Contracting and Involutive Negations of Probability Distributions

A dozen papers have considered the concept of negation of probability distributions (pd) introduced by Yager. Usually, such negations are generated point-by-point by functions defined on a set of probability values and called here negators. Recently it was shown that Yager negator plays a crucial role in the definition of pd-independent linear negators: any linear negator is a function of Yager negator. Here, we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy. We show that any pd-independent negator is non-involutive, and any non-trivial linear negator is strictly contracting. Finally, we introduce an involutive negator in the class of pd-dependent negators that generates an involutive negation of probability distributions.


Introduction
The concept of the negation of a probability distribution (pd) was introduced by Yager [1]. He defined the negation � of a finite probability distribution = ( 1 , … , ) by: . He noted that other negations of probability distributions (pd) could exist. Further, negations of probability distributions were considered in many works [2][3][4][5][6][7][8][9][10][11][12]. The properties of Yager's negation of a probability distribution are studied in [2]. The paper [3] studied uncertainty related to Yager's negation. The authors of [4,5] studied the convergence of the sequence of multiple Yager's negations of pd to the uniform distribution. In [6], Yager's negation is used in a multi-criteria decision-making procedure. In [7], the authors introduced another negation of probability distributions based on Tsallis entropy. The authors of [8] considered a negation of basic probability assignment in Dempster-Shafer theory. The authors of [9] studied the properties of the negation of basic probability assignment based on total uncertainty measure. The paper [10] gives a definition of negation in basic belief assignment in the Dempster-Shafer theory using matrix operators. This matrix negation was also considered in [11]. The authors of the paper [12] studied functions called negators defined on the set of probability values and point-by-point transforming pd in its negation. The results of this paper will be used here. Two types of negators are considered: pd-independent and pddependent, and a class of pd-independent linear negators was introduced [12]. It was shown [12] that Yager's negator plays a crucial role in the definition of pd-independent linear negators: any linear negator is a function of Yager's negator.
In this paper, we study contracting and involutive negators of pd. Involutive negations in fuzzy logic were considered in [13]. Contracting negations were introduced in [14] for lexicographic valuations of plausibility values used in various expert systems with qualitative expert opinions [15]. Further such negations have been studied in fuzzy logic [16][17][18]. In this paper, we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy. We show that any pd-independent negator is non-involutive, and any non-trivial linear negator is strictly contracting. Finally, we introduce an involutive negator in the class of pd-dependent negators that generates an involutive negation of probability distributions. The paper has the following structure. Section 2 considers basic definitions of negators and negations of pd generated by negators. Section 3 describes pd-independent and linear negators' properties from [12] used in the following sections. Section 4 introduces a general form of multiple linear negators and finds the limit of the sequence of such multiple negators. Section 5 shows non-involutivity of pd-independent negators, and shows that non-trivial linear negators are strictly contracting. In Section 6, we introduce an involutive pd-dependent negator that defines the involutive negation of probability distribution. The last section contains conclusions.

Negations of Discrete Probability Distributions
A set = { 1 , … , }, ( ≥ 2), of n real values satisfying for all = 1, … , , the properties: is referred to as a (finite discrete) probability distribution (pd) of the length n. One can consider as a probability of an outcome in some experiment with outcomes { 1 , … , }, ≥ 2. Let be the set of all possible probability distributions of the length n defined on X. For simplicity of the interpretation; we will fix the ordering of outcomes and the ordering of corresponding probability values according to their indexing = 1, … , , and represent the probability distribution = { 1 , … , } as n-tuple = ( 1 , … , ).
We will say that a negator generates (point-by-point) a negation ( ) = � ( 1 ), … , ( )� of pd P. A negator is called a pd-independent [12] if for any pd = ( 1 , … , ) in the negator ( ) depends only on the value but not on other values from P. A negator that is not pd-independent will be referred to as pd-dependent. A negation ( ) = � ( 1 ), … , ( )� of a probability distribution = ( 1 , … , ) will be called a pd-independent negation of pd if it is generated by pd-independent negator N.
In the following sections, we will show that all pd-independent negators are non-involutive. We will introduce an involutive negator in the class of pd-dependent negators. This involutive negator will generate an involutive negation of probability distributions satisfying the property: � ( )� = .

Properties of Pd-Independent and Linear Negators
In [12] it was shown that Yager's negator plays a crucial role in the construction of pd-independent linear negators: any linear negator is a function of Yager's negator. Let us consider some properties of pdindependent and linear negators that will be used further in this paper.
An element in [0,1] is called a fixed point of a negator if ( ) = . A probability distribution is called a fixed point of a negation if ( ) = .
The paper [12] formulates an Open Problem: Prove or disprove a hypothesis that any pd-independent negator is linear. We think that any pd-independent negator is linear. In such a case, the properties established in [12] and in this paper for pd-independent negators will be fulfilled for linear negators and vise versa.
The following section shows that the sequence of multiple linear negations of a pd converges to the uniform distribution with the maximal entropy. , after equivalent transformations, we obtain another representation of linear negator:
As it follows from Theorem 5, the multiple negations of probability distributions have as the limit the uniform distribution with the maximal entropy value [1]: Results obtained for linear negators also fulfilled for Yager's negator, because it is a linear negator.

Involutive Negators and Involutive Negations
Let = ( 1 , … , ) be a probability distribution from n , and ( ) be a negation of pd P. It will be called an involutive negation if the following property is satisfied:  Fig. 2. Contracting negator as a sequence of ( ) with the limit 1 A negator will be called an involutive negator if for any pd = ( 1 , … , ) it satisfies the following property: ( ( )) = , for all = 1, … , .
It is clear that a negation will be involutive if it is generated by an involutive negator N. Indeed: All examples of negators considered in the previous section are non-involutive. Here we introduce an involutive negator that will generate the involutive negation of finite discrete probabilistic distributions. As it follows from Theorem 7, an involutive negator cannot be pd-independent. Let = ( 1 , … ) be a probability distribution. Denote max( ) = max{ } = max{ 1 , … , }, min( ) = min{ } = min{ 1 , … , } and = max( ) + min( ).

Conclusion
The paper studied negators generating element-by-element negations of probability distributions (pd). We showed that the sequence of multiple negations of pd generated by a pd-independent linear negator converges to the uniform distribution with maximal entropy. We showed that all pd-independent negators are non-involutive, and non-trivial linear negators are strictly contracting; hence, we need to look for an involutive negator in the class of pd-dependent negators. Finally, we introduced an involutive negator in the class of pd-dependent negators. It generates involutive negation of probability distributions. Such involutive negation can formalize a probability distribution NOT P, where P is some linguistic concept like HIGH defined on a set of probability distributions. The involutivity of negation, like NOT(NOT(P)) = P, is a common property for many logical systems and can be used in retrieval or reasoning systems operating with terms represented by probability distributions. We plan to apply the proposed involutive negation in Dempster-Shaffer theory as it was proposed in the original work of Yager [1].