪VA, CON A =A 2 , DIL A = A 0.5 etc.
Comment. Along with the above-considered basic values of the linguistic variable "Thickness" (T =("Small thickness", "Medium thickness", "Large thickness")) values depending on the definition area X are possible. in this case the values of the linguistic variable "Product thickness" can be defined as "about 20 mm", "about 50 mm", "about 70 mm", i.e. in the form of fuzzy numbers.
The term-set and the extended term-set under the conditions of the example can be characterized by the membership functions shown in Fig. 1.5 and 1.6.
Rice. 1.5. Membership functions of fuzzy sets: "Small thickness" = A 1,"Average thickness" = A 2, "Large thickness" = A 3
Rice. 1.6. Membership function of fuzzy set "Small or medium thickness" = A 1 ∪ A 2
Fuzzy numbers
Fuzzy numbers- fuzzy variables defined on the numerical axis, i.e. a fuzzy number is defined as a fuzzy set A on the set of real numbers ℝ with the membership function μ A(X) ϵ, where X — real number, i.e. X ϵ ℝ.
Fuzzy number It's okay if max μ A(x) = 1; convex, if for any X ≤ at ≤ z performed
μ A (x)≥ μ A(at) ˄ μ A(z).
A bunch of α -level fuzzy number A defined as
Aα= {x/μ α (x) ≥ α } .
Subset S A⊂ ℝ is called the support of a fuzzy number A, if
S A= { x/μ A(x) > 0 }.
Fuzzy number And unimodal if condition μ A(X) = 1 is only valid for one point on the real axis.
Convex fuzzy number A called fuzzy zero, if
μ A(0) = sup ( μ A(x)).
Fuzzy number And positively, if ∀ xϵ S A, X> 0 and negatively, if ∀ X ϵ S A, X< 0.
Operations on fuzzy numbers
Extended binaries arithmetic operations(addition, multiplication, etc.) for fuzzy numbers are determined through the corresponding operations for crisp numbers using the principle of generalization as follows.
Let A and V- fuzzy numbers, and - a fuzzy operation corresponding to an arbitrary algebraic operation * over ordinary numbers. Then (using here and in what follows the notation instead of instead of) we can write
Fuzzy numbers (L-R) -Type
Fuzzy numbers (L-R) -type are a kind of fuzzy numbers special kind, i.e. asked by certain rules in order to reduce the amount of calculations during operations with them.
Membership functions of fuzzy numbers (L-R) -type are specified using functions of a real variable L ( x) and R ( x) satisfying the properties:
a) L (- x) = L ( x), R (- x) = R ( x);
b) L (0) = R (0).
Obviously, the class of (L-R) -functions includes functions whose graphs have the form shown in Fig. 1.7.
Rice. 1.7. Possible view(L-R) -functions
Examples of analytical task (L-R) -functions can be
Let L ( at) and R ( at) - (L-R) -type functions (specific). Uni-modal fuzzy number A With fashion a(i.e. μ A(a) = 1) using L ( at) and R ( at) is set as follows:
where a is the fashion; α > 0, β > 0 - left and right fuzzy coefficients.
Thus, for given L ( at) and R ( at) a fuzzy number (unimodal) is given by a triple A = (a, α, β ).
Tolerant fuzzy number is set, respectively, by four parameters A = (a 1 , a 2 , α, β ), where a 1 and a 2 - the boundaries of tolerance, i.e. in the interim [ a 1 , a 2], the value of the membership function is 1.
Examples of graphs of membership functions of fuzzy numbers (L-R) -type are shown in Fig. 1.8.
Rice. 1.8. Examples of graphs of membership functions of fuzzy numbers (L-R) -type
Note that in specific situations the functions L (y), R (y), and also parameters a, β fuzzy numbers (a, α, β ) and ( a 1 , a 2 , α, β ) should be selected in such a way that the result of an operation (addition, subtraction, division, etc.) is exactly or approximately equal to a fuzzy number with the same L (y) and R (y), and the parameters α" and β" the result did not go beyond the limitations on these parameters for the initial fuzzy numbers, especially if the result will later participate in operations.
Comment... Solving problems of mathematical modeling of complex systems using the apparatus of fuzzy sets requires performing a large amount of operations on different kinds linguistic and other fuzzy variables. For the convenience of performing operations, as well as for input-output and data storage, it is desirable to work with membership functions of a standard type.
Fuzzy sets, which have to operate in most problems, are, as a rule, unimodal and normal. One of possible methods approximation of unimodal fuzzy sets is approximation by means of (L-R) -type functions.
Examples of (L-R) -representations of some linguistic variables are given in table. 1.2.
Table 1.2. Possible (L- R) -representation of some linguistic variables
