Direct fuzzy inference

Introduction

In 1965, the journal “Information and Control” published the work of L. Zadeh entitled “Fuzzy sets”. This name is translated into Russian as fuzzy sets . The motive was the need to describe such phenomena and concepts that are ambiguous and inaccurate. The previously known mathematical methods that used the classical set theory and two-valued logic did not allow solving problems of this type.



Using fuzzy sets, one can formally define inaccurate and ambiguous concepts, such as “high temperature” or “big city”. To formulate the definition of a fuzzy set, it is necessary to specify the so-called area of ​​reasoning. For example, when we evaluate the speed of a car, we limit ourselves to the range X = [0, Vmax], where Vmax is the maximum speed that a car can develop. It must be remembered that X is a clear set.

Basic concepts

A fuzzy set A in some nonempty space X is the set of pairs

where

is the membership function of the fuzzy set A. This function assigns to each element x the degree of its membership in the fuzzy set A.

Continuing the previous example, we consider three inaccurate formulations:
- “Low speed of the car”;
- “Average vehicle speed”;
- "High speed car."
The figure shows fuzzy sets corresponding to the above formulations using membership functions.

At a fixed point, X = 40km / h. the membership function of the fuzzy set "low speed car" takes a value of 0.5. The same value is taken by the membership function of a fuzzy set “average vehicle speed”, while for the set “high vehicle speed” the value of the function at this point is 0.

The function T of two variables T: [0, 1] x [0, 1] -> [ 0, 1] is called a T-norm if:
- is not increasing with respect to both arguments: T (a, c) <T (b, d) for a <b, c <d;
- is commutative: T (a, b) = T (b, a);
- satisfies the connected condition: T (T (a, b), c) = T (a, T (b, c));
- satisfies the boundary conditions: T (a, 0) = 0, T (a, 1) = a.

Direct fuzzy conclusion

A fuzzy conclusion is a process in which some consequences, possibly also fuzzy, are obtained from fuzzy premises. Approximate reasoning underlies a person’s ability to understand natural language, understand handwriting, play games that require mental effort, in general, make decisions in a complex and incompletely defined environment. This ability to reason in qualitative, inaccurate terms distinguishes human intelligence from the intelligence of a computer.

The basic rule of inference in traditional logic is the modus ponens rule, according to which we judge the truth of the statement B by the truth of the statements A and A -> B. For example, if A is the statement “Stepan is an astronaut,” B is the statement “Stepan flies into space” , then if the statements “Stepan is an astronaut” and “If Stepan is an astronaut, then he flies into space,” the statement “Stepan flies into space” is true.

However, unlike traditional logic, the main tool of fuzzy logic is not the modus ponens rule, but the so-called compositional inference rule, a very special case of which is the modus ponens rule.

Suppose that there is a curve y = f (x) and a value x = a is given. Then from the fact that y = f (x) and x = a, we can conclude that y = b = f (a).

We now generalize this process, assuming that a is an interval and f (x) is a function whose values ​​are intervals. In this case, to find the interval y = b corresponding to the interval a, we first construct the set a 'with the base a and find its intersection I with the curve whose values ​​are the intervals. Then we project this intersection on the OY axis and obtain the desired value of y in the form of an interval b. Thus, from the fact that y = f (x) and x = A is a fuzzy subset of the OX axis, we get the value of y as a fuzzy subset B of the OY axis.

Let U and V be two universal sets with the base variables u and v, respectively. Let A and F be fuzzy subsets of the sets U and U x V. Then the compositional rule of inference states that fuzzy sets A and F imply a fuzzy set B = A * F.

Let A and B be fuzzy statements and m (A), m (B) the corresponding membership functions. Then the implication A -> B will correspond to some membership function m (A -> B). By analogy with traditional logic, we can assume that

Then

, however, this is not the only generalization of the implication operator, there are others.

Implementation

To implement the direct fuzzy inference method, we need to choose the implication operator and the T-norm.
Starting T-norm will be a minimum function:

1. def t_norm(v1, v2):
2.     return min(v1, v2)

and the Gödel function will be the implication operator:

1. def impl(v1, v2):
2.     '''
       Godel implication
       '''
3.     if v1 <= v2:
4.         return 1.0
5.     else:
6.         return v2

The input data will contain knowledge (fuzzy sets) and rules (implications), for example:
A = {(x1, 0.0), (x2, 0.2), (x3, 0.7), (x4, 1.0)}.
B = {(x1, 0.7), (x2, 0.4), (x3, 1.0), (x4, 0.1)}.
A => B. The

implication will be presented in the form of a Cartesian matrix, each element of which is calculated using the selected implication operator (in this example, the Gödel function):

1. def compute_impl(set1, set2):
2.     '''
       Computing implication
       '''
3.     relation = {}
4.     for i in set1.items():
5.         relation[i] = {}
6.         for j in set2.items():
7.             v1 = set1.value(i)
8.             v2 = set2.value(j)
9.             relation[i][j] = impl(v1, v2)
10.     return relation

For the data above, this will be:
Conclusion:
A => B.
    x1 x2 x3 x4
x1 1.0 1.0 1.0 1.0
x2 1.0 1.0 1.0 0.1
x3 1.0 0.4 1.0 0.1
x4 0.7 0.4 1.0 0.1

Next, the result is a new set:

1. def conclusion(set, relation):
2.     '''
       Conclusion  
       '''
3.     conl_set = []
4.     for i in relation:
5.         l = []
6.         for j in relation[i]:
7.             v_set = set.value(i)
8.             v_impl = relation[i][j]
9.             l.append(t_norm(v_set, v_impl))
10.         value = max(l)
11.         conl_set.append((i, value))
12.     return conl_set

Result:
B '= {(x1, 1.0), (x2, 0.7), (x3, 1.0), (x4, 0.7)}.

Sources

• Rutkovskaya D., Pilinsky M., Rutkovsky L. Neural networks, genetic algorithms and fuzzy systems: Per. from polish I. D. Rudinsky. - M .: Hot line - Telecom, 2006. - 452 p.: Ill.
• Zadeh LA Fuzzy Sets, Information and Control, 1965, vol. 8, s. 338-353