2.9.1. Definition. The methods of the theory of fuzzy sets describe semantic concepts, for example, for the concept of "reliability of the operation of a node" one can define such components as "not great value node reliability”, “average value of node reliability”, “large value of node reliability”, which are defined as fuzzy sets on the base set defined by all possible values of reliability values.
A generalization of the description of linguistic variables from a formal point of view is the introduction of fuzzy and linguistic variables.
H fuzzy variable is called a triple of sets, where a- the name of the fuzzy variable, X- domain of definition, - fuzzy subset in the set X, describing restrictions on possible values variable a.
linguistic variable is called a set of sets
It follows from the definition that a linguistic variable is a variable given on a quantitative (measured) scale and taking values that are words or phrases of the natural language of communication. Fuzzy variables describe the values of a linguistic variable. On fig. 2.20 shows the relationship of the basic concepts.
Thus, linguistic variables can describe concepts that are difficult to formalize in the form of a qualitative, verbal description. A linguistic variable and all its values are associated in the description with a specific quantitative scale, which, by analogy with the base set, is sometimes called the base scale.
Using linguistic variables, it is possible to formalize qualitative information in management systems, which is formulated by specialists (experts) in verbal form. This allows you to build fuzzy models of control systems (fuzzy controllers).
2.9.2. Type of membership functions. Consider the requirements that are put forward to the type of membership functions of fuzzy sets that describe the terms of linguistic variables.
Let the linguistic variable
A bunch of T order according to the expression
"T i ,T j нT i>j"($xнC i)("yнC j)(x>y). (2.5)
Expression (2.5) requires that the term, which has support located to the left, receive a lower number. Then the term-set of any linguistic variable must satisfy the following conditions:
("T i нT)($xнX)( ); (2.8)
("b)($x 1 ОR 1)($x 2 ОR 2)("xОX)(x 1
Condition (2.6) requires that the values of the membership functions of the extreme terms (T1 and T2) at points x 1 and x2 respectively equal to one and to prevent the appearance of bell-shaped curves, as shown in Fig. 2.21.
Fig.2.21
Condition (2.7) forbids in the base set X pair of terms of type T1 and T2, T2 and T3. For a couple T1 and T2 there is no natural differentiation of concepts. For a couple T2 and T3 segment no concept matches. Condition (2.7) forbids the existence of terms of type T4, since every concept has at least one typical object. Condition (2.8) defines a physical constraint (within the framework of the problem) on the numerical values of the parameters.
On fig. 2.22 shows an example of setting membership functions for the terms "small price value", "small price value", "average price value", "sufficiently large price value", "large price value" of the linguistic variable "goods price".
2.9.3. Universal scales. Membership functions are built based on the results of expert surveys. However, the procedure for using fuzzy sets built based on the results of a survey of experts has a drawback, which lies in the fact that changing the conditions for the functioning of the model (object) requires adjustment of fuzzy sets. Adjustment can be made based on the results of a second survey of experts.
One of the ways to overcome this shortcoming is the transition to universal scales for measuring the values of the estimated parameters. The well-known technique for constructing universal scales involves describing the frequency of phenomena and processes, which is qualitatively defined in natural language by the following words and phrases: “never”, “extremely rarely”, “rarely”, “neither seldom nor often”, “often”, “ very often", "almost always" (or the like). A person uses these concepts to assess the subjective frequencies of events (the ratio of the number of events characterized by the concept to the total number of events).
The universal scale is built on a segment and is a series of intersecting bell-shaped curves corresponding to scaled frequency estimates. A universal scale of a linguistic variable for a given estimated parameter of the control object is built according to the following procedure.
1. According to the expert survey, the minimum xmin and maximum xmax variable scale values X.
2. Based on the results of an expert survey, membership functions of fuzzy sets are constructed that describe the values of a linguistic variable defined on a scale X. On fig. 2.23 shows an example of building membership functions , where a 1 , a 2 , a 3- some names of fuzzy variables.
3. Dots ( xmin,0) and ( xmax,1) are connected by a straight line p0, which is a display function p 0:X®.
4. The transition from the scale of relative frequencies of occurrence of events to frequency estimates, called quantifiers, occurs as follows.
For an arbitrary point z on the universal scale its prototype is built on the scale X. Then, according to the membership functions of the fuzzy sets corresponding to the terms a 1 , a 2 , a 3, the values are determined which are taken as the values of the corresponding membership functions at the point z on the universal scale . Function p (p=p0 in the considered example) is determined by an expert survey, because its choice affects the adequacy of the model to the object under study.
2.9.4. Multiple display functions. Unambiguous mapping function definition p limit the possibility of simultaneously taking into account different criteria in the control system, which may even be in antagonism with respect to each other, as well as the possibility of simultaneously taking into account various control conditions determined by the properties of the managed object.
Accounting for various conditions and criteria is determined by a subjective approach to solving the problem. If, however, we accept the function of displaying an unambiguous form, then various points of view will be reduced to a “common denominator” or, in fact, rejected. Practice shows that when managing processes that are difficult to formalize, taking into account all variants of the subjective view improves the quality of management, increasing resistance to various kinds of disturbances. However, it should be noted that it is almost never possible to take into account in people all the conditions that affect the choice of control, and all the characteristics of the object. Let us consider how the formalized consideration of control conditions is carried out when polling experts in the form of multiple display functions.
Let the expert surveys quantitatively and qualitatively determine the composition of the states of the object under study. The evaluation of the object's states is carried out according to the values of the signs y i ОY=(y 1 ,y 2 ,…,y p ).
It is impossible to take everything into account, therefore, when assessing states, it is better to use fuzzy categories, and fuzzy definitions of parameter values should be made with a certain degree of uncertainty about the correctness of the definitions. Indeed, one can always assume that there are some set of signs 
, not indicated by experts for various reasons: they were forgotten about; experts believe that these features do not affect accuracy; these parameters cannot be estimated, a consequence of technical difficulties.
Display functions p i ОP=(p 1 ,p 2 ,…,p b ) compared with the degree of confidence b(p i)н, which are set by experts. Also each display function pi weight matched a(pi), which corresponds to the level of expert competence. Weight values a(pi) are determined by the numbers of the segment . So the multiple display function P=(p 1 ,p 2 ,…,p b ) consists of a set of mapping functions pi, each of which is assigned a degree g(pi), defined as the conjunction of degrees of competence and confidence in the correct definition of mapping functions pi, i.e. g(pi)=a(pi)&b(pi).
The practical use of multiple functions has shown that, within a certain competence of experts, the constructed multiple display function is in good agreement with their individual opinions about the most plausible correspondence of fuzzy concepts to the points of the subject scale. X.
FUZZY LOGIC
Fuzzy operation "AND"
The specification of fuzzy sets allows one to generalize clear logical operations into their fuzzy counterparts. A fuzzy extension of the AND operation is the triangular norm T, other name T– norms are S–conorm. On fig. 3.1 shows a circuit representation T– norms.
The fuzzy operation "AND" in general form is defined as a mapping:
for which the axioms hold:
Axioms of boundary conditions T– norms:
Axiom of order:
In the theory of fuzzy sets, there are countless fuzzy operations "AND", which are determined by the ways of specifying the operation (T) under conditions (3.1) - (3.2). In the theory of fuzzy control, the following methods of specifying the operation (T) are applicable, listed below.
Boolean product[Zade, 1973]:
, "xО R. (3.6)
Algebraic product[Bandler, Kohout, 1980]:
, "xО R, (3.7)
where "." is a product accepted in classical algebra.
boundary product[Lukashevich, Giles, 1976]:
, (3.8)
where is the symbol of the boundary product.
Strong, or drastically (drastic), product[Weber, 1983]:
(3.9)
where D is the strong product symbol.
On fig. 3.2 shows the membership function for logical, algebraic, boundary and strong product of fuzzy sets.
Fuzzy operation "OR"
A fuzzy extension of the "OR" operation is S-norm. Sometimes the name is used T–conorm. On fig. 3.3 shows a circuit representation S– norms.
The fuzzy operation "OR" is defined as a mapping
for which mappings are performed:
Axioms of boundary conditions T– norms:
, ; (3.10)
Union axioms (intersections):
Axiom of order:
From an infinite number fuzzy operations that satisfy axioms (3.10) - (3.14), the following operations have found application in control theory, listed below.
Boolean sum[Zade, 1973]:
, "xО R. (3.15)
Algebraic sum[Bandler & Kohout, 1980]:
, "xО R, (3.16)
Limit amount[Lukashevich, Giles, 1976]:
, (3.17)
Strong, or drastic, sum[Weber, 1983]:
(3.18)
Comparison of axioms T-norms with axioms S-norms shows that the difference in them is only in the axioms of the boundary conditions.
On fig. 3.4 shows the membership function for a logical, algebraic, boundary and strong sum of fuzzy sets.
Fuzzy operation "NOT"
The fuzzy "NOT" operation is defined as a mapping for which the following axioms hold:
The set of mappings satisfying axioms (3.19) - (3.21) is a fuzzy negation. The operation of fuzzy negation in the form of a scheme is shown in fig. 3.5.
Of the infinite number of fuzzy operations "NOT" that satisfy the axioms (3.19) - (3.21), the following operations listed below have found application in control theory.
Fuzzy "NOT" by Zada(1973) is defined as subtraction from unity:
. (3.22)
Fuzzy "NOT" according to Sugeno(1977) or the l-complement is defined as the formula
. (3.23)
At l=0 equation (3.23) coincides with equation (3.22).
Fuzzy "NOT" according to Yager(1980) is defined as:
, (3.24)
where p>0- parameter. At p=1 equation (3.24) coincides with equation (3.22).
For T- norms and S- norms may exist various options negations due to an infinite number of possible fuzzy "NOT" operations. However, it is desirable to choose such negation options that satisfy the conditions:
These conditions, by analogy with clear logic, are called de Morgan's fuzzy laws. Operations (3.25) and (3.26) are called mutually dual, because in fuzzy set theory, it is proved that (3.25) implies (3.26) and, conversely, (3.26) implies (3.25).
The following are also mutually dual. fuzzy operations:
; (3.29)
Algebra of fuzzy inference
3.4.1. Base of fuzzy rules. In fuzzy logic, there is the concept of a fuzzy proposition. A fuzzy sentence is defined as a statement "". Symbol " x” denotes a physical quantity (current, voltage, pressure, speed, etc.), the symbol “ ” denotes a linguistic variable (LP), and the symbol “ p"- abbreviation of proposition - a proposal. For example, in the statement "the magnitude of the current is large" of the physical variable x is the "current value" that can be measured by the current sensor. The fuzzy set is defined by the LP "big" and formalized by the membership function m A (x). The link "is" corresponds to the ordering operation in the form of equality, which is denoted by the symbol "=". Gets the formalized form of the sentence " » .
A fuzzy sentence may consist of several separate fuzzy sentences connected by links "AND", "OR". The choice of logical connectives "AND", "OR" from the meaning and context of the sentences, from the relationship between them. Note that the operations of fuzzy "AND" and "OR" according to Zadeh (formulas (3.6) and (3.15)) in control theory are preferable in relation to the rest, because they have no redundancy. When fuzzy sentences are not equivalent, but are correlated and interrelated, then it is possible to use T- norms and S- norms according to Lukashevich (formulas (3.8) and (3.17)).
Offer p can be represented as a fuzzy relation R with membership function: . To compile a fuzzy sentence consisting of several separate fuzzy sentences connected by "AND" links, the "if" indicator is used. As a result, we obtain a system of conditional fuzzy statements:
.
Fuzzy sentences are called conditions or prerequisites.
The set of conditions allows us to construct a set conclusions or conclusions. In this case, the "then" indicator is used.
Production fuzzy rule(fuzzy rule) is a set of conditions and conclusions:
R 1: if x 1 = and x 2 = and …, then y 1 = and y 2 = and …
……………………………………………………………,
where symbol R1- the abbreviation "rule" - the rule.
For example, the rule for water temperature control is formulated as follows: R1: if the water temperature is cold and the air temperature is cold, then turn the valve hot water to the left at a large angle and the cold water valve to the right at a large angle.
Fuzzy conditions for solving the problem:
-x 1- water temperature (measured by a sensor); - cold;
-x2- air temperature (measured by a sensor); - cold;
Fuzzy inference conditions:
-y 1- angle of rotation of the valve to the left, - large;
-y2- angle of rotation of the valve to the right, - large.
This linguistic fuzzy rule corresponds to a formalized notation:
R 1: if x 1 = and x 2 = , then y 1 = and y 2 = , (3.31)
where , , and are fuzzy sets, given by functions accessories.
The set of fuzzy production rules forms the base of fuzzy rules , where R i: if …, then …;. For the base of fuzzy rules, following properties: continuity, consistency, completeness.
Continuity is defined by the following concepts: an ordered set of fuzzy sets; adjacent fuzzy sets.
Set of fuzzy sets (A i ) called orderly, if they have an order relation: «<»:A 1 <…
If the set of fuzzy sets { } ordered, then the sets and , and are called adjacent provided that these fuzzy sets are overlapping.
The fuzzy rule base is called continuous, if for the rules
Rk: if x 1 = and x 2 = , then y= and k'¹k
conditions are met:
‑ Ù and are adjacent;
‑ Ù and are adjacent;
- and are adjacent.
We will consider the consistency of the fuzzy rule base using an example. The base of fuzzy rules for controlling the robot is given as:
………………………………….
R i: if the obstacle is ahead, then move to the left,
R i +1: if the obstacle is ahead, then move to the right,
……………………………………
The rule base is inconsistent.
An example of a consistent fuzzy rule base is as follows:
R 1: if x 1 = or x 2 = , then y= ;
R 2: if x 1 = or x 2 = , then y= ;
R 3: if x 1 = or x 2 = , then y= .
If the rules contain two conditions and one output, then these rules are a system with two inputs x 1 and x2 and one exit y. This system can be represented in matrix form:
| x2 | x 1 | ||
| y= | |||
| y= | |||
| y= |
The base of fuzzy rules is consistent.
The concept of fuzzy and linguistic variables is used in describing objects and phenomena using fuzzy sets.
Fuzzy variable characterized by the triple (α, X, A), where
α — the name of the variable;
X— universal set (domain α);
BUT is a fuzzy set on x, describing restrictions (i.e. µ A(x) ) on the values of the fuzzy variable α.
Linguistic a variable (LP) is 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 with 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) character 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 suggests that the context allows for possible ambiguities to be resolved.
Example. Let the expert determine the thickness of the manufactured product using the concepts "Small thickness", "Medium thickness" and "Large thickness", while the 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 is 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 thin thickness", etc.;
M- procedure for setting X= fuzzy subsets BUT 1 = "Small thickness", BUT 2 = "Medium thickness", A 3 = "Large thickness", as well as fuzzy sets for terms from G (T) in accordance with the translation rules of fuzzy connectives and modifiers "and", "or", "not", "very", "slightly" and other operations on fuzzy sets of the form: BUT⋂AT,A∪ATA, CON A =A 2 , DIL A \u003d A 0.5 etc.
Comment. Along with the basic values discussed above, the linguistic variable "Thickness" (T =(“Small thickness”, “Medium thickness”, “Large thickness”)), values are possible depending on the X definition area. 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 ,"Medium thickness" = BUT 2 , "Greater thickness" = BUT 3
Rice. 1.6. Membership function of the fuzzy set "Small or medium thickness" = A 1 ∪ BUT 2
fuzzy numbers
fuzzy numbers- fuzzy variables defined on the number axis, i.e. a fuzzy number is defined as a fuzzy set BUT on the set of real numbers ℝwith membership function μ A(X) ϵ , where X is a real number, i.e. X ϵ ℝ.
fuzzy number And it's ok 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 - fuzzy number level BUT defined as
Aα= {x/μ α (x) ≥ α } .
Subset S A⊂ ℝ is called the carrier of the fuzzy number BUT, if
S A= { x/μ A(x) > 0 }.
fuzzy number And unimodal if condition μ A(X) = 1 is valid only for one point of the real axis.
convex fuzzy number BUT called fuzzy zero, if
μ A(0) = sup ( µ A(x)).
fuzzy number And positively if ∀ xϵ S A, X> 0 and negative if ∀ X ϵ S A, X< 0.
Operations on fuzzy numbers
Extended binary arithmetic operations (addition, multiplication, etc.) for fuzzy numbers are defined through the corresponding operations for crisp numbers using the generalization principle as follows.
Let be BUT and AT- fuzzy numbers, and - fuzzy operation corresponding to an arbitrary algebraic operation * on ordinary numbers. Then (using here and below the notation instead of instead of ) we can write
Fuzzy Numbers (L-R)-type
(L-R)-type fuzzy numbers are a kind of fuzzy numbers of a special kind, i.e. set according to certain rules in order to reduce the amount of calculations during operations on them.
Membership functions of (L-R)-type fuzzy numbers are specified using functions of the 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 form of (L-R)-functions
Examples of analytical specification of (L-R)-functions can be
Let L( at) and R( at) are functions of (L-R)-type (concrete). Uni-modal fuzzy number BUT with fashion a(i.e. μ A(a) = 1) using L( at) and R( at) is given as follows:
where a is the mode; α > 0, β > 0 — left and right fuzziness coefficients.
Thus, for given L( at) and R( at) the fuzzy number (uni-modal) is given by the triple BUT = (a, α, β ).
The tolerant fuzzy number is given, respectively, by the four parameters BUT = (a 1 , a 2 , α, β ), where a 1 and a 2 - the limits of tolerance, i.e. in the interim [ a 1 , a 2 ] the value of the membership function is equal to 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), as well as 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 the operation (addition, subtraction, division, etc.) is exactly or approximately equal to a fuzzy number with the same L (y) and R (y), and parameters α" and β" of the result did not go beyond the limits on these parameters for the original 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 various kinds of 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 form.
The fuzzy sets that one has to operate in most problems are, as a rule, unimodal and normal. One of the possible methods for approximating unimodal fuzzy sets is approximation using (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
S.D. Shtovba "Introduction to the theory of fuzzy sets and fuzzy logic"
1.7. fuzzy logic
Fuzzy logic is a generalization of traditional Aristotelian logic to the case when truth is considered as a linguistic variable that takes values like: "very true", "more or less true", "not very false", etc. The indicated linguistic values are represented by fuzzy sets.
1.7.1. Linguistic variables
Recall that a variable is called linguistic if it takes values from a set of words or phrases of some natural or artificial language. The set of admissible values of a linguistic variable is called a term-set. Setting the value of a variable in words, without using numbers, is more natural for a person. Every day we make decisions based on linguistic information such as: "very high temperature"; "long trip"; "quick response"; "beautiful bouquet"; "harmonious taste", etc. Psychologists have established that in the human brain, almost all numerical information is verbally recoded and stored as linguistic terms. The concept of a linguistic variable plays an important role in fuzzy inference and in decision making based on approximate reasoning. Formally, a linguistic variable is defined as follows.
Definition 44.linguistic variable is given by five , where - ; variable name; - ; term-set, each element of which (term) is represented as a fuzzy set on the universal set; - ; syntactic rules, often in the form of a grammar, that generate the name of terms; - ; semantic rules defining the membership functions of fuzzy terms generated by syntactic rules .
Example 9. Consider a linguistic variable named "room temperature". Then the remaining quadruple can be defined as follows:
Table 4 - Rules for calculating membership functions
Graphs of the membership functions of the terms "cold", "not very cold", "comfortable", "more or less comfortable", "hot" and "very hot" of the linguistic variable "room temperature" are shown in Fig. thirteen.
Figure 13 - Linguistic variable "room temperature"
1.7.2. fuzzy truth
A special place in fuzzy logic is occupied by the linguistic variable "truth". In classical logic, truth can only take two values: true and false. In fuzzy logic, truth is "fuzzy". Fuzzy truth is defined axiomatically, and different authors do it differently. The interval is used as a universal set to set the linguistic variable "truth". Ordinary, clear truth can be represented by fuzzy singleton sets. In this case, a clear concept of true will correspond to the membership function 
, and a clear concept is false - ; 
, .
To set the fuzzy truth, Zade proposed the following membership functions for the terms "true" and "false":
;
where - ; a parameter that determines the carriers of fuzzy sets "true" and "false". For a fuzzy set "true" the carrier will be the interval , and for a fuzzy set false" - ; .
The membership functions of the fuzzy terms "true" and "false" are shown in fig. 14. They are built with the value of the parameter . As you can see, the graphs of the membership functions of the terms "true" and "false" are mirror images.
Figure 14 - Linguistic variable "truth" according to Zadeh
To set fuzzy truth, Baldwin proposed the following membership functions of fuzzy "true" and "false":
The quantifiers "more or less" and "very" are often applied to fuzzy sets "true" and "false", thus obtaining the terms "very false", "more or less false", "more or less true", "very true" , "very, very true", "very, very false", etc. Membership functions of new terms are obtained by performing operations of concentration and stretching of fuzzy sets "true" and "false". The concentration operation corresponds to squaring the membership function, and the stretching operation corresponds to raising to the power of ½. Therefore, the membership functions of the terms "very, very false", "very false", "more or less false", "more or less true", "true", "very true", and "very, very true" are given as follows:
Graphs of the membership functions of these terms are shown in fig. fifteen.
Figure 15 - Linguistic variable "truth" according to Baldwin
1.7.3. Fuzzy Boolean Operations
First, briefly recall the main provisions of ordinary (Boolean) logic. Consider two statements A and B, each of which can be true or false, i.e. take the values "1" or "0". For these two statements, there are in total various logical operations, of which only five are meaningfully interpreted: AND (), OR (), exclusive OR (), implication () and equivalence (). The truth tables for these operations are given in Table. 5.
Table 5 - Boolean logic truth tables
Suppose that a logical statement can take not two truth values, but three, for example: "true", "false" and "indeterminate". In this case, we will deal not with two-valued, but with three-valued logic. The total number of binary operations, and, consequently, truth tables, in three-valued logic is . Fuzzy logic is a kind of multivalued logic in which truth values are given by linguistic variables or terms of the linguistic variable "truth". Rules for performing fuzzy logical operations are obtained from Boolean logical operations using the principle of generalization.
Definition 45. Let's designate fuzzy logical variables through and , and the membership functions that define the truth values of these variables through and , . Fuzzy Boolean Operations AND (), OR (),
NOT () and implication () are performed according to the following rules:
;
In multivalued logic, logical operations can be specified by truth tables. In fuzzy logic, the number of possible truth values can be infinite, therefore, in general, a tabular representation of logical operations is impossible. However, in tabular form it is possible to present fuzzy logical operations for a limited number of truth values, for example, for a term-set ("true", "very true", "not true", "more or less false", "false"). For three-valued logic with fuzzy truth values T - ; "true", F - ; "false" and T+F - "unknown" L Zadeh proposed the following linguistic truth tables:
By applying the rules for performing fuzzy logical operations from definition 45, truth tables can be extended for more terms. How to do this, consider the following example.
Example 10 The following fuzzy truth values are given:
Applying the rule from definition 45, we find the fuzzy truth of the expression "almost true OR true":
Let's compare the resulting fuzzy set with the "more or less true" fuzzy set. They are almost equal, which means:
As a result of performing logical operations, a fuzzy set is often obtained, which is not equivalent to any of the previously introduced fuzzy truth values. In this case, among the fuzzy truth values, it is necessary to find one that corresponds to the result of performing the fuzzy logical operation to the maximum extent. In other words, it is necessary to carry out the so-called linguistic approximation, which can be considered as an analogue of the approximation of the empirical statistical distribution by standard distribution functions of random variables. As an example, we present the linguistic truth tables proposed by Baldwin for those shown in Figs. 15 fuzzy truth values:
| indefinitely |
indefinitely |
||
| indefinitely |
indefinitely |
||
| indefinitely |
indefinitely |
indefinitely |
indefinitely |
| very true |
very true |
||
| more or less true |
more or less true |
1.7.3. Fuzzy knowledge base
Definition 46.Fuzzy knowledge base is a set of fuzzy rules "If - then" that determine the relationship between the inputs and outputs of the object under study. The generalized format of fuzzy rules is as follows:
If asending rule,thenrule conclusion.
The premise of the rule or antecedent is a statement like "x is low", where "low" is a term (linguistic value) defined by a fuzzy set on the universal set of the linguistic variable x. Quantifiers "very", "more or less", "not", "almost", etc. can be used to modify antecedent terms.
The conclusion or consequence of the rule is a statement like "y is d", in which the value of the output variable (d) can be given:
- fuzzy term: "y is high";
- decision class: "y has bronchitis"
- a clear constant: "y=5";
- clear function of the input variables: "y=5+4*x".
If the value of the output variable in the rule is given by a fuzzy set, then the rule can be represented by a fuzzy relation. For the fuzzy rule "If x is , then y is", the fuzzy relation is given on the Cartesian product , where - ; universal set of input (output) variable. To calculate the fuzzy ratio, you can use fuzzy implication and the t-norm. When using the operation of finding the minimum as a t-norm, the fuzzy ratio is calculated as follows:
Example 11. The following fuzzy knowledge base describes the relationship between the age of the driver (x) and the possibility of a traffic accident (y):
If ax = Young,theny = high;
If ax = Medium,theny = low;
If ax = very old,theny = High.
Let the membership functions of terms have the form shown in fig. 16. Then the fuzzy relations corresponding to the rules of the knowledge base will be the same as in fig. 17.
Figure 16 - Membership functions of terms
Figure 17 - Fuzzy relationships corresponding to the rules of the knowledge base from example 11
To specify multidimensional dependencies "inputs-outputs", fuzzy logical operations AND and OR are used. It is convenient to formulate the rules in such a way that within each rule the variables are combined by the logical AND operation, and the rules in the knowledge base are connected by the OR operation. In this case, the fuzzy knowledge base connecting the inputs 
with output , can be represented in the following form.
An important aspect of the concept linguistic variable is that this variable is of a higher order than the fuzzy variable , in the sense that the values linguistic variable are fuzzy variables. For example, the values linguistic variable"AGE" can be: "YOUNG, OLDER, OLD, VERY OLD, NOT YOUNG AND NOT OLD", etc. Each of these values is the name fuzzy variable. If is the name of the fuzzy variable, then the constraint imposed by this name can be interpreted as the meaning fuzzy variable .
Another important aspect of the concept linguistic variable is that linguistic variable there are two rules:
- Syntactic, which can be given in the form of a grammar that generates the name of the variable's values;
- Semantic, which defines an algorithmic procedure for computing the meaning of each value.
Definition. linguistic variable characterized by a set of properties , in which:
Variable name;
Denotes the term-set of the variable , i.e. a set of names of linguistic values of a variable , and each of these values is fuzzy variable with values from the universal set with the base variable ;
Syntactic rule that generates the names of the values of the variable ;
The semantic rule that matches each fuzzy variable its meaning, i.e. fuzzy subset universal set .
The specific name generated by the syntactic rule is called a term. A term that consists of one word or of several words always appearing together with each other is called an atomic term. A term that consists of more than one atomic term is called compound term.
Example. Consider linguistic variable named "ROOM TEMPERATURE". Then the remaining four 
, can be defined like this:
