The Neutrosophic Entropy and its Five Components

. This paper presents two variants of penta-valued representation for neutrosophic entropy. The first is an extension of Kaufmann's formula and the second is an extension of Kosko's formula. Based on the primary three-valued information represented by the degree of truth, degree of falsity and degree of neutrality there are built some penta-valued representations that better highlights some specific features of neutrosophic entropy. Thus, we highlight five features of neutrosophic uncertainty such as ambiguity, ignorance, contradiction, neutrality and saturation. These five features are supplemented until a seven partition of unity by adding two features of neutrosophic certainty such as truth and falsity. The paper also presents the particular forms of neutroso-phic entropy obtained in the case of bifuzzy representations, intuitionistic fuzzy representations, paraconsistent fuzzy representations and finally the case of fuzzy representations.


Introduction
Neutrosophic representation of information was proposed by Smarandache [10], [11], [12] as an extension of fuzzy representation proposed by Zadeh [16] and intuitionistic fuzzy representation proposed by Atanassov [1], [2].Primary neutrosophic information is defined by three parameters: degree of truth  , degree of falsity  and degree of neutrality  .Fuzzy representation is described by a single parameter, degree of truth  , while the degree of falsity  has a de- fault value calculated by negation formula : and the degree of neutrality has a default value that is 0   .Fuzzy intuitionistic representation is described by two explicit parameters, degree of truth  and degree of falsity  , while the degree of neutrality has a default value that is 0   .Atanassov considered the incomplete variant taking into account that 1    .This allowed defining the index of ignorance (incompleteness) with the formula: thus obtaining a consistent representation of information, because the sum of the three parameters is 1 , namely: Hence, we get for neutrality the value 0   .For paraconsistent fuzzy information where 1    , the index of contradiction can be defined: and for neutrality it results: 0   .For bifuzzy information that is defined by the pair ) , (   , the net truth, the index of ignorance (incompleteness), index of contradiction and index of ambiguity can be defined, by: Among of these information parameters the most important is the construction of some measures for information entropy or information uncertainty.This paper is dedicated to the construction of neutrosophic entropy formulae.
In the next, section 2 presents the construction of two variants of the neutrosophic entropy.This construction is based on two similarity formulae; Section 3 presents a penta-valued representation of neutrosophic entropy based Vasile Pătraşcu, The Neutrosophic Entropy and its Five Components on ambiguity, ignorance, contradiction, neutrality and saturation; Section 4 outlines some conclusions.

The Neutrosophic Entropy
For neutrosophic entropy, we will trace the Kosko idea for fuzziness calculation [5].Kosko proposed to measure this information feature by a similarity function between the distance to the nearest crisp element and the distance to the farthest crisp element.For neutrosophic information the two crisp elements are ) 0 , 0 , . We consider the following vector: . We will compute the distances: The neutrosophic entropy will be defined by the similarity between these two distances.
Using the Czekanowskyi formula [3] it results the similarity C S and the neutrosophic entropy The neutrosophic entropy defined by (2.4) can be particularized for the following cases: For 0   , it result the bifuzzy entropy, namely: For 0   and 1    it results the intuitionistic fuzzy entropy proposed by Patrascu [8], namely: , it results the fuzzy entropy proposed by Kaufmann [4], namely: Using the Ruzicka formula [3] it result the formulae for the similarity R S and the neutrosophic entropy or its equivalent form: The neutrosophic entropy defined by (2.12) can be particularized for the following cases: For 0   , it results the bifuzzy entropy proposed by Patrascu [7], namely: For 0   and 1    it results the intuitionistic fuzzy entropy proposed by Szmidt and Kacprzyk [14], [15], explicitly: , it results the fuzzy entropy proposed by Kosko [5], namely: (2.17) We notice that the neutrosophic entropy is a strictly de-

Vasile Pătraşcu, The Neutrosophic Entropy and its Five Components
The neutrosophic entropy verify the following conditions:

Penta-valued Representation of Neutrosophic Entropy
The five components of neutrosophic entropy will be: ambiguity a , ignorance u , contradiction c , neutrality n and saturation s [6].We construct formulas for these features both for the variant defined by formula (2.4) and for the variant defined by formula (2.12).For each decomposition, among the four components , always two of them will be zero.In the neutrosophic cube, we consider the entropic rectangle defined by the points: In addition, we consider the point    Also, this point is located midway between the point F (false) and the point T (true).In other words, the point A (ambiguous) represents the center of the Belnap square TUFC (true-uknown-false-contradictory).
If the projection of the point ) , , (

  
on the rectangle UNSC (unknown-neutral-saturated-contradictory) is in- side the triangle UNA (unknown-neutral-ambiguous) then saturation and contradiction will be zero ) 0 (   c s , if the projection is inside the triangle ANS (ambigouous- neutral-saturated) then ignorance and contradiction will be zero ) 0 (   c u and if the projection is inside the triangle ASC (ambiguous-saturated-contradictory) then ignorance and neutrality will be zero ) 0 (   n u .We consider first the first version defined by formula (2.4), namely: The five features must verify the following condition: First, we start with ambiguity formula defined by: The prototype for the ambiguity is the point Then, the other four features will verify the condition: We analyze three cases depending on the order relation among the three parameters    , , where  is the bifuzzy ignorance and  is the bifuzzy contradiction [9].It is obvious that: It results for ignorance and contradiction these formulas: and for saturation and neutrality: Vasile Pătraşcu, The Neutrosophic Entropy and its Five Components It results for ignorance and contradiction these formulas: then for neutrality and saturation it results: It results for ignorance and contradiction these values: Next we obtain: The sum (3.19) can be split in the following manner: Combining formulas previously obtained, it results the final formulas for the five components of the neutrosophic entropy defined by (3.1): The prototype for ignorance is the point ) 0 , 0 , 0 ( and formula (3.23) defines the similarity between the points ) , , ( The prototype for contradiction is the point ) (3.24) defines the similarity between the points ) , , ( The prototype for neutraliy is the point ) 0 , 1 , 0 ( and formula (3.25) defines the similarity between the points ) , , ( The prototype for saturation is the point ) mula (3.26) defines the similarity between the points ) , , ( In addition we also define: The index of truth The prototype for the truth is the point ) .27) defines the similarity between the points ) , , (

The index of falsity
The prototype for the falsity is the point ) 1 , 0 , 0 ( and formula (3.28) defines the similarity between the points ) , , ( Thus, we get the following hepta-valued partition for the neutrosophic information: Vasile Pătraşcu, The Neutrosophic Entropy and its Five Components Formula ( 3.29 ) shows that neutrosophic information can be structured so that it is related to a logic where the information could be: true, false, ambiguous, unknown, contradictory, neutral or saturated [6] (see Figure 2).Next, we also deduce the five components for the variant defined by the formula (2.12).
Using (1.5) formula (2.12) becomes: In this case, the five formulas are obtained from formulas (3.22) -(3.26) by changing the denominator, thus: Also in this case, the 7 parameters verify the condition of partition of unity defined by formula (3.29 ).
For bifuzzy information when 0   , neutrality and saturation are zero and formula (3.29) becomes: Thus, we obtained for bifuzzy information a penta-valued representation.We can conclude that the bifuzzy information is related to a penta-valued logic where the information could be: true, false, ambiguous, unknown and contradictory [6] (see Figure 3).Thus, we obtained for intuitionistic fuzzy information a tetra-valued representation.We can conclude that the intuitionistic fuzzy information is related to a tetra-valued logic where the information could be: true, false, ambiguous and unknown [6] (see Figure 4).The paraconsistent fuzzy information is related to a tetravalued logic where the information could be: true, false, ambiguous and contradictory [6] (see Figure 5).Thus, we obtained for fuzzy information a three-valued representation.We can conclude that the fuzzy information is related to a three-valued logic where the information could be: true, false and ambiguous [6] (see Figure 6).

Conclusion
In this article, there are constructed two formulas for neutrosophic entropy calculating.For each of these, five components are defined and they are related to the following features of neutrosophic uncertainty: ambiguity, ignorance, contradiction, neutrality and saturation.Also, two components are built for neutrosophic certainty: truth and falsity.All seven components, five of the uncertainty and two of certainty form a partition of unity.Building these seven components, primary neutrosophic information is transformed in a more nuanced representation .
midway between the point U (unknown) and the point C (contradiction).

Figure 2 .
Figure 2. The structure of the neutrosophic information.

Figure 3 .For
Figure 3.The structure of the bifuzzy information.

Figure 4 .For
Figure 4.The structure of the intuitionistic fuzzy information.

Figure 5 .For
Figure 5.The structure of the paraconsistent fuzzy information.

Figure 6 .
Figure 6.The structure of the fuzzy information.