Analysis of Fuzzy Logic Models

Of course, Kolmogorov stated probability and statistics on a new and very effective foundation set theory. For the first time in the history basic notions of probability theory have been defined precisely but simply. So a random event has been defined as a subset of a space, a random variable as a measurable function and its mean value as an integral. More precisely, abstract Lebesgue integral. It is hopeful to wait some new stimuls from the fuzzy generalization of the classical set theory. The aim of the chapter is a presentation of some results of the type.


Introduction
One of the most important results of mathematics in the 20th century is the Kolmogorov model of probability and statistics. It gave many impulses for research and develop so in theoretical area as well as in applications in a large scale of subjects.
It is reasonable to ask why the Kolmogorov approach played so important role in the probability theory and in mathematical statistics. In disciplines which have been very successfull for many centuries.
Of course, Kolmogorov stated probability and statistics on a new and very effective foundation -set theory. For the first time in the history basic notions of probability theory have been defined precisely but simply. So a random event has been defined as a subset of a space, a random variable as a measurable function and its mean value as an integral. More precisely, abstract Lebesgue integral. It is hopeful to wait some new stimuls from the fuzzy generalization of the classical set theory. The aim of the chapter is a presentation of some results of the type.

Fuzzy systems and their algebraizations
Any subset A of a given space Ω can be identified with its characteristic function if ω / ∈ A. From the mathematical point of view a fuzzy set is a natural generalization of χ A (see [73]). It is a function ϕ A : Ω → [0, 1].
Evidently any set (i.e. two-valued function on Ω, χ A →{ 0, 1}) is a special case of a fuzzy set (multi-valued function), ϕ A : Ω → [0, 1]. 2 Will-be-set-by- IN-TECH There are many possibilities for characterizations of operations with sets (union A ∪ B and intersection A ∩ B). We shall use so called Lukasiewicz characterization: (Here ( f ∨ g)(ω)=max( f (ω), g(ω)), ( f ∧ g)(ω)=min( f (ω), g(ω)).) Hence if ϕ A , ϕ B : Ω → [0, 1] are fuzzy sets, then the union (disjunction ϕ A or ϕ B of corresponding assertions) can be defined by the formula the intersection (conjunction ϕ A and ϕ B of corresponding assertions) can be defined by the formula In the chapter we shall work with a natural generalization of the notion of fuzzy set so-called IF-set (see [1], [2]), what is a pair Evidently a fuzzy set ϕ A : Ω → [0, 1] can be considered as an IF-set, where Here we have μ A + ν A = 1, while generally it can be μ A (ω)+ν A (ω) < 1 for some ω ∈ Ω. Geometrically an IF-set can be regarded as a function A : Ω → Δ to the triangle 4 Will-be-set-by-IN-TECH Example 1.4. Consider the unit interval [0, 1] in the set R of all real numbers. It will stay an MV-algebra, if we shall define two binary operations ⊕, ⊙ on [0, 1], one unary operation ¬ and the usual ordering ≤ by the following way: It is easy to imagine that a ⊕ b corresponds to the disjunction of the assertions a, b, a ⊙ b to the conjunction of a, b and ¬a to the negation of a.
By the Mundici theorem ( [48])any MV-algebra can be defined similarly as in Example 1.4, only the group R must be substitute by an arbitrary l-group.

Definition 1.2.
By an l-group we consider an algebraic system (G, +, ≤) such that

and the lattice operations
Put u =(1, 0) and define the MV-algebra Connections with the family of IF-sets (Definition 1.1) is evident. Hence we can formulate the main result of the section.
is partially ordered set with the smallest element 0 and the largest element 1, − is a partially binary operation satisfying the following statements:

Probability on IF-events
In IF-events theory an original terminology is used. The main notion is the notion of a state ( [21], [22], [57], [58], [61][, [62]). It is an analogue of the notion of probability in the Kolmogorov classical theory. As before F is the family of all IF-sets Of course, also the notion with the name probability has been introduced in IF-events theory.  Will-be-set-by-IN-TECH It is easy to see that the following property holds. Hence it is sufficient to characterize only the states ( [4], [5], [54]).
Now we prove two identities. First the implication: It can be proved by induction. The second identity is the following First it can be proved by induction the equality holding for every q ∈ N. Therefore Combining (2), (3), and the definition of P we obtain is an arbitrary S-measurable function, then there exists a sequence ( f n ) of simple measurable functions such that f n ր f . Evidently,

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Analysis of Fuzzy Logic Models for any measurable f : Now take our attention to the second term m(0, f + g) in the right side of the equality (1). First define M : S→[0, 1] by the formula As before it is possible to prove that M is a measure. Of course, Define Q : S→[0, 1] by the formula As before, it is possible to prove for any h : Ω → [0, 1], S-measurable. Combining (1), (4), and (5) we obtain A simple consequence of the representation theorem is the following property of the mapping P − αQ : S→R.  Proof. It is easy to see that any element (μ A , ν A ) ∈Mcan be presented in the form Generally, we can define m : so that m is an extension of m. Of course, we must prove that m is a state. First we prove that m is additive.
Before the continuity of m we shall prove its monotonicity.
Of course, as an easy consequence o Proposition 2.1 we obtain the inequality Analysis of Fuzzy Logic Models www.intechopen.com for any non-negative measurable f : Ω → R. Therefore

Observables
In the classical probability there are three main notions: probability = measure random variable = measurable function mean value = integral.
The first notion has been studied in the previous section. Now we shall define the second two notions.
Classically a random variable is such function ξ : is the σ-algebra generated by the family J of all intervals). Now instead of a σ-algebra S we have the family F of all IF-events, hence we must give to any Borel set A an element of F . Of course, instead of random variable we shall use the term observable ( [15], [16], [18], [32], [35]).
Therefore, we shall construct similarly Fix ω ∈ Ω and define μ, ν : Let μ × ν be the product of the probability measures μ, ν. Put Then Now we shall present two applications of the notion of the joint observable. The first is the definition of function of a finite sequence of observables, e.g. their sum. In the classical case Hence ξ + η can be defined by the help of pre-images: ., x n , and g : R n → R is defined by the equality g(u 1 , ..., u n )=u 1 + ... + u n .
The second application of the joint observable is in the formulation of the independency.
for any n ∈ N and any C 1 , ..., C n ∈ σ(J ). Now let us return to the notion of mean value of an observable. In the classical case where F is the distribution function of ξ. Example 3.2. Let x be discrete, i.e. there exist x i ∈ R, p i ∈ (0, 1], i = 1, ..., k such that The second classical case is the continuous distribution, where
Here we have two possibilities. The first i.e.
in the discrete case, and in the continuous case. The second possibility is the equality Since a = E(x) is known, it is sufficient to compute E(x 2 ). In the case we have g(t)=t 2 , hence In the discrete case we have in the continuous case we obtain

Sequences
In the section we want to present a method for studying of limit properties of some sequences (x n ) n , x n : B(R) →Fof observables ( [7], [25], [31], [32], [49]). The main idea is a representation of the given sequence by a sequence of random variables (ξ n ) n , ξ n : (Ω, S, P) → R. Of course, the space (Ω, S) depends on a concrete sequence (x n ) n , for different sequences various spaces (Ω, S, P) can be obtained.
The main instrument is the Kolmogorov consistency theorem ( [67]). It starts with a sequence of probability measures (μ n ) n , μ n : where k ∈ N, B ∈B(R k )=σ(J k ). Then by the Kolmogorov consistency theorem there exists exactly one probability measure P : σ(C) → [0, 1] such that P(A)=μ k (B).
If we denote by π n the projection π n : R N → R n , then we can formulate the assertion (6) by the equality for any B ∈C. Proof. Let C 1 , C 2 , ..., C n ∈B(R). Then by Definition 3.3. and Definition 3.1 hence μ n+1 |(J n × R)=μ n |J n . Of course, if two measures coincide on J n then they coincide on σ(J n ), too.
Now we shall formulate a translation formula between sequences of observables in (F , m) and corresponding random variables in (R N , σ(C), P) ( [67]).
As an easy corollary of Theorem 4.2 we obtain a variant of central limit theorem. In the classical case lim n→∞ P({ω; Of course, we must define for observables the element It is sufficient to put Let (x n ) n be a sequence of square integrable, equally distributed, independent observables, E(x n )=a, σ 2 (x n )=σ 2 (n = 1, 2, ...). Then Proof. We shall use the notation from the last two theorems. Then for C ∈ σ(J ) Moreover, hence ξ 1 , ..., ξ n are independent for every n. Put g n (u 1 , ..., Therefore by the classical central limit theorem Let us have a look to the previous theorem from another point of view, say, categorial. We had We can say that (η n ) n converges to φ in distribution. Of course, there are important possibilities of convergencies, at least in measure and almost everywhere.
A sequence (η n ) n of random variables (= measurable functions) converges to 0 in measure μ : for every ε > 0. And the sequence converges to 0 almost everywhere, if Analysis of Fuzzy Logic Models www.intechopen.com And η n → 0 almost everywhere, if the set {ω; η(ω) → 0} has measure 1. (iii) if (η n ) n converges P-almost everywhere to 0, then (y n ) n m-almost everywhere converges to 0.

Conditional probability
Conditional entropy (of A with respect to B) is the real number P(A|B) such that Generally P(A|S 0 ) can be defined for any σ-algebra S 0 ⊂S ,a sa nS 0 -measurable function such that P(A ∩ C)= C P(A|S 0 )dP, C ∈S o .
If S 0 = S, then we can put P(A|S 0 )=χ A , since χ A is S 0 -measurable, and C χ A dP = P(A ∩ C).
An important example of S 0 is the family of all pre-images of a random variable ξ : In this case we shall write P(A|S 0 )=P(A|ξ), hence By the transformation formula And exactly this formulation will be used in our IF-case, Of course, we must first prove the existence of such a mapping p(A|x) : R → R ( [34], [70], [72]). Recall that the product of IF-events is defined by the formula for any B ∈ σ(J ).

Algebraic world
At the end of our communication we shall present two ideas. The first one is in some algebraizations of the product The second idea is a presentation of a dual notion to the notion of IF-event.
In MV-algebras the product was introduced independently in [56] and [47]. Let us return to Definition 1.3 and Example 1.5.
is an MV-algebra with product. Indeed, On the other hand Denote 1. ∀a ∈ D : a * 1 = a; 2. ∀a, b ∈ D, a ≤ b, ∀c ∈ D : a * c ≤ b * c; 4. ∀(a n ) n ⊂ D, a n ր a, ∀b ∈ D : a n * b ր a * b.
Evidently every IF-family F can be embedded to an MV-algebra with product and it is a special case of a Kôpka D-poset, hence any result from the Kôpka D-poset theory can be applied to our IF-events theory ( [26], [64]). Now let us consider a theory dual to the IF-events theory, theory of IV-events. A prerequisity of IV-theory is in the fact that it considers natural ordering and operations of vectors. On the other hand the IV-theory is isomorphic to the IF-theory ( [65], [43]). Definition 6.3. Let (Ω, S) be a measurable space, S be a σ-algebra. By an IV-event a pair A =(μ A , ν A ) : Ω → [0, 1] 2 is considered such that  Proof. It is almost straightforward. Of course, the using of the family V is more natural and the results can be applied immediately to probability theory on F .

Conclusion
The structures studied in this chapter have two aspects: the first one is practical, the second theoretical one. Fuzzy sets and their generalization -Atanassov intuitionistic fuzzy sets -in both directions new possibilities give.
The main contribution of the presented theory is a new point of view on human thinking and creation. We consider algebraic models for multi valued logic: IF-events, and more generally MV-algebras, D-posets, and effect algebras. They are important for many valued logic as Boolean algebras for two valued logic. Of course, we presented also some results about entropy ( [11], [12], [40 -42], [59]), or inclusion -exclusion principle ( [6], [26], [30])for an illustration. But the more important idea is in building the probability theory on IF-events.
The theoretical description of uncertainty has two parts in the present time : objectiveprobability and statistics, and subjective -fuzzy sets. We show that both parts can be considered together.

Acknowledgement
This work was partially supported by the Agency of Slovak Ministry of Education for the structural Funds of the EU, uder project ITMS 26220120007 and the Agency VEGA 1/0621/11