A t-Norm Fuzzy Logic for Approximate Reasoning

A t-norm fuzzy logic is presented, in which a triangular norm (t-norm) plays the role of a graduated conjunction operator. Based on this fuzzy logic we develop methods for fuzzy reasoning in which antecedents and consequents involve fuzzy conditional propositions of the form “If x is A then y is B”, with A and B being fuzzy concepts (fuzzy sets). In this study, we present a systemic approach toward fuzzy logic formalization for approximate reasoning. We examine statistical characteristics of the proposed fuzzy logic. As the matter of practical interest, we construct a set of fuzzy conditional inference rules on the basis of the proposed fuzzy logic. Important features of these rules are investigated.


Introduction
In our daily life we often make inferences whose antecedents and consequents contain fuzzy concepts.Such an inference cannot be made adequately by the methods which are based either on classical two valued logic or on many valued logic.In order to make such an inference, Zadeh suggested an inference rule called "compositional rule of inference".Using this inference rule, he, Mamdani, Mizumoto et al., R. Aliev and A. Tserkovny suggested several methods for fuzzy reasoning in which the antecedent contain a conditional proposition with fuzzy concepts: Ant 1: If x is P then y is Q Ant 2: x is P' ----------------------------------(1.1) Cons: y is Q'.
Those methods are based on an implication operator in various fuzzy logics.
This matter has been under discussion for the last couple decades [1]- [46].
In (1.1) x and y are the names of objects, and P, P', Q and Q' are fuzzy concepts represented by fuzzy sets in universe of discourse U, U, V and V, respectively.This form of inference may be viewed as a generalized modus ponens which reduces to modus ponens when P P ′ = and Q Q ′ = .Let P and Q be fuzzy sets in U and V respectively and correspondent fuzzy sets be represented as such [ ] : 0,1 , .
A necessary consideration for this discussion is that with the only few exceptions for S-logic (1.6) and G-logic (1.7), and L1-L4(1.3)-(1.6) all other known fuzzy logics don't satisfy either the classical "modus-ponens" principal, or other criteria which fit human intuition and first formulated in [32].The proposed fuzzy logic has an implication operator, which satisfies the "modus-ponens" principal and criteria, which fit human intuition.
The second section of the article will cover some initial fuzzy logic creation considerations.In third section a set of operations in proposed fuzzy logic is presented.The fourth section is devoted to an introduction of a t-norm as a graduated conjunction operator in presented fuzzy logic.The Section five will cover a power sets based features of proposed fuzzy logic.The statistical analysis of the fuzzy logic is completed in Section six.Section Seven covers the issue of fuzzy conditional inference rules based on proposed fuzzy logic and extended investigation of its features.

Preliminary Considerations
In order to start formulating of a fuzzy logic major implication operator, we are proposing the following function as a part of it: An implication function is a continuous function , , , , I p I q r I q I p r = , (Exchange Principle); , , I p q I n q n p = .(Contra positive Symmetry Principle), where
n -is a negation, which could be defined for ( ) n q as ( ) ( ) Before proving that ( ) , F p q is from (2.1) satisfies (I1)-(I6) axioms, let us show some basic op- erations in proposed fuzzy logic.

The Fuzzy Logic
Let us designate the truth values of logical antecedent P and consequent Q as ( ) = respectively.Then relevant set of proposed fuzzy logic operators are shown in Table 1.
In other words we propose a new many-valued system, characterized by the set of base union (  ) and intersection (  ) operations with relevant complement, defined as ( ) ( ) . In addition, the operators ↓ and ↑ are , p q p q p q q p q p expressed as negations of the  and  correspondingly.It is a well-known fact that the operation implication in a fuzzy logic was the foundation of decision making procedure for numerous approximate reasoning tasks.Therefore let us prove that proposed implication operation from (2.2) satisfies axioms (I1)-(I6).For this matter let us pose the problem very explicitly.We are working in many-valued system, which for present purposes is all or some of the real interval [ ] 0,1 ℜ = . As was mentioned in [1], the rationales there are more than ample for our purposes in very much of practice, the following set { } 0, 0.1, 0.2, , 0.9,1  of 11 values is quite sufficient, and we shall use this set V 11 in our illustration.
F p q p q p q p q I p q p q p q Where function
Where function

( )
, norm F p q is defined in (1), then axioms (I1)-(I6) are satisfied and, therefore it is an implication operation. Proof: , q p p I p q I p q I p q I p q 0 I p q I p q p q p q p q q q q ′ ′ ′ , , p q q I p q I p q I p q I p q q p q p p q p q I n q n p q p q p p q In addition proposed fuzzy logic is characterized by the following features: Commutativity for both conjunction (  )and disjunction (  ) operations, i.e.: p q q p ∧ = ∧ and p q q p ∨ = ∨ ; Assotiativity for these operations: ( ) ( ) To prove the feature (3.4) note that 1 , 1 p q r p q p q r r p q p q r p q r p q r r r p q r On the other hand , 1, 0, 1 q r q r q r q r To prove the Feature (3.5) by using (3.7) we have q r q r p q r p q r p q r p q r p q r Therefore the Expression (3.8) equals (3.9) Q.E.D.
DeMorgan theorems, which are extrapolated over the [ ] To prove these theorems notice that On the other hand p q p q p q p q p q p q p q Therefore the Expression (3.10) equals (3.11) Q.E.D.By analogous On the other hand , 1, 0, 1 Therefore the Expression (3.12) equals (3.13) Q.E.D. It should be mentioned that proposed fuzzy logic could also be characterized by yet another featured p p ¬ = .As a conclusion we should admit that all above features confirm that resulting system can be applied to V 11 for every finite and infinite n up to that (V 11, , , , ¬ ∧ ∨ → ) is then closed under all its operations.

The t-Norm
Proposition.
In proposed fuzzy logic the operation of conjunction ( ) is a t-norm. Proof: The function ( ) 2) Associativity: ( ) ( ) 4) Neutrality: Associativity: Case: ( ) Conj p q p q p q Conj p q r Conj p q r p q r p q r Conj p q r p q r Conj p q r From where we have that ( ) For the case ( ) Using (4.3) we are getting similar to (4.2) results

Fuzzy Power Sets and the Fuzzy Logic
Given a fuzzy implication operator → and a fuzzy subset Q of a crisp un- iverse U , the fuzzy power set PQ of Q is given by the membership function PQ µ , with ( ) ( ) The degree to which Where conditions are as in Definition 2, the degree to which the fuzzy sets P and Q is the same, or their degree of sameness, is ( ) ( ) ( ) The following is then immediate.
Proposition 1. [9] ( ) ( ) , , , As it was mentioned in [9] there seem to be two plausible ways to define the degree to which sets P and Q may be said to be disjointed.One is the degree to which each is a subset of the other's complement.The second is the degree to which their intersection is empty.Definition 4. [9] The degree of disjointness of P and Q , or degree to which P and Q are disjointed, in the first and second sense, are For the case (1) ) ) Therefore from (5.5) and (5.6) definition (1) looks like ( For the case ( (5.8) Definition 5. [9] (Degree to which a set is a subset of its complement).The expression ( ) 1, 0.5, [9] (Degree to which a set is disjointed from its complement, in the two senses).From (5.7), (5.8) the following is taking place The value of ( )

Statistical Property of the Fuzzy Logic
In this chapter we discuss some properties of proposed fuzzy implication opera- Then the value of the implication ( ) is some function ( ) Because p and q are assumed to be uniformly and independently distributed across [0, 1], the expected value of the implication is And its variance is ≤ ≤ ≤ ≤ .From (6.1) and given Expression (3.1) and the fact that , , , , , , we have the following But from Table 2 it is clear that for ( ) because of the following ( ) 0.8 0.9 0.8 0.9 I p q p q p q p q q p p q From (6.6) finally we have , d d 0.056898.3 2 3 p q q p I p q q p p q p p E I I p q p q p Var I demonstrate that the proposed fuzzy im- plication operator could be considered as a second of the fuzziest implication from the list [34] of known so far.In addition to that feature it satisfies the set of important Criteria I-IV, which is not the case for the most above mentioned implication operators.

The Fuzzy Logic and Fuzzy Conditional Inference
As it was mentioned in [32] in the semantics of natural language there exist vast amounts of concepts and we humans very often make inferences antecedents and consequences of which contain fuzzy concepts.Therefore, from the standpoint of artificial intelligence, it seems that formalization of inference methods for such inferences is very important.Following a well-known pattern, established a couple of decades ago and the standard approaches toward such formalization, let U and V (from now on) be two universes of discourses and correspondent fuzzy sets be represented as such where ( ) Whereas given (7.1) and (7.2) a binary relationship for the fuzzy conditional proposition of the type: "If x is P then y is Q" for proposed fuzzy logic is defined as , , , , .
It is well known that given a unary relationship , , .
In order that Criterion I is satisfied, that is must be satisfied for arbitrary v in V and in order that the equality (7.6) is satisfied, it is necessary that the inequality holds for arbitrary u U ∈ and v V ∈ .Let us define new methods of fuzzy conditional inference of the following type: Cons: y is Q'.
Where , , R A x A y is defined by the following .
From (7.10) and given subsets from (7.11) we have 2 1 1 Let us introduce the following function (as a part of implication operation) Then the following is taking place: , , , , , , , From (7.16) and given subsets from (7.11) we have .
Apparently the following is taking place 5.17 1 1 1

If fuzzy sets P U
⊆ and Q V ⊆ are defined as (7.1) and (7.2) respectively and , R A x A y is defined by the following . where .
Apparently the following is taking place For many real practical applications a decision making apparatus could be based not on a fuzzy conditional proposition of the type: "If x is P then y is Q", but rather on a rule of the following type If is and is and and is then is And correspondent fuzzy sets are represented as such that Given (7.1) and (7.2) a binary relationship for a fuzzy conditional proposition of the Type (7.28) for proposed fuzzy logic is defined as ( , Theorem 4. If a fuzzy conditional proposition is defined as (7.28), correspondent fuzzy sets of antecedents and consequent are presented as (7.29) and a binary relationship for a fuzzy conditional proposition is from (7.30), and "elementary" binary relationships are defined as following ( ) ( ) ( ) 1, , then the following expression is taking place and as a result the following is also true.
In other words from (7.59) we see that the membership function of a Consequent from fuzzy conditional inference rule (7.41) ( ) is poly modal one.

 
Here we also see that the membership function of a Consequent from fuzzy conditional inference rule (7.41) ( ) is poly modal one.

 
In a meantime one can see that In order to make EK from (7.42)  U V , defined in (7.47) and (7.48) correspondingly, would be empty, i.e. ,

Concluding Remarks
In this paper we proposed new t-norm fuzzy logic in which: 1) Truth values of an implication operator are based on truth values of both antecedent and consequent; 2) Implication operator could be considered as one of the fuzziest im- plication from the list [34] of known so far; 3) The suggested implication operator is a base for fuzzy conditional inference rules and satisfies the set of important human intuition Criteria; 4) Important features of these rules are investigated.
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tor ( 3 . 1 )
, assuming that the two propositions (antecedent/consequent) in a given compound proposition are independent of each other and the truth values of the propositions are uniformly distributed [20] on the interval [0,1].In other words we assume that the propositions P and Q are independent of each other and the truth values ( ) v P and ( ) v Q are uniformly distributed across the interval [0, 1].Let( ) . Let us notice that for the most implications we have the following traditional way (7.41) looks like: 42) by "elementary knowledge" (EK).Each EK is characterized by the follow- 7.49) *From now on upper and lower indices of based variables u and v denote not a membership functions, but their correspondent singletons It is clear, that for 2U from(7.45) In other words we see that there are two fuzzy sets , used as an Antecedents in fuzzy conditional inference rule (7.41), induce the same Consequent, in other words, based on the Definition 7 we have logically contradictive EK from (7.42).(Q.E.D.)Based on the results of this Theorem we have to present the following Corollary.

Table 1 .
Relevant set of proposed fuzzy logic operators.

Table 2
shows the operation implication in proposed fuzzy logic.

Table 2 .
The operation implication in proposed fuzzy logic.
which requires the satisfaction of Criteria I-IV from Appendix.It is clear that (6.8) is translated by Expression (7.5), and , logically contradictive or fruitless if a membership function of a consequent in fuzzy conditional rule (7.41) is non-unimodal or when two different antecedents induct the same consequent. called logically non-contradictive, both member-