Fuzzy Mnesors

Fuzzy Mnesors Gilles C HAMPENOIS Collège Saint-André, Saint-Maur, France gilles_champenois@yahoo.fr ABSTRACT . A fuzzy mnesor space is a semimodule over the positive real numbers. It can be used as theoretical framework for fuzzy sets. Hence we can prove a great number of properties for fuzzy sets without refering to the membership functions. "Classical logic has erred in devoting so little attention to approximate reasoning and focusing to such a high degree on exact reasoning. So when you take a course in logic, you learn all kinds of things which are of very little use in everyday life." Lotfi Zadeh I. INTRODUCTION According to Zadeh’s fuzzy theory [1], € k - € 16 ( € k - € i here represents the number of years a student attend school before leaving) falls under the fuzzy set of € HIGH − EDUCATED with a membership value of € 1 . € k - € 12 FIGURE 1 . The membership function of the fuzzy set € HIGH € k - € 8 € k - € 16 € HIGH € 1 € 0 We can now imagine an operator on any fuzzy set which we will call external multiplication: multiplying a fuzzy set € A by the positive real number € λ returns the fuzzy set (written € A λ ) whose membership function is: € µ A λ ⋅ ( ) = µ A ⋅ ( ) ( ) λ − 1 To the previous example we add the fuzzy set € HIGH λ with € λ = 1 2 : € 0 € k - € 12 FIGURE 2 . The membership functions of the fuzzy sets € HIGH and € HIGH 1 2 € k - € 8 € k - € 16 € 1 € HIGH 1 2 € HIGH You can see that € HIGH 1 2 is more selective than € HIGH . We want here to embed fuzzy set theory in a larger framework called the mnesor theory and based on the semimodule over the positive real numbers. II. SEMIMODULE OVER THE POSITIVE REAL NUMBERS The set of the positive real numbers (denoted € R + ) is considered with two operations: the multiplication and the maximum operator (written € ⊕ ) which will be called addition: for any € x , y ∈ R + , € x ⊕ y = max x , y ( ) . Let € M denote a commutative monoid with the identity element € s 0 . We add an external multiplication returning € A λ ∈ M from € A ∈ M and € λ ∈ R + . € M is called a semimodule over € R + iff the four next properties hold for any € λ , µ ∈ R + and € A , B ∈ M : (1) € A 1 = A (2) € A λ + A µ = A λ ⊕ µ ( ) (3) € A + B ( ) λ = A λ + B λ (4) € A λ ( ) µ = A λ × µ ( ) We first derive some basic properties. Idempotence . The addition of fuzzy mnesors is idempotent. PROOF . For any € A ∈ M , € A + A = A 1 + A 1 = A 1 ⊕ 1 ( ) = A 1 = A Ordering . An order relation is naturally defined by the addition: € A ⊆ B iff € A + B = B . PROOF . € ⊆ is an order relation indeed, since € M is a commutative idempotent monoid. Note that if € B = A λ with € λ ≤ 1 , then € B is inferior to € A ( € A + A λ = A 1 + A λ = A 1 ⊕ λ ( ) = A 1 = A ) and if € B = A λ with € λ ≥ 1 , then € B is superior to € A . In the fomer case, we say that € B is more selective than € A (see the example of € HIGH 1 2 ). Positivity . All mnesors are positive ( € A + s 0 = A ) and € M is zero-sum-free ( € A + B = s 0 implies that € A = B = s 0 ) PROOF . € A = A + s 0 = A + A + B = A + B = s 0 Empty mnesor . € s 0 λ = s 0 for all € 0 ≤ λ ≤ 1 PROOF . € A + s 0 λ = A + s 0 + s 0 λ = A + s 0 1 + s 0 λ = A + s 0 1 ⊕ λ ( ) = A + s 0 1 = A + s 0 = A , for any € A ∈ M . Thus € s 0 λ is itself the identity element. III. COMPLEMENT We suppose now that the top element of € M exists (written € s 1 and called full mnesor) and we add an unary operator on € M called complement ( € A is the compement of € A ) that satisfies the following properties: (5) € A = A (6) € s 1 = s 0 (7) € A λ = A λ − 1 (8) € A ⊆ B iff € B ⊆ A Note that € s 1 λ = s 1 for all numbers € λ ≥ 1 ( € s 1 λ = s 0 λ = s 0 λ − 1 = s 0 = s 1 ). Intersection . We define the intersection of the two mnesors € A , B by € A o B = A + B and we prove that the external multiplication distributes over the intersection. PROOF . € A λ ( ) o B λ ( ) = A λ + B λ = A λ − 1 + B λ − 1 = A + B ( ) λ − 1 = A + B ( ) λ = A o B ( ) λ The intersection is idempotent ( € A o A = A + A = A = A ) Absorption . € A o B is inferior to € A and € B ( € A o B ⊆ A since € A o B = A + B ⊇ A ). Thus € A + A o B ( ) = A . Then by complementing we get € A + A o B ( ) = A or € A o A + B ( ) = A . Lattice . € M , + , o ( ) is a lattice. PROOF . € + , o are commutative, associative and idempotent. Absorption holds. IV. INTERPRETATION OF FUZZY SETS We want to prove that fuzzy sets satisfy the definition of mnesors. But we first substitute the function € c k x ( ) = e k ln x ( ) for € x → 1 − x in the definition of the complement. Like € 1 − x , € c k x ( ) takes values in € 0 1 [ ] , decreases monotonously, exchanges € 0 and € 1 (that is, € c k 0 ( ) = 1 and € c k 1 ( ) = 0 ) and is involutive. Moreover € c k x ( ) values are very close to € 1 − x if € k ≈ 0 , 4 as you can see below. 1 0 1 0,5 0 1-x k=0,4 k=0,5 FIGURE 3 . € 1 − x function, € c 0 ,4 x ( ) and € c 0 ,5 x ( ) functions Note that € c k x ( ) ( ) n = c k × n x ( ) and € c k x n ( ) = c k n x ( ) . The reunion of fuzzy sets and the external multiplication satisfy the linear properties and the complement properties. PROOF . (1) € µ A 1 ⋅ ( ) = µ A ⋅ ( ) ( ) 1 − 1 = µ A ⋅ ( ) ( ) 1 = µ A ⋅ ( ) (2) € µ A ∪ B ( ) λ ⋅ ( ) = µ A ∪ B ⋅ ( ) ( ) λ − 1 = µ A ⋅ ( ) ⊕ µ B ⋅ ( ) ( ) λ − 1 = µ A ⋅ ( ) ( ) λ − 1 ⊕ µ B ⋅ ( ) ( ) λ − 1 = µ A λ ⋅ ( ) ⊕ µ B λ ⋅ ( ) (3) € µ A λ ⊕ δ ( ) ⋅ ( ) = µ A ⋅ ( ) ( ) λ ⊕ δ ( ) − 1 = µ A ⋅ ( ) ( ) λ − 1 ⊕ µ A ⋅ ( ) ( ) δ − 1 = µ A λ ⋅ ( ) ⊕ µ A δ ⋅ ( ) (4) € µ A λ ( ) δ ⋅ ( ) = µ A λ ⋅ ( ) ( ) δ − 1 = µ A ⋅ ( ) ( ) λ − 1       δ − 1 = µ A ⋅ ( ) ( ) λ ⊗ δ ( ) − 1 = µ A λ ⊗ δ ( ) ⋅ ( ) (5) € µ A ⋅ ( ) = µ A ⋅ ( ) because € c k is involutive (6) € µ s 1 ⋅ ( ) = c k µ s 1 ⋅ ( ) ( ) = c k 1 ( ) = 0 = µ s 0 ⋅ ( ) (7) € µ A λ ⋅ ( ) = c k µ A λ ⋅ ( ) ( ) = c k µ A ⋅ ( ) ( ) λ − 1       = c k × λ µ A ⋅ ( ) ( ) = c k µ A ⋅ ( ) ( ) ( ) λ = µ A ⋅ ( ) ( ) λ = µ A λ − 1 ⋅ ( ) (8) If € µ A ⋅ ( ) ≤ µ B ⋅ ( ) , then € c k µ A ⋅ ( ) ( ) ≥ c k µ B ⋅ ( ) ( ) , since the € c k function decreases. Hence € µ A ⋅ ( ) ≥ µ B ⋅ ( ) REFERENCES 1. ZADEH L. (1965), Fuzzy Sets