Neutrosophic Sets and Systems, Vol. 7, 2015 40 Vasile Ptracu, The Neutrosophic Entropy and its Five Components The Neutrosophic Entropy and its Five Components Vasile Ptracu Tarom Information Technology, 224F Bucurestilor Road, Otopeni, 075150, Romania. E-mail: patrascu.v@gmail.com Abstract.Thispaperpresentstwovariantsofpenta-valuedrepresentationforneutrosophicentropy.Thefirst isanextensionofKaufman'sformulaandthesecondis an extension of Kosko's formula. Basedontheprimarythree-valuedinformationrepre-sented by the degree of truth, degree of falsity and degree ofneutralitytherearebuiltsomepenta-valuedrepresen-tationsthatbetterhighlightssomespecificfeaturesof neutrosophic entropy. Thus, we highlight five features of neutrosophicuncertaintysuchasambiguity,ignorance, contradiction,neutralityandsaturation.Thesefivefea-tures are supplemented until a seven partition of unity by addingtwofeaturesofneutrosophiccertaintysuchas truth and falsity. The paper also presents the particular forms of neutroso-phicentropyobtainedinthecaseofbifuzzyrepresenta-tions,intuitionisticfuzzyrepresentations,paraconsistent fuzzy representationsand finally the case offuzzy rep-resentations. Keywords: Neutrosophic information,neutrosophic entropy, neutrosophic uncertainty, ambiguity, contradiction, neutrality, igno-rance, saturation.1 Introduction Neutrosophicrepresentationofinformationwasproposed bySmarandache[10],[11],[12]asanextensionoffuzzy representationproposedbyZadeh[16]andintuitionistic fuzzyrepresentationproposedbyAtanassov[1],[2].Pri-mary neutrosophic information is defined by three parame-ters:degreeoftruth ,degreeoffalsityvanddegreeof neutralitye . Fuzzyrepresentationisdescribedbyasingleparameter, degreeoftruth ,whilethedegreeoffalsity vhasade-fault value calculated by negation formula : v =1(1.1) andthedegreeofneutralityhasadefaultvaluethatis 0 = e . Fuzzyintuitionisticrepresentationisdescribedbytwoex-plicitparameters,degreeoftruthanddegreeoffalsity v , while the degree of neutrality has a default value that is0 = e . Atanassovconsideredtheincompletevarianttakinginto account that1 s +v . This alloweddefiningtheindexof ignorance (incompleteness) with the formula: v t =1 (1.2) thus obtaining a consistent representation of information, because the sum of the three parameters is1, namely: 1 = + + v t (1.3) Hence, we get for neutrality the value0 = e . Forparaconsistentfuzzyinformationwhere1 > +v ,the index of contradiction can be defined: 1 + = v k (1.4) and for neutrality it results:0 = e . Forbifuzzyinformationthatisdefinedbythepair) , ( v , thenettruth,theindexofignorance(incompleteness),in-dex of contradiction and index of ambiguity can be defined, by: v t =(1.5) ) 1 , min( 1 v t + =(1.6) 1 ) 1 , max( + = v k (1.7) | 1 | | | 1 + = v v o(1.8) Among of these information parameters the most important istheconstructionofsomemeasuresforinformation entropy or information uncertainty. This paper is dedicated to the construction of neutrosophic entropy formulae. Inthenext,section2presentstheconstructionoftwo variantsoftheneutrosophicentropy.Thisconstructionis basedontwosimilarityformulae;Section3presentsa penta-valuedrepresentationofneutrosophicentropybased Neutrosophic Sets and Systems, Vol. 7, 201541 Vasile Ptracu, The Neutrosophic Entropy and its Five Componentsonambiguity,ignorance,contradiction,neutralityand saturation; Section 4 outlines some conclusions. 2 The Neutrosophic Entropy For neutrosophic entropy, we will trace the Kosko idea for fuzzinesscalculation[5].Koskoproposedtomeasurethis informationfeaturebyasimilarityfunctionbetweenthe distance to the nearest crisp element and the distance to the farthestcrispelement.Forneutrosophicinformationthe twocrispelementsare) 0 , 0 , 1 (and) 1 , 0 , 0 ( .Weconsider thefollowingvector:) , 1 , ( e v v + = V .For ) 0 , 0 , 1 ( and) 1 , 0 , 0 ( itresults:) 0 , 0 , 1 ( =TVand ) 0 , 0 , 1 ( =FV . We will compute the distances: e v v + + + = | 1 | | 1 | ) , (TV V D (2.1)e v v + + + + = | 1 | | 1 | ) , (FV V D(2.2) The neutrosophic entropywill be defined by thesimilarity between these two distances. Using the Czekanowskyi formula[3] it results the similar-ity CSand the neutrosophic entropy CE :

) , ( ) , (| ) , ( ) , ( |1F TF TCV V D V V DV V D V V DS+ = (2.3) e v v + + + =| 1 | 1| |1CE (2.4) or in terms ofk t e t , , , : e k t t+ + + =1| |1CE (2.5) Theneutrosophicentropydefinedby(2.4)canbeparticu-larized for the following cases: For0 = e , it result the bifuzzy entropy, namely: | 1 | 1| |1 + + =v v CE (2.6) For0 = eand1 s +v itresultstheintuitionisticfuzzy entropy proposed by Patrascu [8], namely: tv + =1| |1CE (2.7) For0 = eand1 > +v it results the paraconsistent fuzzy entropy,namely: kv + =1| |1CE (2.8) For1 = +v and0 = e , it resultsthefuzzy entropy pro-posed by Kaufman [4], namely: | 1 2 | 1 = CE (2.9) Using the Ruzicka formula [3] it result the formulae for the similarity RSand the neutrosophic entropy RE : ( ) ) , ( ), , ( max| ) , ( ) , ( |1F TF TRV V D V V DV V D V V DS =(2.10) e v v v + + + + =| 1 | | | 1| | 21RE(2.11) or its equivalent form: e v v e v v + + + ++ + + =| 1 | | | 1| 1 | | | 1RE (2.12) or in terms ofk t e t , , , : e k t tt+ + + + =| | 1| | 21RE (2.13) The neutrosophic entropy defined by (2.12) can be particu-larized for the following cases: For0 = e ,itresultsthebifuzzyentropyproposedby Patrascu [7], namely: | 1 | | | 1| 1 | | | 1 + + + + + =v v v v RE (2.14) For0 = eand1 s +v itresultstheintuitionisticfuzzy entropyproposedbySzmidtandKacprzyk[14],[15],ex-plicitly:t v t v + ++ =| | 1| | 1RE (2.15) For0 = eand1 > +v it results the paraconsistent fuzzy entropy, explicitly : k v k v + ++ =| | 1| | 1RE (2.16) For1 = +v and0 = e ,itresultsthefuzzyentropypro-posed by Kosko [5], namely:| 1 2 | 1| 1 2 | 1 + =RE (2.17) Wenoticethattheneutrosophicentropyisastrictlyde-creasingfunctionin| | v andnon-decreasingineand in| 1 | +v . 42Neutrosophic Sets and Systems, Vol. 7, 2015 Vasile Ptracu, The Neutrosophic Entropy and its Five ComponentsThe neutrosophic entropy verify the following conditions: (i)0 ) , , ( = v e Eif) , , ( v e isacrispvalue, namely if{ } ) 1 , 0 , 0 ( ), 0 , 0 , 1 ( ) , , ( e v e .(ii)1 ) , , ( = v e Eifv = . (iii)) , , ( ) , , ( e v v e E E =(iv)) , , ( ) , , (2 1v e v e E E sif 2 1e e s . (v)) , , , ( ) , , , (2 1k t e t k t e t E E < if| | | |2 1t t > . (vi)) , , , ( ) , , , (2 1k t e t k t e t E E sif 2 1e e s . (vii)) , , , ( ) , , , (2 1k t e t k t e t E E sif 2 1t t s .(viii)) , , , ( ) , , , (2 1k t e t k t e t E E s if 2 1k k s . 3Penta-valuedRepresentationofNeutrosophic Entropy The five components of neutrosophic entropywill be: am-biguitya ,ignoranceu ,contradictionc ,neutralitynand saturations[6].Weconstructformulasforthesefeatures bothforthevariantdefinedbyformula(2.4)andforthe variantdefinedbyformula(2.12). Foreachdecomposi-tion,amongthefourcomponentsc s n u , , , , alwaystwoof them will be zero. Intheneutrosophiccube,weconsidertheentropicrectan-gledefinedbythepoints:) 0 , 0 , 0 ( = U , ) 0 , 1 , 0 ( = N , ) 1 , 1 , 1 ( = S , ) 1 , 0 , 1 ( = C . Inaddition,weconsiderthepoint( ) 5 . 0 , 0 , 5 . 0 = AwhichislocatedmidwaybetweenthepointU(unknown)andthe pointC (contradiction).Figure 1. The neutrosophic cube TUFCTNFS and its entropic rectangle UNSC. Also,thispointislocatedmidwaybetweenthepointF(false) and the pointT(true). In other words, the pointA (ambiguous)representsthecenteroftheBelnapsquare TUFC (true-uknown-false-contradictory). Iftheprojectionofthepoint) , , ( v e ontherectangle UNSC(unknown-neutral-saturated-contradictory)isin-sidethetriangleUNA(unknown-neutral-ambiguous)then saturation and contradiction will be zero) 0 ( = = c s , if the projectionisinsidethetriangleANS(ambigouous-neutral-saturated) then ignorance and contradiction will be zero) 0 ( = = c uandiftheprojectionisinsidethetriangle ASC (ambiguous-saturated-contradictory)thenignorance and neutrality will be zero) 0 ( = = n u .We consider first the first version defined by formula (2.4), namely: e v v + + + =| 1 | 1| |1CE (3.1) The five features must verify the following condition:

CE s n c u a = + + + +(3.2) First, we start with ambiguity formula defined by:

e v v v + + + + =| 1 | 1| 1 | | | 1a(3.3) The prototype for the ambiguity is the point) 5 . 0 , 0 , 5 . 0 (and theformula(3.3) definesthesimilarity between thepoints ) , , ( v e and) 5 . 0 , 0 , 5 . 0 ( . Then, the other four features will verify the condition:

e v e v + + ++ += + + +| 1 | 1| 1 | 2s n c u(3.4) Weanalyzethreecasesdependingontheorderrelation amongthethreeparametersk t e , ,wheretisthe bifuzzy ignorance andkis the bifuzzy contradiction [9].It is obvious that: 0 = k t(3.5) Case (I) 0 = > > k e t(3.6) It results for ignorance and contradiction these formulas: k t ee t+ + +=12 2u(3.7) 0 = c(3.8) and for saturation and neutrality: 0 = s(3.9) Neutrosophic Sets and Systems, Vol. 7, 201543 Vasile Ptracu, The Neutrosophic Entropy and its Five Components

k t e e+ + +=13n(3.10) Case (II) 0 = > > t e k(3.11) It results for ignorance and contradiction these formulas: 0 = u(3.12) k t ee k+ + +=12 2c(3.13) then for neutrality and saturation it results: 0 = n(3.14)

k t e e+ + +=13s(3.15) Case (III) ) , max( k t e >(3.16) It results for ignorance and contradiction these values: 0 = u(3.17) 0 = c(3.18) Next we obtain: k t ek t k t e+ + ++ + = +13 3 ) (n s(3.19) The sum (3.19) can be split in the following manner:

k t etk t e+ + ++ =132n(3.20) k t ekk t e+ + ++ =132s(3.21) Combining formulas previously obtained, it results thefi-nalformulasforthefivecomponentsoftheneutrosophic entropy defined by (3.1): ambiguity

k t ek t v + + + =1| | 1a(3.22) ignorance

k t ee e t+ + + =1) , max(2 u (3.23) Theprototypeforignoranceisthepoint) 0 , 0 , 0 (andfor-mula(3.23)definesthesimilaritybetweenthepoints ) , , ( v e and) 0 , 0 , 0 ( . contradiction k t ee e k+ + + =1) , max(2 c(3.24)Theprototypeforcontradictionisthepoint) 1 , 0 , 1 (and formula(3.24)definesthesimilaritybetweenthepoints ) , , ( v e and) 1 , 0 , 1 ( . neutrality

k t et ek t e+ + ++ =1) , min( 32) 0 , max(n (3.25) The prototype for neutraliy is the point) 0 , 1 , 0 (and formula (3.25)definesthesimilaritybetweenthepoints) , , ( v e and) 0 , 1 , 0 ( . saturation k t ek ek t e+ + ++ =1) , min( 32) 0 , max(s (3.26) Theprototypeforsaturationisthepoint) 1 , 1 , 1 (andfor-mula(3.26)definesthesimilaritybetweenthepoints ) , , ( v e and) 1 , 1 , 1 ( . In addition we also define: The index of truth k t ev v + + +=1) , max(t(3.27) The prototype for the truth is the point) 0 , 0 , 1 (and formula (3.27)definesthesimilaritybetweenthepoints) , , ( v e and) 0 , 0 , 1 ( . The index of falsity k t e v+ + +=1) , max(f (3.28) Theprototypeforthefalsityisthepoint) 1 , 0 , 0 (andfor-mula(3.28)definesthesimilaritybetweenthepoints ) , , ( v e and) 1 , 0 , 0 ( . Thus,wegetthefollowinghepta-valuedpartitionforthe neutrosophic information: 1 = + + + + + + s n c u a f t (3.29) 44Neutrosophic Sets and Systems, Vol. 7, 2015 Vasile Ptracu, The Neutrosophic Entropy and its Five ComponentsFormula ( 3.29 ) shows that neutrosophic information can be structured so that it is related to a logic where the in-formation could be: true, false, ambiguous, unknown, con-tradictory, neutral or saturated [6] (see Figure 2). Figure 2. The structure of the neutrosophic information. Next, we also deduce the five components for the variant defined by the formula (2.12). Using (1.5)formula (2.12) becomes: e k t tt+ + + + =| | 1| | 21RE (3.30) or e k t te k t t+ + + ++ + + =| | 1| | 1RE(3.31) In this case, the five formulas are obtained from formulas (3.22) - (3.26) by changing the denominator, thus: ambiguity

k t e tk t t+ + + + =| | 1| | 1a(3.32) ignorance

k t e te t+ + + +=| | 1) 0 , max( 2u (3.33) contradiction

k t e te k+ + + +=| | 1) 0 , max( 2c (3.34) neutrality

k t e tt ek t e+ + + ++ =| | 1) , min( 32) 0 , max(n (3.35) saturation k t e tk ek t e+ + + ++ =| | 1) , min( 32) 0 , max(s (3.36) From (3.27) and (3.28) it results: Index of truth k t e tt+ + + +=| | 1) 0 , max( 2t (3.37) Index of falsity k t e tt+ + + +=| | 1) 0 , max( 2f(3.38) Also in this case, the 7 parameters verify the condition of partition of unity defined by formula (3.29 ). For bifuzzy information when0 = e , neutrality and satu-ration are zero and formula (3.29) becomes: 1 = + + + + c u a f t (3.39) Thus, we obtained for bifuzzy information a penta-valued representation. We can conclude that the bifuzzy infor-mation is related to a penta-valued logic where the infor-mation could be: true, false, ambiguous, unknown and con-tradictory [6] (see Figure 3). Figure 3. The structure of the bifuzzy information. Neutrosophic Sets and Systems, Vol. 7, 201545 Vasile Ptracu, The Neutrosophic Entropy and its Five ComponentsForintuitionisticfuzzyinformationwhen0 = eand 1 s +v ,neutrality,saturationandcontradictionarezero and formula (3.29) becomes: 1 = + + + u a f t (3.40) Thus, we obtained for intuitionistic fuzzy information a tet-ra-valuedrepresentation.Wecanconcludethattheintui-tionistic fuzzy information is related to a tetra-valued logic where the information could be: true, false, ambiguous and unknown [6] (see Figure 4). Figure 4. The structure of the intuitionistic fuzzy information. For paraconsistent fuzzy information when0 = eand 1 > +v , neutrality, saturation and ignorance are zero and formula (3.29) becomes: 1 = + + + c a f t (3.41) The paraconsistent fuzzy information is related to a tetra-valued logic where the information could be: true, false, ambiguous and contradictory [6] (see Figure 5). Figure 5. The structure of the paraconsistent fuzzy information. For fuzzy information when0 = eand1 = +v , neu-trality, saturation ignorance and contradiction are zero and formula (3.29) becomes: 1 = + + a f t (3.42) 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). Figure 6. The structure of the fuzzy information. 4Conclusion Inthisarticle,thereareconstructedtwoformulasfor neutrosophicentropycalculating.Foreachofthese,five components are defined and they are related to the follow-ingfeaturesofneutrosophicuncertainty:ambiguity,igno-rance,contradiction,neutralityandsaturation.Also,two componentsarebuiltforneutrosophiccertainty:truthand falsity.Allsevencomponents,fiveoftheuncertaintyand twoofcertaintyformapartitionofunity.Buildingthese sevencomponents,primaryneutrosophicinformationis transformed in a more nuanced representation . References [1]K. Atanassov. 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Received: November 24, 2014. Accepted: January 2, 2015