^{1}

^{*}

^{1}

^{*}

The aim of this paper is to present, by axiomatic way, an idea about the general conditional information of a single, fixed fuzzy set when the conditioning fuzzy event is variable. The properties of this conditional information are translated in a system of functional equations. Some classes of solutions of this functional system have been found.

In these last years, on crisp setting, measures of information J, have been studied by many authors [ 1 ] [ 2 - 4 ]. Later particular researches have been done to these information measures defined without probability. The last information measures are called general because they are defined without any probability [ 5 - 8 ]. Analogous studies were presented in [ 9 ] on fuzzy setting.

In order to study the integration in information theory without probability, in [ 10 - 12 ] the families N , F , I ∞ , I 0 have been introduced. The mentioned classes of crisp sets have replaced the families of null sets in the classical theory of integration (i.e. with respect to an additive measure or a probability). A detailed overview can be found in [ 13 ].

Later, since 2005, on fuzzy setting, measures of conditional information, when the conditional event is fixed, have been considered and studied (see [ 14 - 16 ]).

Indeed, in this paper, a definition of the general conditional information of a fixed fuzzy set A, when the conditioning fuzzy set H is variable will be presented and it will be indicated by J ( A | H ) . For this reason, the families mentioned above will be adapted to the fuzzy setting and the corresponding class H A will be introduced.

Example: let A be the fuzzy set of old men, H 1 , H 2 , H 3 the fuzzy set of those old men, who seem ill, who are ill, who are seriously ill, respectively. These different conditions H 1 , H 2 , H 3 are the conditioning variable events. The conditional information: J ( A | H 1 ) , J ( A | H 2 ) , J ( A | H 3 ) measure the influence of the grade of the illness in the old men.

The paper is organized in the following way: in Sect. 2 some preliminaires are recalled; in Sect. 3 the definition of general information conditioned by a variable event is given. The statement of the problem is presented in Sect. 4 and in Sect. 5 the properties of the form of conditional information are translated in a system of functional equations [ 17 ], for which some classes of solutions are shown. Sect. 6 is devoted to the conclusions.

In this paragraph, the definition of general information for fuzzy sets is recalled [ 9 ]. The concept of fuzzy set was introduced by Zadeh in [ 18 ], for all knowledge see [ 19 , 20 ].

Let X be an abstract space and A a-algebra of all fuzzy sets of X, such that ( X , A ) is a measurable space.

Definition 2.1 In the fuzzy setting measure of the general information is a map

J ( ⋅ ) : A → [ 0, + ∞ ]

such that ∀ A 1 , A 2 ∈ A :

(i) A 1 ⊃ A 2 ⇒ J ( A 1 ) ≤ J ( A 2 ) ,

(ii) J ( ∅ ) = + ∞ , J ( X ) = 0.

Following the idea presented in [ 10 , 12 ], assigned an information measure J, the following families are introduced:

I 0 = { N ∈ A / J ( N ) = 0 } ; (1)

I + ∞ = { F ∈ A / J ( F ) = + ∞ } . (2)

The family (1) is not empty because it contains the whole set X and all supersets N ′ of N ∈ I 0 :

N ∈ I 0 , J ( N ) = 0 , ∀ N ′ ∈ A , N ⊂ N ′ , J ( N ′ ) ≤ J ( N ) = 0 ⇒ N ′ ∈ I 0 .

I 0 is not an ideal [ 21 , 22 ] because it is not stable with respect to the union between fuzzy sets.

The family (2) is not empty because it contains the empty set ∅ and all subsets F' of F ∈ I + ∞ :

F ∈ I + ∞ , J ( F ) = + ∞ , ∀ F ′ ∈ A , F ′ ⊂ F , J ( F ) ≥ J ( F ′ ) = + ∞ ⇒ F ′ ∈ I + ∞ .

I + ∞ is not an filter [ 21 , 22 ] because it is not stable with respect to the intersection between fuzzy sets.

From now on, the family H = A − I + ∞ shall be considered and measure of general conditional information of a fixed fuzzy set A ∈ A , J ( A | ⋅ ) defined on the family H will be introduced.

Definition 3.1 Measure of general information of a fixed A ∈ A , conditioned by a variable event H ∈ H is a map

J ( A | ⋅ ) : H → [ 0, + ∞ ]

such that

(j) H ⊃ H ′ ⇒ J ( A | H ) ≤ J ( A | H ′ ) , ∀ H , H ′ ∈ H ,

(jj) J ( A | N ) = J ( A ) if N ∈ I 0 .

From the previous axioms, it follows that J ( A | X ) = J ( A ) .

The condition (j) is the monotonicity, the (jj) means that all null sets N ∈ N 0 don’t condition any fuzzy set A ∈ A .

Taking into account the previous axiomatic statement, fixed an information measure J on A and any fuzzy set A ∈ A , some classes of measures J ( A | H ) , will be sought by supposing that J ( A | H ) depends only on J ( A ) , J ( H ) and J ( A ∩ H ) . Now it is necessary to specify the family where H belongs.

Fixed A, our definition is restricted to the following family

H A = { H ∈ H / J ( A ∩ H ) ≠ + ∞ }

at least H A contains the whole space X, so this family is not empty. The condition J ( A ∩ H ) ≠ + ∞ ensures that J ( H ) ≠ + ∞ : in fact if J ( H ) = + ∞ from the monotonicity of J, it is J ( A ∩ H ) = + ∞ .

So, the information J ( A | ⋅ ) is the function

J ( A | ⋅ ) : H A → [ 0, + ∞ ]

expressed by a function Φ : V → ( 0, + ∞ ) , such that

J ( A | H ) = Φ ( J ( A ) , J ( H ) , J ( A ∩ H ) )

with

V = { ( x , y , z ) / x = J ( A ) , y = J ( H ) , z = J ( A ∩ H ) , x , y , z ∈ [ 0 , + ∞ ] , x ≤ z , y ≤ z } .

This justifies the domain of the function Φ .

From (3), (j), (jj), ∀ H 1 , H 2 ∈ H , the function Φ shall satisfy the following properties:

(I) H 1 ⊃ H 2 ⇒ Φ ( J ( A ) , J ( H 1 ) , J ( A ∩ H 1 ) ) ≤ Φ ( J ( A ) , J ( H 2 ) , J ( A ∩ H 2 ) ) ,

(II) Φ ( J ( A ) , J ( N ) , J ( A ∩ N ) ) = J ( A ) , ∀ N ∈ I 0 .

Setting J ( A ) = x , J ( H 1 ) = y 1 , J ( H 2 ) = y 2 , J ( A ∩ H 1 ) = z 1 ,

J ( A ∩ H 2 ) = z 2 , J ( A ∩ N ) = t with x , y 1 , y 2 , z 1 , z 2 ∈ [ 0 , + ∞ ) and x ≤ z 1 ,

y 1 ≤ z 1 , x ≤ z 2 , y 1 ≤ z 2 , x ≤ t , the following system of functional equations is obtained:

{ ( 1 ) Φ ( x , y 1 , z 1 ) ≤ Φ ( x , y 2 , z 2 ) , x ≤ z 1 , y 1 ≤ z 1 , x ≤ z 2 , y 2 ≤ z 2 ( 2 ) Φ ( x , 0 , t ) = x , x ≤ t .

A function Φ continuous defined in the following set:

D = { ( x , y , z ) : x , y , z ∈ [ 0 , + ∞ ) , x ≤ z , y ≤ z } (3)

will be sought as an universal law in the sense that the equation and the inequality about the function of the system must be satisfied for all values and variables in their proper space, which satisfies the system [(1)-(2)].

Now, the following results are proved:

Proposition 4.1 A class of solutions of the system [(1)-(2)] is:

Φ h ( x , y , z ) = h − 1 ( [ h ( x ) + h ( y ) ] ∧ h ( z ) ) , (4)

where h : [ 0, + ∞ ) → [ 0, + ∞ ) is any continuous function, strictly increasing with h ( 0 ) = 0.

Proof. The condition (1) follows by the monotonicity of the function h. The second one (2) results from the value h ( 0 ) = 0. and the property of the function h.

Proposition 4.2 A class of solutions of the system [(1)-(2)] is:

Φ h ( x , y , z ) = h − 1 ( [ h ( x ) ∨ h ( y ) ] ∧ h ( z ) ) = h − 1 ( h ( x ) ∨ h ( y ) ) , (5)

where h : [ 0, + ∞ ) → [ 0, + ∞ ) is any continuous function, strictly increasing with h ( 0 ) = 0.

Proof. The proof is immediate. ,

Proposition 4.3 A class of solutions of the system [(1)-(2)] is:

Φ h ( x , y , z ) = h − 1 ( [ h ( x ) ∧ h ( y ) ] ∨ h ( z ) ) = z , (6)

where h : [ 0, + ∞ ) → [ 0, + ∞ ) is any continuous function, strictly increasing with h ( 0 ) = 0.

Proof. The proof is immediate. ,

From (4) (5) and (6), the following expressions of conditional information have been obtained, respectively:

J h ( A | H ) = h − 1 ( [ h ( J ( A ) ) + h ( J ( H ) ) ] ∧ h ( J ( A ∩ H ) ) ) ; (7)

J h ( A | H ) = h − 1 ( h ( J ( A ) ) ) ∨ h ( J ( H ) ) , (8)

where h : [ 0, + ∞ ) → [ 0, + ∞ ) is any continuous function, strictly increasing with h ( 0 ) = 0 , and

J ( A | H ) = J ( A ∩ H ) . (9)

In this paper, for the first time, we present an axiomatic definition of the information J ( A | H ) , when the conditioning event is variable.

We think that this axiomatic approach could be useful for future applications.

For the first author, this research is in the framework of GNFM of MIUR (Italy) and University “Sapienza” Roma.

The authors declare that they have no conflict of interest.