_{1}

^{*}

In this article, we present a systemic approach toward a fuzzy logic based formalization of an approximate reasoning methodology in a fuzzy resolution, where we derive a truth value of A from both values of B → A and B by some mechanism. For this purpose, we utilize a t-norm fuzzy logic, in which an implication operator is a root of both graduated conjunction and disjunction operators. Furthermore by using an inverse approximate reasoning, we conclude the truth value of A from both values of B → A and B, applying an altogether different mechanism. A current research is utilizing an approximate reasoning methodology, which is based on a similarity relation for a fuzzification, while similarity measure is utilized in fuzzy inference mechanism. This approach is applied to both generalized modus-ponens/modus-tollens syllogisms and is well-illustrated with artificial examples.

This study is a continuation of a research, which is based on a proposed t-norm fuzzy logic, presented in [

Taking into account the above mentioned, we present the following.

Definition 1 [

A fuzzy resolvent of two fuzzy clauses C 1 ˜ and C 2 ˜ , containing the complementary literals x i and ¬ x i respectively, is defined as

R ( C 1 ˜ , C 2 ˜ ) = L 1 ∨ L 2 (1.1)

where C 1 ˜ = x i ∨ L 1 and C 2 ˜ = ¬ x i ∨ L 2 . L 1 , L 2 are fuzzy clauses, which don’t contain x i and ¬ x i respectively. The operator ∨ is understood as the disjunction of the literals present in them. It is also a logical consequence of C 1 ˜ ∧ C 2 ˜ . A resolution deduction of a clause C ˜ from a set S of clauses is a finite sequence of clauses C 1 ˜ , C 2 ˜ , ⋯ , C n ˜ = C ˜ such that each C i ˜ is either a member of or is a resolvent of two clauses taken from the resolution principle in propositional logic we deduce that, if S is true under some truth valuation v , then v ( C i ) = TRUE for all i , and in particular, v ( C ) = TRUE [

Example 1: Here is a derivation of a clause from a set of clauses presented by means of a resolution Tree in

In first order logic, resolution condenses the traditional syllogism of logical inference down to single rule.

A simple resolution scheme is:

Antecedent 1: a ∨ b

Antecedent 2: ¬ b (1.2)

Consequent: a .

The entire historical analysis of this approach toward applying of a resolution principal to a logical inference is presented in [

First, Let us consider that A, A', B and B' are fuzzy concepts represented by fuzzy sets in universe of discourse U, U, V and V, respectively and correspondent fuzzy sets be represented as such A ⊂ U | μ A : U → [ 0 , 1 ] , B ⊂ V | μ B : V → [ 0 , 1 ] , where

A = ∫ U μ A ( u ) / u , B = ∫ V μ B ( v ) / v (2.1)

Given (2.1) let us formulate the argument form of simple Fuzzy Resolution as follows.

A ∨ B ¬ B

---------- (2.2)

A

Given (2.2) the scheme for Generalized Fuzzy Resolution looks like that

Antecedent 1: If x is A OR y is B Antecedent 2: y is B'

---------------------------------- (2.3)

Consequent: x is A'.

In case (2.3), we can say that the Disjunctive Syllogism holds if B' is close to not B, whereas A' is close to A. The second approach is called Inverse Approximate Reasoning. Its scheme looks like that:

Antecedent 1: If x is A, then y is B Antecedent 2: y is B'

---------------------------------- (2.4)

Consequent: x is A'.

We shall transform the disjunction form of rule into fuzzy implication from fuzzy logic, introduced in [

Let us consider Generalized Fuzzy Resolution first. The key operation used in this method is disjunction. The disjunction operation ∨ is presented in

A ∨ B = { A ⋅ B , A + B < 1 , 1 , A + B ≥ 1 (2.5)

Whereas correspondent conjunction operation ∧ is also presented in

A ∧ B = { A ⋅ B , A + B > 1 , 0 , A + B ≤ 1 (2.6)

Taking into account (2.5) and (2.6) and the fact that ¬ A = 1 − A , let us formulate the following

Lemma 1.

If there are two fuzzy clauses C 1 ˜ , C 2 ˜ and R ( C 1 ˜ , C 2 ˜ ) is a fuzzy resolvent of them with keyword x i , then the following inequality holds

T ( C 1 ∧ C 2 ˜ ) ≤ T ( R ( C 1 ˜ , C 2 ˜ ) ) (2.7)

where T(x) is a truth value of an x.

Proof: Since C 1 ˜ = x i ∨ L 1 and C 2 ˜ = ¬ x i ∨ L 2 , where L 1 , L 2 ≠ 0 , then

C 1 ∧ C 2 ˜ = ( x i ∨ L 1 ) ∧ ( ¬ x i ∨ L 2 ) = ( x i ∧ ¬ x i ) ∨ ( L 1 ∧ ¬ x i ) ∨ ( x i ∧ L 2 ) ∨ ( L 1 ∧ L 2 ) (2.8)

whereas from (1.1) R ( C 1 ˜ , C 2 ˜ ) = L 1 ∨ L 2 . From (2.8) let define the following values of truth:

T 1 = ( x i ∧ ¬ x i ) = { x i ⋅ ¬ x i , x i + ¬ x i > 1 , 0 , x i + ¬ x i ≤ 1 (2.9)

Since x i + ¬ x ≡ 1 , then from (2.9) we are getting that

T 1 ≡ 0 (2.10)

In a meantime from the same (2.8) we have

T 2 = ( L 1 ∧ ¬ x i ) = { L 1 ⋅ ¬ x i , L 1 > x i , 0 , L 1 ≤ x i (2.11)

From (2.11) let’s take a note that

S u p ( T 2 ) ≡ 1 | L 1 = 1 , x i = 0 (2.12)

Also from (2.11) let

T 3 = ( x i ∧ L 2 ) = { x i ⋅ L 2 , x i + L 2 > 1 , 0 , x i + L 2 ≤ 1 (2.13)

Continuing from (2.11) let

T 4 = ( L 1 ∧ L 2 ) = { L 1 ⋅ L 2 , L 1 + L 2 > 1 , 0 , L 1 + L 2 ≤ 1 (2.14)

And finally from (1.1) we have

L 1 ∨ L 2 = { L 1 ⋅ L 2 , L 1 + L 2 < 1 , 1 , L 1 + L 2 ≥ 1 (2.15)

Let’s rewrite (2.8) in the following way

T ( C 1 ∧ C 2 ˜ ) = T 1 ∨ T 2 ∨ T 3 ∨ T 4 (2.16)

From (2.16) given both (2.10) and (2.12) we have

T 12 = T 1 ∨ T 2 = 0 ∨ T 2 = { 0 , T 2 < 1 , 1 , T 2 ≥ 1 = { 0 , T 2 < 1 , 1 , T 2 = 1 (2.17)

Taking into account (2.12) finally we are getting

T 12 = { 0 , T 2 < 1 , 1 , L 1 = 1 , x i = 0. (2.18)

Furthermore from the same (2.16) let

T 34 = T 3 ∨ T 4 = { T 3 ⋅ T 4 , T 3 + T 4 < 1 , 1 , T 3 + T 4 ≥ 1 (2.19)

Note that from (2.18) x i = 0 , which means that from (2.13) T 3 ≡ 0 , therefore from (2.19) we have

T 34 = { 0 , T 4 < 1 , 1 , T 4 ≥ 1 (2.20)

But from (2.14) and (2.20) T 4 = 1 , when L 1 ⋅ L 2 = 1 or equally

L 1 = L 2 = 1 (2.21)

Finally from (2.16), given (2.18) and (2.20), we are getting the following

T ( C 1 ∧ C 2 ˜ ) = T 12 ∨ T 34 = { T 12 ⋅ T 34 , T 12 + T 34 < 1 , 1 , T 12 + T 34 ≥ 1 = { 0 , T 12 + T 34 < 1 , 1 , T 12 + T 34 ≥ 1 (2.22)

Taking into account (2.21) from (2.15) and (1.1)

R ( C 1 ˜ , C 2 ˜ ) = L 1 ∨ L 2 ≡ 1 (2.23)

From (2.22) and (2.23) we are getting

T ( C 1 ∧ C 2 ˜ ) ≤ T ( R ( C 1 ˜ , C 2 ˜ ) ) (Q. E. D.).

Corollary 1

Let C 1 ˜ , C 2 ˜ are two fuzzy clauses. If T ( R ( C 1 ˜ , C 2 ˜ ) ) < 0.25 , then the following is true:

T ( C 1 ∧ C 2 ˜ ) < T ( R ( C 1 ˜ , C 2 ˜ ) ) (2.24)

Proof: First note that for

L 1 , L 2 ∈ [ 0 , 1 ] | L 1 + L 2 < 1 ⇒ L 1 ⋅ L 2 < 0.25 (2.25)

From (1.1) and (2.15), given (2.25) we are getting the following

T ( R ( C 1 ˜ , C 2 ˜ ) ) = L 1 ∨ L 2 = { L 1 ⋅ L 2 , L 1 + L 2 < 1 , 1 , L 1 + L 2 ≥ 1 = { 0.25 , L 1 + L 2 < 1 , 1 , L 1 + L 2 ≥ 1 (2.26)

First from (2.14) we have the following

T 4 ≡ 0 | L 1 + L 2 ≤ 1 ⇒ T 34 = T 3 ∨ 0 = { 0 , T 3 < 1 , 1 , T 3 = 1 (2.27)

But from (2.13) we have the following T 3 = 1 | x i = L 2 = 1 ⇒ L 1 ≡ 0 , but L 1 + L 2 < 1 ⇒ T 3 ≡ 0 , therefore

T 34 ≡ 0 ⇒ T 12 ∨ T 34 ≡ 0 , i . e . < 0.25 , in other words the following is true.

T ( C 1 ∧ C 2 ˜ ) < T ( R ( C 1 ˜ , C 2 ˜ ) ) (Q. E. D.).

Corollary 2

Let C 1 ˜ , C 2 ˜ are two fuzzy clauses. If T ( R ( C 1 ˜ , C 2 ˜ ) ) ≥ 0.25 , then the following is true:

T ( C 1 ∧ C 2 ˜ ) = T ( R ( C 1 ˜ , C 2 ˜ ) ) (2.28)

Proof:

From (2.22) we have T ( ( C 1 ∧ C 2 ˜ ) = { 0 , T 12 + T 34 < 1 , 1 , T 12 + T 34 ≥ 1 , which means that

T ( C 1 ∧ C 2 ˜ ) ≥ 0.25 ⇒ 1 . From (2.14), (2.17) and (2.19) we are getting T ( C 1 ∧ C 2 ˜ ) ≡ 1 ⇒ L 1 = L 2 ˜ = 1 , but from (2.26) T ( R ( C 1 ˜ , C 2 ˜ ) ) ≡ 1 | L 1 = L 2 = 1 , which means that T ( C 1 ∧ C 2 ˜ ) = T ( R ( C 1 ˜ , C 2 ˜ ) ) (Q. E. D.).

Based on above presented results let formulate the following

Theorem 1 (Deduction):

Let A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ and B ˜ are fuzzy concepts. A fuzzy concept B ˜ is a logical consequent of A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ if and only if the following inequality holds

T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) < T ( B ˜ ) . (2.29)

Or equally

{ [ A 1 ˜ and A 2 ˜ and ⋯ and A n ˜ ] ⇒ B ˜ } (2.30)

Which means a fuzzy formula (2.30) is UV. Note, that a fuzzy formula f ∈ F is called UV, if T ( f ) ≥ 0.5 . Before providing a proof of this theorem let us give the following

Definition 2

A fuzzy concept B ˜ is a logical consequent of A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ , i.e. A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ | = B ˜ , if and only when UV of A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ has caused UV of a fuzzy concept B ˜ , in other words the following is true.

T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≥ 0.5 ⇒ T ( B ˜ ) ≥ 0.5 , where ∩ ∀ i ∈ [ 1 , n ] A i ˜ = min i { A i ˜ }

Proof: Let fuzzy concept B ˜ is a logical consequent of fuzzy concepts A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ .

If fuzzy concepts A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ are UV, i.e. T ( A i ˜ ) ≥ 0.5 , i = 1 , n − − − , then in accordance with Definition 2 a fuzzy concept B ˜ is also UV, i.e. T ( B ˜ ) ≥ 0.5 . An implication operator in a fuzzy logic, used in this article is defined as the following (see

T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) = { 1 , T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≤ T ( B ˜ ) , ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ⋅ T ( B ˜ ) , T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) > T ( B ˜ ) (2.31)

Let us consider a set of cases.

・ If T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≤ T ( B ˜ ) , then T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) ≡ 1 , therefore UV of a fuzzy formula (2.30) is apparent.

・ If T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) > T ( B ˜ ) , then T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) ≡ ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ⋅ T ( B ˜ ) , but since T ( A i ˜ ) ≥ 0.5 , i = 1 , n − − − and T ( B ˜ ) ≥ 0.5 , then ¬ T ( A i ˜ ) < 0.5 , i = 1 , n − − − and ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) < T ( B ˜ ) . Therefore it is obvious then ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ⋅ T ( B ˜ ) < 0.5 , i.e. a fuzzy formula (2.30) is not UV, a logical contradiction takes place.

・ If ∃ i = i * | T ( A i * ˜ ) < 0.5 , then T ( ∩ i A i ˜ ) = min i = 1 , n − − − { T ( A i ˜ ) } < 0.5 a fuzzy sub formula (antecedent) from (2.30) [ A 1 ˜ and A 2 ˜ and ⋯ and A n ˜ ] is not UV, i.e. contradictive, but in a meantime T ( B ˜ ) ≥ 0.5 , therefore T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≤ T ( B ˜ ) ⇒ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) ≡ 1 , i.e. a fuzzy formula (2.30) is UV.

・ If

T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) < 0.5 , T ( B ˜ ) < 0.5 (2.32)

and if T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≤ T ( B ˜ ) ⇒ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) ≡ 1 , therefore formula (2.30) is UV, whereas if T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) > T ( B ˜ ) , then again T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) ≡ ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ⋅ T ( B ˜ ) and given conditions (2.32) we have ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≥ 0.5 ⇒ ¬ T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ⋅ T ( B ˜ ) ≤ 0.5 , which means that a fuzzy formula (2.30) is not UV or is contradictive.

At last let a fuzzy formula (2.30) be UV and also let T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≤ T ( B ˜ ) . Then if a fuzzy sub formula (antecedent) from (2.30) [ A 1 ˜ and A 2 ˜ and ⋯ and A n ˜ ] is also UV, i.e. T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) ≥ 0.5 , then from (2.31) we have T ( ∩ ∀ i ∈ [ 1 , n ] A i ˜ ) → T ( B ˜ ) ≡ 1 ⇒ T ( B ˜ ) ≥ 0.5 . In other words a fuzzy concept B ˜ is UV. Therefore from Definition 2 a fuzzy concept B ˜ is a logical consequent of fuzzy concepts A 1 ˜ , A 2 ˜ , ⋯ , A n ˜ (Q. E. D.).

Based on these results we formulate the following

Theorem 2

If there are two fuzzy clauses C 1 ˜ , C 2 ˜ and R ( C 1 ˜ , C 2 ˜ ) is a fuzzy resolvent of them with keyword x i , then R ( C 1 ˜ , C 2 ˜ ) is logical consequent of both C 1 ˜ and C 2 ˜ i.e.

C 1 ˜ , C 2 ˜ | = R ( C 1 ˜ , C 2 ˜ ) (2.33)

Proof:

Let C 1 ˜ = x i ∨ L 1 and C 2 ˜ = ¬ x i ∨ L 2 , where L 1 , L 2 ≠ 0 , whereas R ( C 1 ˜ , C 2 ˜ ) = L 1 ∨ L 2 . By Definition 1 and in accordance with (2.5)

T ( R ( C 1 ˜ , C 2 ˜ ) ) = { L 1 ⋅ L 2 , L 1 + L 2 < 1 , 1 , L 1 + L 2 ≥ 1 (2.34)

T ( C 1 ˜ ) = { x i ⋅ L 1 , x i + L 1 < 1 , 1 , x i + L 1 ≥ 1 (2.35)

T ( C 2 ˜ ) = { ¬ x i ⋅ L 2 , ¬ x i + L 2 < 1 , 1 , ¬ x i + L 2 ≥ 1 (2.36)

Let C 1 ˜ , C 2 ˜ are both UV, i.e. T ( C 1 ˜ ) ≥ 0.5 and T ( C 2 ˜ ) ≥ 0.5 . Since from (2.35) and (2.36) the following is taking place x i ⋅ L 1 < 0.5 | x i + L 1 < 1 and ¬ x i ⋅ L 2 < 0.5 | ¬ x i + L 2 < 1 then UV of C 1 ˜ , C 2 ˜ is in reality means that

x i + L 1 ≥ 1 ¬ x i + L 2 ≥ 1 } (2.37)

Taking into account that T ( ¬ x i ) = 1 − T ( x i ) let sum both inequalities (2.37) together and get the following L 1 + L 2 ≥ 1 . From (2.34) T ( R ( C 1 ˜ , C 2 ˜ ) ) = 1 | L 1 + L 2 ≥ 1 ⇒ T ( R ( C 1 ˜ , C 2 ˜ ) ) ≥ 0.5 i.e. R ( C 1 ˜ , C 2 ˜ ) is UV. Therefore by Definition 2 we are getting a fact that if C 1 ˜ , C 2 ˜ are both UV, and then R ( C 1 ˜ , C 2 ˜ ) is also UV. (Q. E. D.).

Let us present some considerations about using a notion of similarity, which plays a fundamental role in theories of knowledge and behavior and has been dealt with extensively in psychology and philosophy. A careful analysis of the different similarity measures reveals that it is impossible to single out one particular similarity measure that works well for all purposes. We will utilize a consistent approach toward definition of a similarity measure, based on the same fuzzy logic we used above [

Suppose U be an arbitrary finite set, and ℑ ( U ) be the collection of all fuzzy subsets of U . For A , B ∈ ℑ ( U ) , a similarity index between the pair { A , B } is denoted as S ( A , B ; U ) or simply S ( A , B ) which can also be considered as a function S : ℑ ( U ) × ℑ ( U ) → [ 0 , 1 ] . In order to provide a definition for similarity index, a number of factors must be considered.

Definition 3

A function S ( A , B ) defines a similarity between fuzzy concepts A , B if it satisfies the following axioms:

P1. S ( B , A ) = S ( A , B ) , S ( ¬ A , ¬ B ) = S ( A , B ) , where ¬ A = 1 − A ,

P2. 0 ≤ S ( A , B ) ≤ 1 ,

P3. S ( A , B ) = 1 , iff A = B ,

P4. For two fuzzy concepts A , B | A ≠ ∅ , B ≠ ∅ , S ( A , B ) = 0 ⇒ min ( μ A ( u ) , μ B ( u ) ) = 0 , ∀ u ∈ U , i.e. A ∩ B = ∅ ,

P5. A ⊆ B ⊆ C ( A ⊇ B ⊇ C ) ⇒ S ( A , C ) ≤ min ( S ( A , B ) , S ( B , C ) ) .

Lemma 2

If a function S ( A , B ) is defined as operation equivalence from

Proof:

From

S ( A , B ) = { ( 1 − b ) ⋅ a , a < b , 1 , a = b , ( 1 − a ) ⋅ b , a > b , (2.38)

P1. From (2.38)

S ( ¬ A , ¬ B ) = { b ⋅ ¬ a , ¬ a < b , 1 , ¬ a = ¬ b , a ⋅ ¬ b , ¬ a > b , = { ( 1 − a ) ⋅ b , a > b , 1 , a = b , ( 1 − b ) ⋅ a , a < b , ⇒ S ( ¬ A , ¬ B ) = S ( A , B ) (2.39)

whereas

S ( B , A ) = { b ⋅ ( 1 − a ) , b < a , 1 , b = a , ( 1 − b ) ⋅ a , b > a , = S ( A , B ) (2.40)

Axioms P2 and P3 are trivially satisfied by (2.38).

P4. From (2.38)

S ( A , B ) ≡ 0 | b = 1 , a = 0 or b = 0 , a = 1 ⇒ A ∩ B = ∅ (2.41)

P5. From (2.38)

S ( B , C ) = { ( 1 − c ) ⋅ b , b < c , 1 , b = c , ( 1 − b ) ⋅ c , b > c , (2.42)

Case: a < b < c

From (2.38) and (2.42) we have

S ( A , B ) = ( 1 − b ) ⋅ a ; S ( B , C ) = ( 1 − c ) ⋅ b ; S ( A , C ) = ( 1 − c ) ⋅ a (2.43)

From (2.43) ( 1 − c ) ⋅ a < ( 1 − c ) ⋅ b ⇒ S ( A , C ) < S ( B , C ) , whereas ( 1 − c ) ⋅ a < ( 1 − b ) ⋅ a ⇒ S ( A , C ) < S ( A , B ) . Since S ( A , C ) < S ( B , C ) and S ( A , C ) < S ( A , B ) , then the following is also true:

S ( A , C ) ≤ min ( S ( A , B ) , S ( B , C ) ) (2.44)

Case: a > b > c

From (2.38) and (2.42) we have

S ( A , B ) = ( 1 − a ) ⋅ b ; S ( B , C ) = ( 1 − b ) ⋅ c ; S ( A , C ) = ( 1 − a ) ⋅ c (2.45)

From (2.45) ( 1 − a ) ⋅ c < ( 1 − a ) ⋅ b ⇒ S ( A , C ) < S ( A , B ) , whereas ( 1 − a ) ⋅ c < ( 1 − b ) ⋅ c ⇒ S ( A , C ) < S ( A , B ) . Since S ( A , C ) < S ( B , C ) and S ( A , C ) < S ( A , B ) , then the following is also true:

S ( A , C ) ≤ min ( S ( A , B ) , S ( B , C ) ) (Q. E. D.).

To illustrate our further research before giving the definition of similarity index, we will present couple examples.

Let A and B be two normal fuzzy sets defined over the same universe of discourse U and presented by unimodal linear monotonic membership functions and s u p p { A } ≥ C a r d U − 1 ; s u p p { B } ≥ C a r d U − 1 . Correspondent linguistic scale could consist of the terms like {“SMALL”…, “MEDIUM”…, “LARGE”}. Let us consider the following cases.

1) A labeled “SMALLER THAN LARGE” = 0.2 / u 1 + 0.3 / u 2 + 0.4 / u 3 + 0.5 / u 4 + 0.6 / u 5 + 0.7 / u 6 + 0.8 / u 7 + 0.9 / u 8 + 1.0 / u 9 + 0.9 / u 10 + 0.8 / u 11

Whereas B labeled “LARGE” = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11

From (2.38) the similarity matrix S ( A , B ) would look like that

2) A labeled “MEDIUM” = 0.5 / u 1 + 0.6 / u 2 + 0.7 / u 3 + 0.8 / u 4 + 0.9 / u 5 + 1.0 / u 6 + 0.9 / u 7 + 0.8 / u 8 + 0.7 / u 9 + 0.6 / u 10 + 0.5 / u 11

And B labeled “LARGE” = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11

From (2.38) the similarity matrix S ( A , B ) would look like that (for simplicity sake we show elements of a matrix with singles only)

3) A labeled “MEDIUM” = 0.5 / u 1 + 0.6 / u 2 + 0.7 / u 3 + 0.8 / u 4 + 0.9 / u 5 + 1.0 / u 6 + 0.9 / u 7 + 0.8 / u 8 + 0.7 / u 9 + 0.6 / u 10 + 0.5 / u 11

And B labeled “SMALL” = 1.0 / u 1 + 0.9 / u 2 + 0.8 / u 3 + 0.7 / u 4 + 0.6 / u 5 + 0.5 / u 6 + 0.4 / u 7 + 0.3 / u 8 + 0.2 / u 9 + 0.1 / u 10 + 0.0 / u 11

From (2.38) the similarity matrix S ( A , B ) would look like that

4) A labeled “LARGE” = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11

And B labeled “SMALL” = 1.0 / u 1 + 0.9 / u 2 + 0.8 / u 3 + 0.7 / u 4 + 0.6 / u 5 + 0.5 / u 6 + 0.4 / u 7 + 0.3 / u 8 + 0.2 / u 9 + 0.1 / u 10 + 0.0 / u 11

From (2.38) the similarity matrix S ( A , B ) would look like that

5) A labeled “MEDIUM” = 0.5 / u 1 + 0.6 / u 2 + 0.7 / u 3 + 0.8 / u 4 + 0.9 / u 5 + 1.0 / u 6 + 0.9 / u 7 + 0.8 / u 8 + 0.7 / u 9 + 0.6 / u 10 + 0.5 / u 11

And B labeled “MEDIUM” = 0.5 / u 1 + 0.6 / u 2 + 0.7 / u 3 + 0.8 / u 4 + 0.9 / u 5 + 1.0 / u 6 + 0.9 / u 7 + 0.8 / u 8 + 0.7 / u 9 + 0.6 / u 10 + 0.5 / u 11

From (2.38) the similarity matrix S ( A , B ) would look like that

Let consider similarity measure as a matrix S ( A , B ) = ‖ s i j ‖ ; i = 1 , n ¯ ; j = 1 , n ¯ ; n = C a r d U . We are presenting the following

Proposition 1

Since A and B are two normal fuzzy sets, then ∃ i * , j * ∈ [ 1 , n ] | s i * j * = 1 ; n = C a r d U . Then the following function could be considered as a similarity index

S I ( A , B ) = max ∑ i = i * + 1 , n ¯ ; j = j * + 1 , n ¯ s i j n − 1 ; n = C a r d U . (2.46)

In

1) For i * = 1 , j * = 3 , G r 1 { i , j } = { 2 , 4 ; 3 , 5 ; 4 , 6 ; 5 , 7 ; 6 , 8 ; 7 , 9 ; 8 , 10 ; 9 , 11 } ⇒ S I ( G r 1 { i , j } ) = 8 10 = 0.8

2) For i * = 10 , j * = 10 , G r 2 { i , j } = ∅ ⇒ S I ( G r 2 { i , j } ) = 0

3) For i * = 11 , j * = 9 , G r 3 { i , j } = ∅ ⇒ S I ( G r 3 { i , j } ) = 0 .

From (2.46) we are getting S I ( A , B ) = max k = 1 , 3 ¯ [ S I ( G r k { i * , j * } ) ] = 0.8 . This value perfectly matches our intuition and perception of a closeness of terms “SMALLER THAN LARGE” and “LARGE” and membership functions of correspondent fuzzy sets.

In

1) For i * = 1 , j * = 6 , G r 1 { i , j } = { 2 , 7 ; 3 , 8 ; 4 , 9 ; 5 , 10 ; 6 , 11 } ⇒ S I ( G r 1 { i , j } ) = 5 10 = 0.5

2) For i * = 7 , j * = 10 , G r 2 { i , j } = ∅ ⇒ S I ( G r 2 { i , j } ) = 0

3) For i * = 8 , j * = 9 , G r 3 { i , j } = ∅ ⇒ S I ( G r 3 { i , j } ) = 0

4) For i * = 9 , j * = 8 , G r 4 { i , j } = ∅ ⇒ S I ( G r 4 { i , j } ) = 0

5) For i * = 10 , j * = 7 , G r 5 { i , j } = ∅ ⇒ S I ( G r 5 { i , j } ) = 0

6) For i * = 11 , j * = 6 , G r 6 { i , j } = ∅ ⇒ S I ( G r 6 { i , j } ) = 0

From (2.46) S I ( A , B ) = max k = 1 , 6 ¯ [ S I ( G r k { i * , j * } ) ] = 0.5 . This value is in a middle of a scale [0, 1] and also perfectly matches our intuition and perception of an average closeness of terms “LARGE” and “MEDIUM” and membership functions of correspondent fuzzy sets.

Similarly in

1) For i * = 6 , j * = 1 , G r 1 { i , j } = { 7 , 2 ; 8 , 3 ; 9 , 4 ; 10 , 5 ; 11 , 6 } ⇒ S I ( G r 1 { i , j } ) = 5 10 = 0.5

2) For i * = 5 , j * = 2 , G r 2 { i , j } = ∅ ⇒ S I ( G r 2 { i , j } ) = 0

3) For i * = 4 , j * = 3 , G r 3 { i , j } = ∅ ⇒ S I ( G r 3 { i , j } ) = 0

4) For i * = 3 , j * = 4 , G r 4 { i , j } = ∅ ⇒ S I ( G r 4 { i , j } ) = 0

5) For i * = 2 , j * = 5 , G r 5 { i , j } = ∅ ⇒ S I ( G r 5 { i , j } ) = 0

6) For i * = 1 , j * = 6 , G r 6 { i , j } = ∅ ⇒ S I ( G r 6 { i , j } ) = 0 .

From (2.46) we are getting S I ( A , B ) = max k = 1 , 6 ¯ [ S I ( G r k { i * , j * } ) ] = 0.5 . This value is also in a middle of a scale [ 0 , 1 ] and also perfectly matches our intuition and perception of an average closeness of terms “SMALL” and “MEDIUM” and membership functions of correspondent fuzzy sets.

In

1) For i * = 11 , j * = 1 , G r 1 { i , j } = ∅ ⇒ S I ( G r 1 { i , j } ) = 0

2) i * = 10 , j * = 2 , G r 2 { i , j } = ∅ ⇒ S I ( G r 2 { i , j } ) = 0

3) i * = 9 , j * = 3 , G r 3 { i , j } = ∅ ⇒ S I ( G r 3 { i , j } ) = 0

4) i * = 8 , j * = 4 , G r 4 { i , j } = ∅ ⇒ S I ( G r 4 { i , j } ) = 0

5) i * = 7 , j * = 5 , G r 5 { i , j } = ∅ ⇒ S I ( G r 5 { i , j } ) = 0

6) i * = 6 , j * = 6 , G r 6 { i , j } = ∅ ⇒ S I ( G r 6 { i , j } ) = 0

7) For i * = 5 , j * = 7 , G r 7 { i , j } = ∅ ⇒ S I ( G r 7 { i , j } ) = 0

8) For i * = 4 , j * = 8 , G r 8 { i , j } = ∅ ⇒ S I ( G r 8 { i , j } ) = 0

9) For i * = 3 , j * = 9 , G r 9 { i , j } = ∅ ⇒ S I ( G r 9 { i , j } ) = 0

10) For i * = 2 , j * = 10 , G r 10 { i , j } = ∅ ⇒ S I ( G r 10 { i , j } ) = 0

11) For i * = 1 , j * = 11 , G r 11 { i , j } = ∅ ⇒ S I ( G r 11 { i , j } ) = 0 .

From (2.46) we are getting S I ( A , B ) = max k = 11 [ S I ( G r k { i * , j * } ) ] = 0 . This value also perfectly matches our intuition and perception of a fact that terms “SMALL” and “LARGE” has nothing in common.

In

1) For i * = 11 , j * = 1 , G r 1 { i , j } = ∅ ⇒ S I ( G r 1 { i , j } ) = 0

2) i * = 10 , j * = 2 , G r 2 { i , j } = ∅ ⇒ S I ( G r 2 { i , j } ) = 0

3) i * = 9 , j * = 3 , G r 3 { i , j } = ∅ ⇒ S I ( G r 3 { i , j } ) = 0

4) i * = 8 , j * = 4 , G r 4 { i , j } = ∅ ⇒ S I ( G r 4 { i , j } ) = 0

5) i * = 7 , j * = 5 , G r 5 { i , j } = ∅ ⇒ S I ( G r 5 { i , j } ) = 0

6) i * = 6 , j * = 6 , G r 6 { i , j } = ∅ ⇒ S I ( G r 6 { i , j } ) = 0

7) For i * = 5 , j * = 7 , G r 7 { i , j } = ∅ ⇒ S I ( G r 7 { i , j } ) = 0

8) For i * = 4 , j * = 8 , G r 8 { i , j } = ∅ ⇒ S I ( G r 8 { i , j } ) = 0

9) For i * = 3 , j * = 9 , G r 9 { i , j } = ∅ ⇒ S I ( G r 9 { i , j } ) = 0

10) For i * = 2 , j * = 10 , G r 10 { i , j } = ∅ ⇒ S I ( G r 10 { i , j } ) = 0

11) For i * = 1 , j * = 11 , G r 11 { i , j } = ∅ ⇒ S I ( G r 11 { i , j } ) = 0 .

12) For i * = 1 , j * = 1 , G r 12 { i , j } = { 2 , 2 ; 3 , 3 ; 4 , 4 ; 5 , 5 ; 6 , 6 ; 7 , 7 ; 8 , 8 ; 9 , 9 ; 10 , 10 ; 11 , 11 } ⇒ S I ( G r 12 { i , j } ) = 10 10 = 1 .

From (2.46) we are getting S I ( A , B ) = max k = 12 [ S I ( G r k { i * , j * } ) ] = 1 . This value is a confirmation of a fact that both fuzzy sets are identical.

Let us remind that the scheme for Generalized Fuzzy Resolution (2.3) looks like that

Antecedent1: If x is A OR y is B Antecedent2: y is B’

---------------------------------- (3.1)

Consequent: x is A’.

First consider the following classical logic equivalence

a ∨ b ≡ b ∨ a ≡ ¬ b → a (3.2)

The classical logic equivalence (3.2) can be extended in fuzzy logic with implication and negation functions. We use the same fuzzy logic, which operations are presented in

Since

a ∨ b = { a ⋅ b , a + b < 1 , 1 , a + b ≥ 1 ⇒ b ∨ a = { b ⋅ a , b + a < 1 , 1 , b + a ≥ 1 ⇒ a ∨ b ≡ b ∨ a (3.3)

And because a → b = { ( 1 − a ) ⋅ b , a > b , 1 , a ≤ b therefore

¬ b → a = { ( 1 − ¬ b ) ⋅ a , ¬ b > a , 1 , ¬ b ≤ a = { b ⋅ a , b + a < 1 , 1 , b + a ≥ 1 ≡ a ∨ b (3.4)

Both (3.3) and (3.4) proofs that classical logic equivalence (3.2) holds. It is very important to show that we transform Generalized Fuzzy Resolution rule (3.1) into its equivalent form

Antecedent1: If y is ØB then x is A Antecedent2: y is B'

---------------------------------- (3.5)

Consequent: x is A'.

Let us formulize an inference method for a rule (3.5). Following a well-known pattern, established a couple of decades ago and the standard approaches toward such formalization, presented and extensively used in [

A = ∫ U μ A ( u ) / u , B = ∫ V μ B ( v ) / v (3.6)

Whereas given (3.6) a binary relationship for the fuzzy conditional proposition of the type: “If y is ¬ B then x is A” for a fuzzy logic is defined as

R ( D 1 ( y ) , D 2 ( x ) ) = ¬ B × V → U × A = ∫ V × U ( 1 − μ B ( v ) ) / ( v , u ) → ∫ V × U μ A ( u ) / ( v , u ) = ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( v , u ) (3.7)

Given an implication operator from

( 1 − μ B ( v ) ) → μ A ( u ) = { μ B ( v ) ⋅ μ A ( u ) , 1 − μ B ( v ) > μ A ( u ) , 1 , 1 − μ B ( v ) ≤ μ A ( u ) . = { μ B ( v ) ⋅ μ A ( u ) , μ A ( u ) + μ B ( v ) < 1 , 1 , μ A ( u ) + μ B ( v ) ≥ 1. (3.8)

It is well known that given a unary relationship R ( D 1 ( y ) ) = B ′ one can obtain the consequence R ( D 2 ( x ) ) by applying compositional rule of inference (CRI) to R ( D 1 ( y ) ) and R ( D 1 ( y ) , D 2 ( x ) ) of type (3.7):

R ( D 2 ( x ) ) = R ( D 1 ( y ) ) ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B ′ ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ μ B ′ ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] / v (3.9)

In order that Criterion I (see Appendix) is satisfied, that is R ( D 2 ( x ) ) = ¬ A from (3.9) the equality

∪ v ∈ V [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] = 1 − μ A ( u ) (3.10)

must be satisfied for arbitrary u ∈ U and in order that the equality (3.10) is satisfied, it is necessary that the inequality

μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ≤ 1 − μ A ( u ) (3.11)

holds for arbitrary u ∈ U and v ∈ V . Let us define new methods of fuzzy conditional inference of the type (3.5), which requires the satisfaction of Criteria I-IV from Appendix.

Theorem 3

If fuzzy sets A ⊂ U | μ A : U → [ 0 , 1 ] , B ⊂ V | μ B : V → [ 0 , 1 ] are defined as (3.6) and R ( D 1 ( y ) , D 2 ( x ) ) is defined by (3.7), where

( 1 − μ B ( v ) ) → μ A ( u ) = { μ B ( v ) ⋅ μ A ( u ) , μ B ( v ) < 1 − μ A ( u ) , 1 , μ B ( v ) ≥ 1 − μ A ( u ) . (3.12)

then Criteria I, II, III and IV-1 are satisfied.

Proof:

For Criteria I-III let R ( D 1 ( y ) ) = B α ( α > 0 ) then

R ( D 2 ( x ) ) = B α ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B α ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ μ B α ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] / u (3.13)

∃ V 1 , V 2 ⊂ V | V 1 ∪ V 2 = V ; | V 1 ∩ V 2 = ∅ ⇒ ∀ v ∈ V 1 | μ B ( v ) < 1 − μ A ( u ) ; ∀ v ∈ V 2 | μ B ( v ) ≥ 1 − μ A ( u ) (3.14)

From (3.13) and given subsets from (3.14) we have

R ( D 2 ( x ) ) = [ ∫ U ∪ v ∈ V 1 [ μ B α ( v ) ∧ ( μ B ( v ) ⋅ μ A ( u ) ) ] / u ] ∨ [ ∫ U ∪ v ∈ V 2 [ μ B α ( v ) ∧ 1 ] / u ] . (3.15)

Let us introduce the following function (as a part of implication operation)

f ( v , u ) = μ B ( v ) ⋅ μ A ( u ) | μ B ( v ) < 1 − μ A ( u ) . (3.16)

Then the following is taking place:

∀ v ∈ V 1 | μ B α ( v ) ∧ f ( v , u ) = { μ B α ( v ) , μ B α ( v ) ≤ f ( v , u ) , f ( v , u ) , μ B α ( v ) > f ( v , u ) , (3.17)

Since from (3.16) μ A ( u ) + μ B ( v ) < 1 ⇒ f ( v , u ) < 0.25 , but μ B α ( v ) ∈ [ 0 , 1 ] , therefore from (3.17) we have

∀ v ∈ V 1 | μ B α ( v ) ∧ f ( v , u ) = f ( v , u ) < 0.25 (3.18)

∀ v ∈ V 2 | μ B α ( v ) ∧ 1 = μ B α ( v ) (3.19)

From (3.16)-(3.19) we have

( 3.15 ) = [ ∫ U ∪ v ∈ V 2 μ B α ( v ) / u ] = ∫ U 1 − μ A α ( u ) / u = ¬ A α . (Q. E. D.).

For Criteria IV-2 let R ( D 1 ( y ) ) = ¬ B then

R ( D 2 ( x ) ) = ¬ B ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V ( 1 − μ B ( v ) ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ ( 1 − μ B ( v ) ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] / u (3.20)

From (3.20) and given subsets from (3.14) we have

R ( D 2 ( x ) ) = [ ∫ U ∪ v ∈ V 1 [ 1 − μ B ( v ) ∧ ( μ B ( v ) ⋅ μ A ( u ) ) ] / u ] ∨ [ ∫ U ∪ v ∈ V 2 [ 1 − μ B ( v ) ∧ 1 ] / u ] = [ ∫ U ∪ v ∈ V 1 [ μ A ( u ) ] / u ] ∨ [ ∫ U ∪ v ∈ V 2 [ 1 − μ B ( v ) ] / u ] = ∫ U μ A ( u ) / u = A (Q. E. D.) (3.21)

To illustrate these results we will present couple examples.

Example 1

Let U and V be two universes of discourses and correspondent fuzzy sets are represented as in (3.6) A ⊂ U | μ A : U → [ 0 , 1 ] , B ⊂ V | μ B : V → [ 0 , 1 ] ; related linguistic scale could consist of the terms like {“SMALL”…, “MEDIUM”…, “LARGE”}. Let us consider the following cases.

A labeled “LARGE” = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11

And B labeled “SMALL” = 1.0 / v 1 + 0.9 / v 2 + 0.8 / v 3 + 0.7 / v 4 + 0.6 / v 5 + 0.5 / v 6 + 0.4 / v 7 + 0.3 / v 8 + 0.2 / v 9 + 0.1 / v 10 + 0.0 / v 11

The negation of a fuzzy set B would look like

¬ B = 0.0 / v 1 + 0.1 / v 2 + 0.2 / v 3 + 0.3 / v 4 + 0.4 / v 5 + 0.5 / v 6 + 0.6 / v 7 + 0.7 / v 8 + 0.8 / v 9 + 0.9 / v 10 + 1.0 / v 11

The binary relationship matrix R ( D 1 ( y ) , D 2 ( x ) ) of a type (3.7) would look like

Let B 2 labeled “very SMALL” = 1.0 / v 1 + 0.81 / v 2 + 0.64 / v 3 + 0.49 / v 4 + 0.36 / v 5 + 0.25 / v 6 + 0.16 / v 7 + 0.09 / v 8 + 0.04 / v 9 + 0.01 / v 10 + 0.0 / v 11

Applying (3.13)

R ( D 2 ( x ) ) = B 2 ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B 2 ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( u , v ) = 1.0 / u 1 + 0.81 / u 2 + 0.64 / u 3 + 0.49 / u 4 + 0.36 / u 5 + 0.25 / u 6 + 0.16 / u 7 + 0.09 / u 8 + 0.04 / u 9 + 0.01 / u 10 + 0.0 / u 11 = ¬ A 2

Example 2

A labeled “LARGE” = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11

And B also labeled “LARGE” = 0.0 / v 1 + 0.1 / v 2 + 0.2 / v 3 + 0.3 / v 4 + 0.4 / v 5 + 0.5 / v 6 + 0.6 / v 7 + 0.7 / v 8 + 0.8 / v 9 + 0.9 / v 10 + 1.0 / v 11

The negation of a fuzzy set B would look like

¬ B = 1.0 / v 1 + 0.9 / v 2 + 0.8 / v 3 + 0.7 / v 4 + 0.6 / v 5 + 0.5 / v 6 + 0.4 / v 7 + 0.3 / v 8 + 0.2 / v 9 + 0.1 / v 10 + 0.0 / v 11

The binary relationship matrix R ( D 1 ( y ) , D 2 ( x ) ) of a type (3.7) would look like

Applying (3.13)

R ( D 2 ( x ) ) = ¬ B ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V 1 − μ B ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( u , v ) = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11 = A

Let us revisit the fuzzy conditional inference rule (3.5). It will be shown that when the membership function of the observation ¬ B is continuous, then the conclusion A depends continuously on the observation; and when the membership function of the relation R ( ¬ B , A ) is continuous then the observation A has a continuous membership function. We start with some definitions. A fuzzy set A with membership function μ A : ℜ → [ 0 , 1 ] = I , is called a fuzzy number if A is normal, continuous, and convex. The fuzzy numbers represent the continuous possibility distributions of fuzzy terms of the following type

A = ∫ ℜ μ A ( x ) / x

Let A be a fuzzy number, then for any θ ≥ 0 we define ϖ A ( θ ) the modulus of continuity of A by

ϖ A ( θ ) = max | x 1 − x 2 | ≤ θ | μ A ( x 1 ) − μ A ( x 2 ) | . (3.22)

An α-level set of a fuzzy interval A is a non-fuzzy set denoted by [ A ] α and is defined by [ A ] α = { t ∈ ℜ | μ A ( t ) ≥ α } for α ∈ ( 0 , 1 ] and [ A ] α = c l ( supp μ A ) for α = 0 . Here we use a metric of the following type

D ( A , B ) = sup α ∈ [ 0 , 1 ] d ( [ A ] α , [ B ] α ) , (3.23)

where d denotes the classical Hausdorff metric expressed in the family of compact subsets of ℜ 2 , i.e.

d ( [ A ] α , [ B ] α ) = max { | a 1 ( α ) − b 1 ( α ) | , | a 2 ( α ) − b 2 ( α ) | } .

whereas [ A ] α = [ a 1 ( α ) , a 2 ( α ) ] , [ B ] α = [ b 1 ( α ) , b 2 ( α ) ] when the fuzzy sets A and B both have finite support { x 1 , ⋯ , x n } then their Hamming distance is defined as

H ( A , B ) = ∑ i = 1 n | μ A ( x i ) − μ B ( x i ) | .

In the sequel we will use the following lemma.

Lemma 3 [

Let δ ≥ 0 be a real number and let A , B be fuzzy intervals. If D ( ¬ B , A ) ≤ δ , then

sup t ∈ ℜ | μ A ( t ) − μ ¬ B ( t ) | ≤ max { ϖ A ( δ ) , ϖ ¬ B ( δ ) } .

Consider the fuzzy conditional inference rule (3.5) with different observations B and B 1 :

Antecedent 1: If y is ØB then x is A Antecedent 2: y is B

----------------------------------

Consequent: x is ØA

Antecedent 1: If y is ØB then x is A Antecedent 2: y is B^{1}

----------------------------------

Consequent: x is ØA^{1}

According to the fuzzy conditional inference rule (3.5), the membership functions of the conclusions are computed as

μ ¬ A ( u ) = ∪ v ∈ ℜ [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] ;

μ ¬ A 1 ( u ) = ∪ v ∈ ℜ [ μ B 1 ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ]

Or

1 − μ A ( u ) = sup v ∈ ℜ [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] ;

1 − μ A 1 ( u ) = sup v ∈ ℜ [ μ B 1 ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] , (3.24)

The following theorem shows the fact that when the observations are closed to each other in the metric D ( . ) of (3.23) type, then there can be only a small deviation in the membership functions of the conclusions.

Theorem 4 (Stability theorem)

Let δ ≥ 0 and let B , B 1 be fuzzy intervals and an implication operation in the fuzzy conditional inference rule (3.5) is from

sup u ∈ ℜ | μ A ( u ) − μ A 1 ( u ) | ≤ max { ϖ ¬ B ( δ ) , ϖ ¬ B 1 ( δ ) }

Proof:

Given an implication operation in the fuzzy conditional inference rule (3.5) is from

∃ V 1 , V 2 ⊂ ℜ | V 1 ∪ V 2 = V ; | V 1 ∩ V 2 = ∅ ⇒ ∀ v ∈ V 1 | μ B ( v ) < 1 − μ A ( u ) ; ∀ v ∈ V 2 | μ B ( v ) ≥ 1 − μ A ( u )

A = B ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → μ A ( u ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u ) ) ] / u (3.25)

From (3.13) and given subsets from (3.14) we have

A 1 = [ ∫ U ∪ v ∈ V 1 [ μ B 1 ( v ) ∧ ( μ B ( v ) ⋅ μ A ( u ) ) ] / u ] ∨ [ ∫ U ∪ v ∈ V 2 [ μ B 1 ( v ) ∧ 1 ] / u ] . (3.26)

Then from (3.16) the following is taking place:

∀ v ∈ V 1 | μ B 1 ( v ) ∧ f ( v , u ) = { μ B 1 ( v ) , μ B 1 ( v ) ≤ f ( v , u ) , f ( v , u ) , μ B 1 ( v ) > f ( v , u ) , (3.27)

Since from (3.16) μ A ( u ) + μ B ( v ) < 1 ⇒ f ( v , u ) < 0.25 , but μ B ( v ) ∈ [ 0 , 1 ] , therefore from (3.17) we have

∀ v ∈ V 1 | μ B 1 ( v ) ∧ f ( v , u ) = f ( v , u ) < 0.25 (3.28)

∀ v ∈ V 2 | μ B 1 ( v ) ∧ 1 = μ B 1 ( v ) (3.29)

From (3.27)-(3.29) we have

( 3.26 ) = [ ∫ U ∪ v ∈ V 2 μ B 1 ( v ) / u ] = ∫ U 1 − μ A 1 ( u ) / u = ¬ A 1

From (3.28) and (3.29), we see that the difference of the values of conclusions for both B and B 1 observations for arbitrary fixed v ∈ ℜ | | μ B ( v ) − μ B 1 ( v ) | is defined as follows

∀ v ∈ V 1 | | μ B ( v ) − μ B 1 ( v ) | = { | μ A ( u ) − μ A 1 ( u ) | , 0 ,

∀ v ∈ V 2 | | μ B ( v ) − μ B 1 ( v ) | = | μ A ( u ) − μ A 1 ( u ) | . Therefore from Lemma 1 we have

sup v ∈ ℜ | μ B ( v ) − μ B 1 ( v ) | = sup u ∈ ℜ | μ A ( u ) − μ A 1 ( u ) | ≤ max { ϖ ¬ B ( δ ) , ϖ ¬ B 1 ( δ ) } (Q. E. D.)

Theorem 5 (Continuity theorem)

Let binary relationship R ( v , u ) = ( 1 − μ B ( v ) ) → μ A ( u ) be continuous. Then A is continuous and ϖ A ( δ ) ≤ ϖ R ( δ ) for each δ ≥ 0 .

Proof:

Let δ ≥ 0 be a real number and let u 1 , u 2 ∈ ℜ such that | u 1 − u 2 | ≤ δ . From (3.22) we have ϖ A ( δ ) = max | u 1 − u 2 | ≤ δ | μ A ( u 1 ) − μ A ( u 2 ) | . Then

| μ A ( u 1 ) − μ A ( u 2 ) | = | sup v ∈ ℜ [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u 1 ) ) ] − sup v ∈ ℜ [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → μ A ( u 2 ) ) ] | ≤ sup v ∈ ℜ [ μ B ( v ) ∧ | ( ( 1 − μ B ( v ) ) → μ A ( u 1 ) ) − ( ( 1 − μ B ( v ) ) → μ A ( u 2 ) ) | ] ≤ sup v ∈ ℜ [ μ B ( v ) ∧ ϖ R ( | u 1 − u 2 | ) ] ≤ sup v ∈ ℜ [ μ B ( v ) ∧ ϖ R ( δ ) ] = ϖ R ( δ ) (Q. E. D.).

Results from Theorem 4 and Theorem 5 could be used for formulating another similarity measure, based on Hamming distance between two fuzzy sets

A ⊆ U and B ⊆ U , which presented by non-linear membership functions, i.e.

S I 1 ( A , B ) = 1 − ∑ i = 1 n | μ A ( u i ) − μ B ( u i ) | n ; n = C a r d U . (3.30)

For instance let us apply (3.29) to fuzzy sets from Example 1 (see

Let B labeled “SMALL” = 1.0 / v 1 + 0.9 / v 2 + 0.8 / v 3 + 0.7 / v 4 + 0.6 / v 5 + 0.5 / v 6 + 0.4 / v 7 + 0.3 / v 8 + 0.2 / v 9 + 0.1 / v 10 + 0.0 / v 11

And B 2 labeled “very SMALL” = 1.0 / v 1 + 0.81 / v 2 + 0.64 / v 3 + 0.49 / v 4 + 0.36 / v 5 + 0.25 / v 6 + 0.16 / v 7 + 0.09 / v 8 + 0.04 / v 9 + 0.01 / v 10 + 0.0 / v 11

S I 1 ( B , B 2 ) = 0.85 . It is important to pay attention to a fact that S I 1 ( ¬ A , ¬ A 2 ) = 0.85 from the same Example 1, which confirms results of both Theorem 4 and Theorem 5.

Let us remind that the scheme for Generalized Modus Tollens (2.3) looks like that

Antecedent 1: If x is A, then y is B Antecedent 2: y is B^{1}

---------------------------------- (4.1)

Consequent: x is A^{1}.

First consider classical logic equivalence

a → b ≡ ¬ b → ¬ a (4.2)

The classical logic equivalence (4.2) can be extended in fuzzy logic with implication and negation functions. We use the same fuzzy logic, which operations are presented in

Since

a → b = { ( 1 − a ) ⋅ b , a > b , 1 , a ≤ b (4.3)

And because

¬ b → ¬ a = { ( 1 − ¬ b ) ⋅ ¬ a , ¬ b > ¬ a , 1 , ¬ b ≤ ¬ a = { b ⋅ ¬ a , a > b , 1 , a ≤ b ≡ ¬ a → ¬ b (4.4)

Both (4.3) and (4.4) proofs that classical logic equivalence (4.2) holds. It is very important to show that we transform Generalized Modus Tollens rule (4.1) into its equivalent form

Antecedent1: If y is ØB then x is ØA Antecedent2: y is B^{1}

---------------------------------- (4.5)

Consequent: x is A^{1}.

Let us formulize an inference method for a rule (4.5). Following a standard approaches toward such formalization, let U and V be two universes of discourses and correspondent fuzzy sets be represented as such A ⊂ U | μ A : U → [ 0 , 1 ] , B ⊂ V | μ B : V → [ 0 , 1 ] .

Whereas given (3.6) a binary relationship for the fuzzy conditional proposition of the type: “If y is ØB then x is ØA” for a fuzzy logic is defined as

R ( D 1 ( y ) , D 2 ( x ) ) = ¬ B × V → U × ¬ A = ∫ V × U ( 1 − μ B ( v ) ) / ( v , u ) → ∫ V × U ( 1 − μ A ( u ) ) / ( v , u ) = ∫ V × U ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) / ( v , u ) (4.6)

Given an implication operator from

( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) = { μ B ( v ) ⋅ ( 1 − μ A ( u ) ) , 1 − μ B ( v ) > 1 − μ A ( u ) , 1 , 1 − μ B ( v ) ≤ 1 − μ A ( u ) . = { μ B ( v ) ⋅ ( 1 − μ A ( u ) ) , μ A ( u ) > μ B ( v ) , 1 , μ A ( u ) ≤ μ B ( v ) .

And again given a unary relationship R ( D 1 ( y ) ) = B ′ one can obtain the consequence R ( D 2 ( x ) ) by applying compositional rule of inference (CRI) to R ( D 1 ( y ) ) and R ( D 1 ( y ) , D 2 ( x ) ) of type (4.6):

R ( D 2 ( x ) ) = R ( D 1 ( y ) ) ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B ′ ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ μ B ′ ( v ) ∧ ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) ] / v (4.7)

In order that Criterion V (see Appendix) is satisfied, that is R ( D 2 ( x ) ) = ¬ A from (3.9) the equality

∪ v ∈ V [ μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) ] = μ A ( u ) (4.8)

must be satisfied for arbitrary u ∈ U and in order that the equality (3.10) is satisfied, it is necessary that the inequality

μ B ( v ) ∧ ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) ≤ μ A ( u ) (4.9)

holds for arbitrary u ∈ U and v ∈ V . Let us define new methods of fuzzy conditional inference of the type (4.6), which requires the satisfaction of Criteria V-VIII from Appendix.

Theorem 6

If fuzzy sets A ⊂ U | μ A : U → [ 0 , 1 ] , B ⊂ V | μ B : V → [ 0 , 1 ] are defined as (3.6) and R ( D 1 ( y ) , D 2 ( x ) ) is defined by (4.6), where

( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) = { μ B ( v ) ⋅ ( 1 − μ A ( u ) ) , μ A ( u ) > μ B ( v ) , 1 , μ A ( u ) ≤ μ B ( v ) . (4.10)

then Criteria V,VI,VII and VIII-2 are satisfied.

Proof:

For Criteria V-VII let R ( D 1 ( y ) ) = B α ( α > 0 ) then

R ( D 2 ( x ) ) = B α ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B α ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ μ B α ( v ) ∧ ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) ] / u (4.11)

∃ V 1 , V 2 ⊂ V | V 1 ∪ V 2 = V ; | V 1 ∩ V 2 = ∅ ⇒ ∀ v ∈ V 1 | μ A ( u ) > μ B ( v ) ; ∀ v ∈ V 2 | μ A ( u ) ≤ μ B ( v ) (4.12)

From (4.11) and given subsets from (4.12) we have

R ( D 2 ( x ) ) = [ ∫ U ∪ v ∈ V 1 [ μ B α ( v ) ∧ ( μ B ( v ) ⋅ ( 1 − μ A ( u ) ) ] / u ] ∨ [ ∫ U ∪ v ∈ V 2 [ μ B α ( v ) ∧ 1 ] / u ] . (4.13)

Let us introduce the following function (as a part of implication operation)

f ( v , u ) = μ B ( v ) ⋅ ( 1 − μ A ( u ) ) | μ A ( u ) > μ B ( v ) . (4.14)

Then the following is taking place:

∀ v ∈ V 1 | μ B α ( v ) ∧ f ( v , u ) = { μ B α ( v ) , μ B α ( v ) ≤ f ( v , u ) , f ( v , u ) , μ B α ( v ) > f ( v , u ) , (4.15)

∀ v ∈ V 2 | μ B α ( v ) ∧ 1 = μ B α ( v ) , (4.16)

From (4.15), (4.16) we have

( 4.13 ) = [ ∫ U ∪ v ∈ V 2 μ B α ( v ) / u ] = ∫ U μ A α ( u ) / u = A α . (Q. E. D.).

For Criteria VIII-2 let R ( D 1 ( y ) ) = ¬ B then

R ( D 2 ( x ) ) = ¬ B ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V ( 1 − μ B ( v ) ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) / ( v , u ) = ∫ U ∪ v ∈ V [ ( 1 − μ B ( v ) ) ∧ ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) ] / u (4.17)

From (4.17) and given subsets from (4.15) and (4.16) we have

R ( D 2 ( x ) ) = [ ∫ U ∪ v ∈ V 1 [ 1 − μ B ( v ) ∧ f ( v , u ) ] / u ] ∨ [ ∫ U ∪ v ∈ V 2 [ 1 − μ B ( v ) ∧ 1 ] / u ] . Since the following is taking place

[ ( 1 − μ B ( v ) ) ∧ f ( v , u ) ] ≤ [ ( 1 − μ B ( v ) ) ∧ 1 ] .

Therefore

( 4.17 ) = [ ∫ U ∪ v ∈ V 2 [ ( 1 − μ B ( v ) ) ∧ 1 ] / u ] = ∫ U 1 − μ A ( u ) / u = ¬ A . (Q. E. D.)

To illustrate these results we will present couple examples. We use similar fuzzy sets as in Examples 1 and 2.

Example 3

Let U and V be two universes of discourses and correspondent fuzzy sets are represented as in (3.6) A ⊂ U | μ A : U → [ 0 , 1 ] , B ⊂ V | μ B : V → [ 0 , 1 ] ; related linguistic scale could consist of the terms like {“SMALL”…, “MEDIUM”…, “LARGE”}. Let us consider the following cases.

A labeled “LARGE” = 0.0 / u 1 + 0.1 / u 2 + 0.2 / u 3 + 0.3 / u 4 + 0.4 / u 5 + 0.5 / u 6 + 0.6 / u 7 + 0.7 / u 8 + 0.8 / u 9 + 0.9 / u 10 + 1.0 / u 11

And B labeled “SMALL” = 1.0 / v 1 + 0.9 / v 2 + 0.8 / v 3 + 0.7 / v 4 + 0.6 / v 5 + 0.5 / v 6 + 0.4 / v 7 + 0.3 / v 8 + 0.2 / v 9 + 0.1 / v 10 + 0.0 / v 11

The negations of fuzzy sets A B would look like

¬ A = 1.0 / u 1 + 0.9 / u 2 + 0.8 / u 3 + 0.7 / u 4 + 0.6 / u 5 + 0.5 / u 6 + 0.4 / u 7 + 0.3 / u 8 + 0.2 / u 9 + 0.1 / u 10 + 0.0 / u 11

¬ B = 0.0 / v 1 + 0.1 / v 2 + 0.2 / v 3 + 0.3 / v 4 + 0.4 / v 5 + 0.5 / v 6 + 0.6 / v 7 + 0.7 / v 8 + 0.8 / v 9 + 0.9 / v 10 + 1.0 / v 11

The binary relationship matrix R ( D 1 ( y ) , D 2 ( x ) ) of a type (4.6) would look like

Let B 2 labeled “very SMALL” = 1.0 / v 1 + 0.81 / v 2 + 0.64 / v 3 + 0.49 / v 4 + 0.36 / v 5 + 0.25 / v 6 + 0.16 / v 7 + 0.09 / v 8 + 0.04 / v 9 + 0.01 / v 10 + 0.0 / v 11

Applying (4.7)

R ( D 2 ( x ) ) = B 2 ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V μ B 2 ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) / ( v , u ) = 0.0 / u 1 + 0.01 / u 2 + 0.04 / u 3 + 0.09 / u 4 + 0.16 / u 5 + 0.25 / u 6 + 0.36 / u 7 + 0.49 / u 8 + 0.64 / u 9 + 0.81 / u 10 + 1.0 / u 11 = A 2

(“very LARGE”).

Example 4

And B also labeled “LARGE” = 0.0 / v 1 + 0.1 / v 2 + 0.2 / v 3 + 0.3 / v 4 + 0.4 / v 5 + 0.5 / v 6 + 0.6 / v 7 + 0.7 / v 8 + 0.8 / v 9 + 0.9 / v 10 + 1.0 / v 11

The negations of fuzzy sets A B would look like

¬ A = 1.0 / u 1 + 0.9 / u 2 + 0.8 / u 3 + 0.7 / u 4 + 0.6 / u 5 + 0.5 / u 6 + 0.4 / u 7 + 0.3 / u 8 + 0.2 / u 9 + 0.1 / u 10 + 0.0 / u 11

¬ B = 1.0 / v 1 + 0.9 / v 2 + 0.8 / v 3 + 0.7 / v 4 + 0.6 / v 5 + 0.5 / v 6 + 0.4 / v 7 + 0.3 / v 8 + 0.2 / v 9 + 0.1 / v 10 + 0.0 / v 11

The binary relationship matrix R ( D 1 ( y ) , D 2 ( x ) ) of a type (4.6) would look like

Applying (4.7)

R ( D 2 ( x ) ) = ¬ B ∘ R ( D 1 ( y ) , D 2 ( x ) ) = ∫ V 1 − μ B ( v ) / v ∘ ∫ V × U ( ( 1 − μ B ( v ) ) → ( 1 − μ A ( u ) ) ) / ( v , u ) = 1.0 / u 1 + 0.9 / u 2 + 0.8 / u 3 + 0.7 / u 4 + 0.6 / u 5 + 0.5 / u 6 + 0.4 / u 7 + 0.3 / u 8 + 0.2 / u 9 + 0.1 / u 10 + 0.0 / u 11 = ¬ A

In this article, we presented a systemic approach toward a fuzzy logic based formalization of an approximate reasoning methodology in a fuzzy resolution. We derived a truth value of A from both values of B → A and B by some mechanism. We used a t-norm fuzzy logic, in which an implication operator is a root of both graduated conjunction and disjunction operators. We investigated features of correspondent fuzzy resolvent, which was based on introduced operators. We proposed two types of Similarity Measures for both linear and non-linear membership functions. We applied this approach to both generalized modus-ponens/modus-tollens syllogisms, for which we formulated a set of Criterion. The content of this investigation is well-illustrated with artificial examples.

Tserkovny, A. (2017) A Fuzzy Logic Based Resolution Principal for Approximate Reasoning. Journal of Software Engineering and Applications, 10, 793-823. https://doi.org/10.4236/jsea.2017.1010045

Criterion I

Antecedent 1: If y is ØB then x is A Antecedent 2: y is B

----------------------------------

Consequent: x is ØA.

Criterion II

Antecedent 1: If y is ØB then x is A Antecedent 2: y is very B

----------------------------------

Consequent: x is Ø(very A).

Criterion III

Antecedent 1: If y is ØB then x is A Antecedent 2: y is more or less B

----------------------------------

Consequent: x is Ø(more or less A).

Criterion IV-1

Antecedent 1: If y is ØB then x is A Antecedent 2: y is ØB

----------------------------------

Consequent: x is unknown

Criterion IV-2

Antecedent 1: If y is ØB then x is A Antecedent 2: y is ØB

----------------------------------

Consequent: x is A.

Criterion V

Antecedent 1: If y is ØB then x is ØA Antecedent 2: y is B

----------------------------------

Consequent: x is A.

Criterion VI

Antecedent 1: If y is ØB then x is ØA Antecedent 2: y is very B

----------------------------------

Consequent: x is very A

Criterion VII

Antecedent 1: If y is ØB then x is ØA Antecedent 2: y is more or less B

----------------------------------

Consequent: x is more or less A

Criterion VIII-1

Antecedent 1: If y is ØB then x is ØA Antecedent 2: y is ØB

----------------------------------

Consequent: x is unknown

Criterion VIII-2

Antecedent 1: If y is ØB then x is ØA Antecedent 2: y is ØB

----------------------------------

Consequent: x is ØA.