The approaches to the solution of various problems of artificial intelligence methods are proposed. In particular, the problem of knowledge representation by means of fuzzy specifications in expert systems, the problem of recognizing the structures of the proteins of different organization levels and the problem of building linguistic models in fuzzy Boolean variables logic are considered. All methods are based on the ideas of inductive mathematics. To investigate a reliability of these methods is possible only with the help of the theory of probability or possibility theory.
It is known that the fuzzy sets [
The fuzzy specification of problem means ordered set of fuzzy instructions (fuzzy rules, linguistic rulles). The fuzzy specification of the problem with the algorithm during fulfilling which the approximate (fuzzy) solution of the problem is received will be called as fuzzy system of logical inference.
Let x1, . . . , xn are input linguistic variables and y – output linguistic variable [
if x1 is Am1 ∧ · · · ∧ xn ∈ Amn then y is Bm, where Aij and Bi – fuzzy sets, symbol “∧” is interpreted as t-norm of fuzzy sets.
′ The al′gorithm of calculating the output of such specification under the inputs
′ ′ αi = min hmax A1(x1) ∧ Ai1(x1) , . . . , max An(xn) ∧ Ain(xn) i ; ′ Bi(y) = min(αi, Bi(y));
′ ′
B(y) = max B1(y), . . . , Bm(y) .
Let X1 = {5, 10, 15, 20}, X2 = {5, 10, 15, 20}, X3 = {35, 36, 37, 38, 39, 40} – spaces for determining the values of linguistic variables: x1 = “Coughing“ = {“weak“, “moderate“, “strong“}, x2 = “Runningnose“ = {“weak“, “moderate“, “strong“}, x3 = “T emperature“ = {“normal“, “raised“, “high“, “veryhigh“}, accordingly.
Determine the elements of these sets:
“Coughing“ : “weak“ = 1/5 + 0.5/10; “Runningnose“ : “weak“ = 1/5 + 0.5/10; “moderate“ = 0.5/5 + 0.7/10 + 1/15; “strong“ = 0.5/10 + 0.7/15 + 1/20. “moderate“ = 0.5/10 + 1/15; “strong“ = 0.7/15 + 1/20. “raised“ = 0.5/37 + 1/38; Let Y = {inf luenza, sharp respiratory disease, angina, pneumonia} – space for determining the value of linguistic variable y. Then the dependence of the patient’s disease on his symptoms can be described by the following systems of specifications:
if x1 is “weak“ ∧ x2 is “weak“ ∧ x3 is “raised“ then y is “0.5/inf luenza + 0.5/sharp respiratory disease + 0.4/angina+ +0.8/pneumonia“; if x1 is “weak“ ∧ x2 is “moderate“ ∧ x3 is “high“ then y is “0.8/inf luenza + 0.7/sharp respiratory disease + 0.8/angina+ +0.3/pneumonia“; if x1 is “weak“ ∧ x2 is “moderate“ ∧ x3 is “very high“ then y is “0.9/inf luenza + 0.7/sharp respiratory disease + 0.8/angina+ +0.2/pneumonia“; ′ If to the input′ x1 of this algorithm to supply value A1 = 1′/5 + 0.7/10, to the input x2 – value A2 = 1/5 + 0.5/10, to the input x3 – value A3 = 1/36 + 0.9/37, then in accordance with procedure of fulfilling the algorithm of the fuzzy system of logical inference the fuzzy solution of the problem will be received
B = 0.5/inf luenza + 0.5/sharp respiratory disease+
+0.4/angina + 0.5/pneumonia.
The problem of searching the symptoms using fuzzy diagnosis can be inverse to this pr′oblem. Specifically, let output of the fuzz′y system of logical in′ference with inputs A1 = x1/5 + x2/10 + x3/15 + x4/20, A2 = 1/5 + 0.5/10, A3 = 1/36 + 0.9/37 is fuzzy set B = 0.5/inf luenza + 0.5/sharp respiratory disease + 0.4/angina + 0.5/pneumonia. Aggregation of the individual outputs leads to the next system of fuzzy relation equations: min [max(x1 ∧ 1, x2 ∧ 0.5), 0.4] = 0.4, min [max(x1 ∧ 1, x2 ∧ 0.5), 0.5] = 0.5.
Solving it the value of the symptom “Coughing“ will be received, which is described by the fuzzy set “Coughing“ = 0.5/5 + 1/10.
To determine the probability of event A in the space of elementary events X, the notion of probable measure is introduces. It is numerical function P , which puts number
0 ≤ P (A) ≤ 1, P (X) = 1, P for any A1, A2, . . . such, as Ai ∩ Aj = ∅, with i 6= j.
The fuzzy event A in space X will be called the fuzzy set ∞ [ Ai i=1 !
∞ = X P (Ai)
i=1
A = {(x, μA(x)), x ∈ X} ,
where μA : X → [
P (A) =
X μA(x)P (x).
x∈A Considering this, it is possible to calculate the probabilities of any fuzzy event with provided probability measure. In particular, the probability of symptom “Coughing“ at probabilities distribution
P (5) = 0.4, P (10) = 0.4, P (15) = 0.1, P (20) = 0.1 can be calculated in the following way:
P (“Coughing“) = 0.5 · 0.4 + 1 · 0.4. 3
It is known [
The method of recognition of the secondary structure of DNA using fuzzy systems of logical inference is proposed. The problem is the following: it is necessary to build the fuzzy system of logical inference which using random amino acid sequence would define (as an fuzzy set) the secondary structure of central remainder (of the amino acid) of the input sequence.
To solve this problem at first it is necessary to design the fuzzy specification of the problem according to learning samples. One of the methods to build the system of fuzzy instructions according to numerical data consists of the following. Let’s the rules base with n inputs and one output is created. There are learning dates (samples) as the sets of pairs for that
(x1(i), x2(i), . . . , xn(i); d(i)), i = 1, 2, . . . , m, where xj (i) – inputs and d(i) – output, at that xj (i) ∈ {a1, a2, . . . , ak}, d(i) ∈ {b1, b2, . . . , bl}. It is necessary to build the fuzzy system of logical inference which would generate the correct output data according to random input values. The algorithm of solving of the provided problem consists in the following sequence of steps: 1. Dividing the space of inputs and outputs for areas (dividing learning data for groups on m1, . . . , mk lines, which means, each input and output is divided for 2N + 1 cuts where N for each input is selected individually. Separate areas (segments) will be called in the following way:
MN (lef t N ), . . . , M1(lef t 1), S(medium), D1(right 1), . . . , DN (right N ). Determination membership function for each areas. 2. Building fuzzy sets on the basis of learning samples (Each group mi learning data (x1(1), x2(1), . . . , xn(1); d(1)) (x1(2), x2(2), . . . , xn(2); d(2)) (x1(mi), x2(mi), . . . , xn(mi); d(mi)) is compared with the fuzzy sets of the form:
A(mi) =
1 A(nmi) = a(1)
1 a(n) 1 mi
/a1 + · · · + mi mi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a(1) k a(n) k mi /ak, /ak, /a1 + · · · + B(mi) = |b1| /b1 + · · · + |bl| /bl,
mi mi (x1(1), x2(1), . . . , xn(1); d(1)) (x1(2), x2(2), . . . , xn(2); d(2))
To build the fuzzy rules of logical inference the learning samples from 15 remains
of the protein MutS [
K V S E G G L I R E G Y D P D e − − − h h h h h h h h h h h V S E G G L I R E G Y D P D L − − − h h h h h h h h h h h h S E G G L I R E G Y D P D L D − − h h h h h h h h h h h h h E G G L I R E G Y D P D L D A − h h h h h h h h h h h h h h G G L I R E G Y D P D L D A L h h h h h h h h h h h h h h h K V S E G G L I R E G Y D P D V S E G G L I R E G Y D P D L; S E G G L I R E G Y D P D L D E G G L I R E G Y D P D L D A;
G G L I R E G Y D P D L D A L; The prediction belongs to the central remain, besides the following denotations are used: h – for spiral, e – for cylinder, “–” – other.
According to the algorithm, the teaching data is divided, for example, for 3 groups: and compared to each group with according fuzzy sets
A(m1), Ai(m2), Ai(m3), B(m1), B(m2), B(m3).
i
R(1) : if x1 isA(1m1) and x2 is A(m1) and . . . and x14 is A(1m41)then y is B(m1), 2 R(2) : if x1 isA(1m2) and x2 is A(m2) and . . . and x14 is A(1m42)then y is B(m2), 2 R(3) : if x1 isA(1m3) and x2 is A(m3) and . . . and x14 is A(1m43)then y is B(m3), 2
Using the algorithm of solving the specification, we will find the output received system of fuzzy instructions, if to the input the following amino acid sequence is supplied:
L K V S E G G L I R E G Y D P.
In accordance with the procedure of executing the algorithm we will get that the secondary structure of the remainder L is h. 4
The algebra of statement is one of the chapters of classic mathematical logic. The
statement means the variable which can be of two possible values – 0 or 1. Such variable is
called Boolean. In some cases generalizing the notion of Boolean variable to the notion
of fuzzy Boolean variable is useful [
Let p and q – fuzzy Boolean variables. Logical operations with such variables are determined like that:
p¯ = 1 − p, p ∧ q = min(p, q), p ∨ q = max(p, q).
p ∨ p¯ = 1, p ∧ p¯ = 0
Let p1, p2, . . . , pn – fuzzy Boolean variables. Function f (p1, . . . , pn) is called a
function of fuzzy Boolean variables if it takes value on the interval [
Function f of fuzzy Boolean variables is called analytical, if it can be presented by the formula, which includes operations ¬, ∧, ∨.
p → q = p¯ · q¯ ∨ p¯ · q ∨ p · q cannot be simplified.
One of the tasks Boolean variables function analysis consists in the following. It
is necessary to find out under the which conditions the values of analytical function,
for example f (p, q) = p ∧ q, includes in a given interval [α, β) of the segment [
It is know, that weather forecasters evaluate their forecasts using probability theory point out their forecasts with expressions: “sunny“ with probability p ∈ [0.7, 0.8); “windy“ with probability q ∈ [0.3, 0.5); “cloudy“ with probability h ∈ [0.8, 0.9).
f (p, q) = p → q = p¯ · q¯ ∨ p¯ · q ∨ p · q.
Let us suppose, that p ∈ [0.7, 0.8), q ∈ [0.3, 0.5). It is necessary to find out in which
intervals of the segment [
Considering,that p → q = p¯ · q¯ ∨ p¯ · q ∨ p · q and that
p ∈ [0.7, 0.8], q ∈ [0.3, 0.5] So we find that Then p¯ ∈ [0.2, 0.3], q¯ ∈ [0.5, 0.7]. p¯ · q¯ = min(p¯, q¯) ∈ [0.2, 0.3], p¯ · q = min(p¯, q) ∈ [0.2, 0.3], p · q = min(p, q) ∈ [0.3, 0.5].
p¯ · q¯ ∨ p¯ · q ∈ [0.2, 0.3], p¯ · q ∨ p¯ · q¯ ∨ p · q ∈ [0.3, 0.5].
Let us consider another option of the problem. Let us suppose, that as earlier,
p ∈ [0.7, 0.8), q ∈ [0.3, 0.5) analytical Boolean function is unknown, but is known
the interval of the segment [
One of the approaches to solve this problem consists in building and researching it is
so called linguistic model [
R : if p ∈ A1 ∧ q ∈ A2 then f ∈ B where A1 =′ 1/[0.7, 0.8], A2 = 1/[0.3, 0.5],′ B = 1/[0.3, 0.5]. ′It is necessary to find the output B of this fuzzy rule with inputs A1 = 1/[0.6, 0.8], A2 = 1/[0.3, 0.5]. After calculating according to the procedure we will have:
′
B (y) = 1/[0.3, 0.5].
Let us calculate the probability of windy under the condition that p → q ∈ [0.3, 0.5].
Let us suppose, that q ∈ [b1, b2] ⊆ [
0.5 ≤ b1 ⇒
5
The proposed approach to solving problems (based on fuzzy models) allows to simplify the methods of solving problems. Other approaches may be based on possibility theory. But there is a necessity for additional studies of the reliability issues in the first and in the second case.