NeutrosophicComputabilityandEnumeration
AhmetÇevik 1 ,SelçukTopal 2,* andFlorentinSmarandache 3
1 GendarmerieandCoastGuardAcademy,DepartmentofScience,06805Ankara,Turkey, a.cevik@hotmail.com
2 DepartmentofMathematics,FacultyofScienceandArts,BitlisErenUniversity,13000Bitlis,Turkey
3 DepartmentofMathematics,UniversityofNewMexico,Gallup,NM87301,USA;smarand@unm.edu
* Correspondence:s.topal@beu.edu.tr;Tel.:+90-532-709-0239
Received:28October2018;Accepted:13November2018;Published:16November2018
Abstract: WeintroduceoracleTuringmachineswithneutrosophicvaluesallowedintheoracle informationandthengivesomeresultswhenoneispermittedtouseneutrosophicsetsandlogic inrelativecomputation.Wealsointroduceamethodtoenumeratetheelementsofaneutrosophic subsetofnaturalnumbers.
Keywords: computability;oracleTuringmachines;neutrosophicsets;neutrosophiclogic;recursive enumerability;oraclecomputation;criterionfunctions
1.Introduction
Inclassicalcomputabilitytheory,algorithmiccomputationismodeledbyTuringmachines,which wereintroducedbyAlanM.Turing[1].A Turingmachine isanabstractmodelofcomputationdefined bya7-tuple (Q, Σ, Γ, δ, q0, F, {L, R}),where Q isafinitesetofstates, Σ isthealphabet, Γ isthetape alphabet, q0 ∈ Q isthestartingstate, F ⊂ Q isasetofhaltingstates,theset {L, R} denotesthepossible left(L)andright(R)moveofthetapehead,and δ isthetransitionfunction,definedas:
Eachtransitionisastepofthecomputation.Let w beastringoverthealphabet Σ.Wesay thataTuringmachineoninput w halts ifthecomputationendswithsomestate q ∈ F.Theoutput ofthemachine,inthiscase,iswhateverwaswrittenonthetapeattheendofthecomputation. IfaTuringmachine M oninput w halts,thenwesaythat M isdefinedon w.Sincethereisa one-to-onecorrespondencebetweenthesetofallfinitestringsover Σ andthesetofnaturalnumbers N = {0,1,2, },withoutlossofgeneralitywemayassumethatTuringmachinesaredefinedfrom N to N
StandardTuringmachinesadmitpartialfunctions,i.e.,functionsthatmaynotbedefinedonevery input.TheclassoffunctionscomputablebyTuringmachinesarecalled partialrecursive(computable) functions.Weshallnotdelveintothedetailsaboutwhatismeantbyafunctionorsetthatiscomputable byaTuringmachine.Weassumethatthereaderisfamiliarwiththebasicterminology.However,fora detailedaccount,thereadermayrefertoReference[2–4].Usingawellknownmethodcalled Gödel numbering,originatedfromGödel’scelebrated1931paper[5],itispossibletohaveanalgorithmic enumerationofallpartialrecursivefunctions.Welet Ψi denotethe ithpartialrecursivefunction,i.e., the ithTuringmachine.
Ifapartialrecursivefunctionisdefinedoneveryargumentwesaythatitis total.Totalrecursive functionsaresimplycalled recursive or computable.SincetherearecountableinfinitelymanyTuring machines,therearecountableinfinitelymanycomputablefunctions.Computablesetsandfunctionsare widelyusedinmathematicsandcomputerscience.However,nearlyallfunctionsarenon-computable.
Symmetry 2018, 10,643;doi:10.3390/sym10110643 www.mdpi.com/journal/symmetry
Sincethereare 2ℵ0 functionsfrom N to N andonly ℵ0 manycomputablefunctions,thereare uncountablymanynon-computablefunctions.
OracleTuringmachines,introducedbyAlanTuring[6],areusedforrelativizingthecomputation withrespecttoagivensetofnaturalnumbers.An oracleTuringmachine isaTuringmachinewithan extra oracletape containingthecharacteristicfunctionofagivensetofnaturalnumbers.The characteristic function ofaset S ⊂ N isdefinedas:
χS (x)= 1if x ∈ S 0if x ∈ S.
Wemayalsothinkof χS asaninfinitebinarysequenceandcallitthe characteristicsequence of S Foraset S ⊂ N,welet S(i) denote χS (i).Sothecharacteristicsequenceofaset S simplygivesthe membershipinformationaboutnaturalnumbersregarding S.ForagivenanoracleTuringmachine withthecharacteristicsequenceofaset S providedintheoracletape,functionsaredenotedas computablebythemachinerelativetothe oracle S.IftheoracleTuringmachinewithanoracle S computesafunction f ,thenwesaythat f is computablein S orwesay S computes f .Wedenotethe ithoracleTuringmachinewithanoracle A by Ψi (A).Then,itmakessensetowrite Ψi (A)= B if A computes B.
Nowweshalllookatanon-standardTuringmachinemodelbasedonneutrosophicsets. Neutrosophiclogic,firstintroducedbySmarandache[7,8],isageneralizationofclassical,fuzzyand intuitionisticfuzzylogic.Thekeyassumptionofneutrosophyisthateveryideanotonlyhasacertain degreeoftruth,asisgenerallytakeninmany-valuedlogiccontexts,butalsohasdegreesoffalsity andindeterminacy,whichneedtobeconsideredindependentlyfromeachother.Aneutrosophicset reliesontheideathatthereisadegreeofprobabilitythatanelementisamemberofthegivenset, adegreethattheverysameelementis not amemberoftheset,andadegreethatthemembership oftheelementisindeterminatefortheset.Forourpurposewetakesubsetsofnaturalnumbers. Roughlyspeaking,if n wereanaturalnumberandif A wereaneutrosophicset,thentherewouldbea probabilitydistribution p∈(n)+ p∈(n)+ pI (n)= 1,where p∈(n) denotestheprobabilityof n being amemberof A, p∈(n) denotestheprobabilityof n notbeingamemberof A,and pI (n) denotesthe degreeofprobabilitythatthemembershipof n isindeterminatein A.Sincetheprobabilitydistribution isexpectedtobenormalized,thesummationofallprobabilitiesmustbeequaltounity.Weshould notehoweverthatthelatterrequirementcanbemodifieddependingontheapplication.
Theaboveinterpretationofaneutrosophicsetcanbeinfactgeneralizedtoanymulti-dimensional collectionofattributes.Thatis,ourattributesdidnotneedtobemerelyaboutmembership, non-membership,andindeterminacy,butitcouldrangeoveranyfinitesetofattributes a0, a1, , ak and b0, b1, ... bk sothatthevalueofanelementwouldrangeover (x, y, I) suchthat x ∈ am and y ∈ bm for 0 ≤ m ≤ k.Thesetofattributescanalsobecountablyinfiniteorevenuncountable. However,wearenotconcernedwiththesecases.Weshallonlyconsiderthemembershipattribute discussedabove.
Weareparticularlyinterestedinsubsetsofnaturalnumbers A ⊂ N,inourstudy.Anyneutrosophic subset A ofnaturalnumbers(weshalloccasionallydenotesuchasetby AN )isdefinedintheformof orderedtriplets: { p∈(0), p∈(0), pI (0) , p∈(1), p∈(1), pI (1) , p∈(2), p∈(2), pI (2) ,...}, where,foreach i ∈ N, p∈(i) denotesthedegreeofprobabilityof i beinganelementof A, p∈(i) denotes theprobabilityof i beingnotanelementof A,and pI (i) denotestheprobabilityof i beingindetermined. Sinceweassumeanormalizedprobabilitydistribution,wehavethatforevery i ∈ N: p∈(i)+ p∈(i)+ pI (i)= 1.
2.OracleTuringMachineswithNeutrosophicValues
Nowwecanextendthenotionofrelativizedcomputationbasedonneutrosophicsetsand neutrosophiclogic.ForthisweintroduceoracleTuringmachineswithneutrosophicoracletape. Thegeneralideaisasfollows.Standardoracletapecontainstheinformationofthecharacteristic sequenceofagivenset A ⊂ N.Weextendthedefinitionofthecharacteristicfunctiontoneutrosophic setsasfollows.
Definition1. Let A ⊂ N beaset.Aneutrosophicoracletapeisacountablyinfinitesequence t0, t1, where ti = a, b, c isanorderedtripletand a, b, c ∈ Q,sothat a istheprobabilityvalueof i suchthat i ∈ A, b isthe probabilityvalueofisuchthati ∈ A,andcistheprobabilityofibeingindeterminateforA.
TheoverallpictureofaneutrosophicoracletapecanbeseeninFigure 1.Nowweneedtomodify thenotionofthecharacteristicsequenceaccordingly.
Figure1. Neutrosophicoracletape.
Definition2. Let S ⊂ N beasetandlet B denotetheblanksymbolinthealphabetoftheoracletape. TheneutrosophiccharacteristicfunctionofSisdefinedby
1,0, I ifp∈(x) > 0 andp∈(x) > 0 andpI (x) > 0 B,0, I ifp∈(x)= 0 andp∈(x) > 0 andpI (x) > 0 B, B, I ifp∈(x)= 0 andp∈(x)= 0 andpI (x) > 0 B, B, B ifp∈(x)= 0 andp∈(x)= 0 andpI (x)= 0 1, B, I ifp∈(x) > 0 andp∈(x)= 0 andpI (x) > 0 1, B, B ifp∈(x) > 0 andp∈(x)= 0 andpI (x)= 0 1,0, B ifp∈(x) > 0 andp∈(x) > 0 andpI (x)= 0 B,0, B ifp∈(x)= 0 andp∈(x) > 0 andpI (x)= 0
Theideabehindthisdefinitionistolabelthedistributionswhichhavesignificantprobability valuewithrespecttoapre-determinedprobabilitythresholdvalue,inthiscaseweassumethisvalue tobe 0 bydefault.Notethatthisthresholdvaluecouldbedefinedforany r ∈ Q sothatinsteadof beinggreaterthan 0,wewouldrequiretheprobabilityforthatattributetobegreaterthan r inorderto belabelled.Wewilltalkaboutthepropertiesofdefininganarbitrarythresholdvalueanditsrelation toneutrosophiccomputationsinthenextsection.
Definition3. AneutrosophicoracleTuringmachineisaTuringmachinewithanadditionalneutrosophic oracletape (Q, Σ, Γ, Γ , δ, q0, F, {L, R}),where Q isafinitesetofstates, Σ isthealphabet, Γ isthetapealphabet, Γ istheneutrosophicoracletapealphabetcontainingtheblanksymbol B, q0 ∈ Q isthestartingstate, F ⊂ Q is asetofhaltingstates,theset {L, R} denotesthepossibleleft(L)andright(R)moveofthetapehead,and δ isthe transitionfunctiondefinedas:
δ : Q × Γ × Γ → Q × Σ ×{L, R}2
Theorem1. AnyneutrosophicoracleTuringmachinecanbesimulatedbyastandardTuringmachine.
Proof. AssumingtheChurch–Turingthesisintheproof,weonlyneedtoarguethatstandardoracle tapescanintheoryrepresentneutrosophicoracletapes.Infactanyneutrosophicoracletapescanbe representedbythreestandardoracletapeseachofwhichcontainsoneandonlyoneattribute.The ith cellofthefirstoracletapecontainstheprobabilityvalue p∈(i).The ithcellofthesecondoracletape containsthevalue p∈(i).Similarly,the ithcellofthethirdoracletapecontains pI (i)
Wealsoneedtoarguethatathree-tapeoraclestandardTuringmachinecanbesimulatedbya singletapeoracleTuringmachine.Let Γ betheoraclealphabet.Wedefineanextension Γ of Γ by introducingadelimetersymbol # toseparateeachattributeforagivennumber i.Wedefineanother delimitersymbol ⊥ toseparateeach i ∈ N.Let Γ = Γ ∪{#, ⊥}.Then,aneutrosophicoracletapecan berepresentedbyasingleoracletapewiththetapealphabet Γ .Theoracletapewillbeintheform: p∈(0)#p∈(0)#pI (0)⊥p∈(1)#p∈(1)#pI (1)⊥ ...
Thesymbol ⊥ determinesacounterfor i,whereasforeach i,thesymbol # determinesacounter fortheattribute.
Aneutrosophicset A computes anotherneutrosophicset B ifusingfinitelymanypiecesof informationofthecharacteristicsequenceof A determinesthe ithentryofthecharacteristicsequence of B givenanyindex i ∈ N.Then,basedonthisdefinition,aset B ⊂ N is neutrosophicallycomputable in A if B = ΨN e (A) forsome e ∈ N,where ΨN e denotesthe e-thneutrosophicoracleTuringmachine. If B = ΨN e (A) forsome e ∈ N,wedenotethisby B ≤N A.If B ≤N A and A ≤N B,thenwesaythat A and B are neutrosophicallyequivalent anddenotethisby A ≡N B.Intuitively, A ≡N B meansthat A and B areneutrosophicsubsetsofnaturalnumbers,andtheyhavethesamelevelofneutrosophic informationcomplexity.Weleavethediscussiononthepropertiesoftheequivalenceclassesinduced by ≡N foranotherstudyasitisbeyondthescopeofthispaper.
3.NeutrosophicEnumerationandCriterionFunctions
Wenowintroducetheconceptofneutrosophicenumerationofthemembersofneutrosophic subsetsofnaturalnumbers.Sincewetalkaboutenumeration,wemustonlytakecountablesets intoconsideration.Itisknownfromclassicalcomputabilitythat,givenaset A ⊂ N, A iscalled recursivelyenumerable ifthereexistssome e ∈ N suchthat A isthedomainof Ψe.Wewanttodefine theneutrosophiccounterpartofthisnotion,butweneedtobecarefulabouttheindeterminatecases, anintrinsicpropertyinneutrosophiclogic.
Definition4. Aset A iscalledneutrosophicTuringenumerableifthereexistssome e ∈ N suchthat A isthe domainof ΨN e restrictedtoelementswhoseprobabilitydegreeofmembershipisgreaterthanagivenprobability threshold.Moreprecisely,if r ∈ Q isagivenprobabilitythreshold,then A isneutrosophicTuringenumerableif A isthedomainof ΨN e (∅) restrictedtothoseelements i suchthat pA ∈ (i) ≥ r,where pA ∈ (i) denotesthedegreeof probabilityofmembershipofiinA.
Ifthe ethTuringmachineisdefinedontheargument i,wedenotethisby Ψe (i) ↓.The haltingset inclassicalcomputabilitytheoryisdefinedas:
K = {e : Ψe (e) ↓}
Itisknownthat K isrecursivelyenumerablebutnotrecursive.UnlikeinclassicalTuring computability,weshowthatneutrosophicallycomputablesetsallowustoneutrosophicallycompute thehaltingset.Thewaytodothisgoesasfollows.Asingleneutrosophicsubsetofnaturalnumbersis notenoughtocomputethehaltingset.Instead,wetaketheunionofallneutrosophicallycomputable subsetsofnaturalnumbersbytakinganinfinitejoinwhichwillcodetheinformationofthehaltingset.
Let {Ai }i∈N beacountablesequenceofsubsetsof N.The infinitejoin isdefinedby {Ai } = { i, x : x ∈ Ai },
where i, j ismappedtoanaturalnumberusingauniformpairingfunction N × N → N. Theorem2. LetAN beaneutrosophicsubsetof N.Then, {AN i }≡N K
Proof. Wefirstshowthat {AN i }≥N K.Theinfinitejoinofallneutrosophicallycomputablesets computesthehaltingset.Let {ΨN i }i∈N beaneffectiveenumerationofneutrosophicTuringfunctionals. Let {AN i } bethecorrespondingneutrosophicsets,eachofwhichiscomputableby ΨN i .Tocompute K,welet {AN i } betheinfinitejoinofall AN i .Toknowwhether Ψi (i) isdefinedornot,wesee if p∈(i)+ p∈(i) > 0.5.Ifso,then Ψi (i) ↓.Otherwiseitmustbethat pI (i) > 0.5.Inthiscase, Ψi (i) isundefined.
Next,weshow {AN i }≤N K.Toprovethis,weassumethatthereexistsanoraclefor K.If i ∈ K, thenthereexistindices x, y ∈ N suchthat x, y = i anditmustbethat p∈(i)+ p∈(i) > 0.5 since Ψi (i) isdefined,butwemaynotknowwhether i ∈ Ai or i ∈ Ai.If i ∈ K,thenthesameargumentholdsto provethiscaseaswell.
Theuseoftheprobabilityratio 0.5 isforconvenience.Thisnotionwillbegeneralizedlateron. Classicallyspeaking,givenasubsetofnaturalnumbers,wecaneasilyconvertittoaneutrosophic setpreservingthemembershipinformationofthegivenclassicalset.Supposethatwearegivena set A ⊂ N andwewanttoconvertittoaneutrosophicsetwiththesamecharacteristicsequence. Theneutrosophiccounterpart AN isdefined,foreach i ∈ N,as:
AN (i)= i,1 i,0 if A(i)= 1 1 i, i,0 otherwise.
Wenowintroducethetreerepresentationofneutrosophicsetsandgiveamethod,usingtrees, toapproximateitsclassicalcounterpart.Supposethatwearegivenaneutrosophicsubsetofnatural numbersintheform:
AN = { p∈(i), p∈(i), pI (i) }i∈N
Weusetheprobabilitydistributiontodecidewhichelementwillbeincludedintheclassical counterpart.If AN isaneutrosophicsubsetofnaturalnumbers,the classicalcounterpartof AN is definedas:
A(i)= 1if p∈(i) > p∈(i) 0if p∈(i) < p∈(i).
Nowweintroduceasimpleconversionusingtrees.Theaimistoapproximatetotheclassical counterpartofagivenneutrosophicset AN inacomputablefashion.Forthiswestartwithafull ternarytree,asgiveninFigure 2,codingallpossiblecombinations.
Figure2. Approximatinganeutrosophicsetwithaclassicalsetthroughaternarytree.
Thecorrectinterpretationofthistreeisasfollows.Eachbranchrepresentapossibleelementofthe setwewanttoconstruct.Forinstance,if p∈(0) hasthelargestprobabilityvalueamong p∈(0), p∈(0), pI (0),thenwechoose p∈(0) anddefine 0 tobeanelementoftheclassicalsetweconstruct.Ifeither p∈(0) or pI (0) isgreaterthan p∈(0),thenweknow 0 isnotanelementoftheconstructedset.Sincewe aredefiningaclassicalset,theonlytimewhensome i ∈ N isintheconstructedsetisif p∈(i) > p∈(i) and p∈(i) > pI (i).Continuingalongthisline,if p∈(1),say,hasthegreatestprobabilityvalueamong p∈(1), p∈(1), pI (1),then 1 willnotbeanelement.Sofar, 0 isanelementand 1 isnotanelement.Soin thetreewechoosetheleftmostbranchandthennextwechoosethemiddlebranch.Repeatingthis procedureforevery i ∈ N,weendupdefiningacomputableinfinitepathonthisternarytreewhich defineselementsofthesetbeingconstructed.Ateachstep,wesimplytakethemaximumprobability valueandselectthatattribute.Theinfinitepathdefinesacomputableapproximationtotheclassical counterpartof AN usingthetreemethod.
EarlierwedefinedTuringmachineswithaneutrosophicoracletape.Supposethatthecharacteristic sequenceofaneutrosophicset A canbeconsideredasanoracle.Then,the e-thneutrosophicoracle Turingmachinecancomputeafunctionofthesamecharacteristic.Thatis,notonlycanTuringmachines withneutrosophicoraclescomputeclassicalsets,buttheycanalsocomputeneutrosophicsets.Itis importanttonotethatweneedtomodifythedefinitionofstandardoracleTuringmachinesinorderto useneutrosophicsets.Weaddthesymbol I tothealphabetoftheoracletape.Thetransitionfunction δN isthendefinedas:
WesaythattheneutrosophicoracleTuringmachine,say ΨN e , computes aneutrosophicset B if ΨN e = B.
Wenowturntotheproblemofenumeratingmembersofaneutrosophicsubset A ofnatural numbers.Normally,generalintuitionsuggeststhatwepickelements i ∈ N suchthat p∈(i) > 0.5.Itis importanttonotethat,given A = { p∈(i), p∈(i), pI (i) }i∈N,notevery i willbeenumeratedifweuse thisprobabilitycriterion.However,changingthecriteriondependingonwhataspectofthesetwe wanttolookatanddependingontheapplication,wouldalsochangetheenumeratedset.Therefore, wewouldneedakindofcriterionfunctiontosetaprobabilitythresholdregardingwhichelementsof theneutrosophicsetaretobeenumerated.
Inpractice,oneoftenencountersasituationwherethegiveninformationisnotdirectlyused butratheranalyzedunderthecriteriondeterminedbyafunction.Weexaminehowthecomputation behaveswhenweimposeafunctionontheneutrosophicoracletape.Thatis,supposethat f : N → {a, b, c} isafunction,where a, b, c ∈ Q,whichmapseachcelloftheneutrosophicoracletapetoa probabilityvalue.Forexample, f couldbedefinedasaconstantnon-membershipfunctionwhich assignseverytripletinthecellstothenon-membership ∈ attribute.Inthiscase,theprobabilityof anynaturalnumbernotbeinganelementoftheconsideredoracle A isjust 1.Whenthesekindsof functionsareusedintheoracleinformationof A,wemaybeabletocomputesomeusefulinformation.
Theintuitioninusingcriterionfunctionsistoselect,underapreviouslydeterminedprobability threshold,anaturalnumberfromtheprobabilitydistributionwhichisavailableinagivenneutrosophic subsetofnaturalnumbers.Asanexample,letusimagineaneutrosophicsubset A ofnaturalnumbers. Supposeforsimplicitythat A isfiniteandisdefinedas:
A = { 0.1,0.4,0.5 , 0.6,0.3,0.1 , 0,0.9,0.1 }
Firstofall,weshouldreadthisasfollows: A hasneutrosophicinformationaboutthefirstthree naturalnumbers 0,1,2.Inthisexample, p∈(0)= 0.1, p∈(0)= 0.4, pI (0)= 0.5.Forthenaturalnumber 1,wehavethat p∈(1)= 0.6, p∈(1)= 0.3, pI (1)= 0.1.Finally,forthenaturalnumber 2,wehave p∈(2)= 0, p∈(2)= 0.9, pI (2)= 0.1.Nowifwewanttoknowwhichnaturalnumbersarein A, normallywewouldonlypickthenumber 2 since p∈(2) > 0.5.Ourcriterionofenumerationinthis caseis 0.5.Ingeneral,thisprobabilityvaluemaynotbealwaysapplicable.Moreover,thisprobability
thresholdvaluemaynotbeconstant.Thatis,wemaywanttohaveadifferentprobabilitythreshold foreverynaturalnumber i.Ifourcriterionweretoselectthe ithelementwhoseprobabilityexceeds pi, wewouldenumeratethosenumbers.Forexample,ifthecriterionisdefinedas
f (0)= 0, f (1)= 0.8, f (2)= 0.2,
thenforthefirsttriple,theprobabilitythresholdis 0,meaningthatweenumeratethenaturalnumber 0 if p∈(0) > 0.Obviously 0 willbeenumeratedinthiscasesince p∈(0)= 0.1 > 0.Theprobability thresholdforenumeratingthenumber 1 is 0.8,soitwillnotbeenumeratedsince p∈(1)= 0.6 < 0.8. Finally,theprobabilitythresholdforenumeratingthenumber 2 is 0.2.Inthiscase, 2 willnotbe enumeratedsince p∈(2)= 0 < 0.2.Sotheenumerationsetfor A underthecriterion f willbe {0}. Wearenowreadytogivetheformaldefinitionofacriterionfunction. Definition5. Acriteriafunctionisamapping f : N → Q which,givenaneutrosophicsubset A ofnatural numbers,determinesaprobabilitythresholdforeachtriple p∈(i), p∈(i), pI (i) inA.
Wefirstnoteasimpleobservationthatifthecriterionfunctionistheconstantfunction f (n)= 0 forany n ∈ N,theenumerationsetwillbeequalto N itself.However,thisdoesnotmeanthatthe enumerationsetwillbeemptyif f (n)= 1.Givenaneutrosophicset A,if p∈(i)= 1 forall i,thenthe enumerationsetfor A willalsobeequalto N.
Weshallnextgivethefollowingtheorem.Firstweremindthereaderthatwecallafunction f strictlydecreasing if f (i + 1) < f (i).
Theorem3. Let f beastrictlydecreasingcriterionfunctionforaneutrosophicset A suchthat p∈(i) < p∈(i + 1) forevery i ∈ N,andlet EA betheenumerationsetfor A underthecriterion f .Then,thereexistssome k ∈ N such that |EA| < k.
Proof. Clearly,given A andthatforeach i, p∈(i) < p∈(i + 1),onlythosenumbers i whichsatisfy p∈(i) > f (i) willbeenumerated.Sincetheprobabilitydistributionofmembershipdegreesofelements of A strictlyincreasesand f isstrictlydecreasing,therewillbesomenumber j ∈ N suchthat p∈(i) ≤ f (j).Moreover,forthesamereason p∈(m) ≤ f (m) forevery m > j.Therefore,thenumberof elementsenumeratedislessthan j.Thatis, |EA | < j
Wedenotethecomplementofaneutrosophicsubsetofnaturalnumbers A by Ac andwedefine itasfollows.Let pA ∈ (i) denotetheprobabilityof i beinganelementof A andlet pA ∈ (i) denotethe probabilityof i beingnotanelementof A.Inaddition, pA I (i) denotestheprobabilityofthemembership of i beingindeterminate.Then: Ac (i)= pA ∈ (i), pA ∈ (i), pA I (i) .
Sothecomplementofaneutrosophicsetinconsiderationisformedbysimplyinterchanging theprobabilitiesofmembershipandnon-membershipforall i ∈ N.Noticethattheprobabilityof indeterminacyremainsthesame.Ournextobservationisasfollows.Supposethat A and Ac are neutrosophicsubsetsofnaturalnumbersand f isacriterionfunction.If EA ⊂EAc ,thenclearly p∈(i) ≥ p∈(i) forall i ∈ N
Theprobabilitydistributionofmembersofaneutrosophicsetcanbealsobegivenbyafunction g(i, j) suchthat i ∈ N and j ∈{1,2,3} where j istheindexfordenotingthe membership probability by 1, non-membership probabilityby 2,and indeterminacy probabilityby 3,respectively.Forinstance, for i ∈ N, g(i,2) denotestheprobabilityofthenon-membershipof i generatedbythefunction g Now g beingacomputablefunctionmeans,forany i, j,thereisanalgorithmtofindthevalueof g(i, j) Wegivethefollowingtheorem.
Theorem4. Let A beaneutrosophicset.If g isacomputablefunction,thenthereexistsacomputablecriterion functionfsuchthat EA istheenumerationsetofAunderthecriterionfunctionfand,moreover, EA = N.
Proof. Supposethatwearegiven A.If g isacomputablefunctionthatgeneratestheprobability distributionofmembersof A,thenwecancomputablyfind g(i,1)= k.Wethensimplylet f (i) be some m ≤ k.Since p∈(i)= k ≥ f (i),every i willbeamemberof EA.Since i isarbitrary, EA = N
Wesaythatafunction f majorizes afunction g if f (x) > g(x) forall x.Supposenowthat g isa quicklygrowingfunctioninthesensethatitmajorizeseverycomputablefunction.Thatis,assume that g(i, j) ≥ f (i) forevery i, j ∈ N andeverycomputablefunction f .Nowinthiscase g isnecessarily non-computable.Otherwisewewouldbeabletoconstructafunction h where h(i) ischosentobe some s > t suchthat g(i, j) isdefinedatstep t.Soif g isnotcomputable,wecannotapplytheprevious theoremon g.Theonlywaytoenumerate A isbyusingrelativecomputabilityratherthangivinga plaincomputableprocedure.Supposethatwearegivensuchafunction g.Let Ψi (A; i) denotethe ithTuringmachinewithoracle A andinput i.Wedefine g = {x : Ψx (g; x) ↓} tobethe jump of g, where x = i, j forauniformpairingfunction N × N → N.Thejumpof g isbasicallythehalting setrelativizedto g.Ifwewanttoenumeratemembersof A,wecanthenuse g asanoracle.Since, bydefinition, g computes g,weenumeratemembersof A computably ing Weshallalsonoteanobservationregardingtherelationshipbetween A and Ac.Givenafunction f ,unless f (i) isstrictlybetween pA ∈ (i) and pA ∈ (i),wehavethat EA = EAc .Thatis,theonlycasewhen EA = EAc isif pA ∈ (i) < f (i) < pA ∈ (i) or pA ∈ (i) < f (i) < pA ∈ (i).Letusexamineeachcase.Inthefirstcase, since f (i) > pA ∈ (i), i willnotbeenumeratedinto EA,butsince pAc ∈ (i) > f (i),itwillbeenumerated into EAc .Thesecondcaseisjusttheopposite.Thatis, i willbeenumeratedinto EA butnotinto EAc
Whataboutthecaseswhere i isenumeratedintobothenumerationsets?Itdependsonhowwe allowourcriteriafunctiontooperateoverprobabilitydistributions.Ifweonlywanttoenumerate thoseelements i suchthat p∈(i) ≥ f (i),thenwemayhaveequalprobabilitydistributionamong membershipandnon-membershipattributes.Wemayhavethat p∈(i)= p∈(i)= 0.5 and pI (i)= 0 Inthiscase,wegettoenumerate i bothinto EA and EAc .However,ifweallowthecriterionfunction tooperateinawaythat i isenumeratedifandonlyif p∈(i) > f (i),thenitmustbethecasethat p∈(i) < f (i) so i willonlybeenumeratedinto EA
Theuseofthecriterionfunctionmayvarydependingontheapplicationandwhichaspectofthe givenneutrosophicsetwewanttoanalyze.
4.Conclusions
WeintroducedtheneutrosophiccounterpartoforacleTuringmachineswithneutrosophicvalues allowedintheoracletape.Forthiswepresentedanewtypeoforacletapewhereeachcellcontainsa tripletofthreeprobabilityvalues,namelyforthemembership,non-membership,andindeterminacy. ThenotionofneutrosophicoracleTuringmachineisinterestinginitsownrightsinceoracleinformation isusedinrelativecomputabilityofsetsandenablesustoinvestigatethecomputabilitytheoretic propertiesofsetsrelativetooneanother.Inthispaper,wealsointroducedamethodtoenumerate theelementsofaneutrosophicsubsetofnaturalnumbers.Forthiswedefinedacriterionfunctionto chooseelementswhichsatisfyacertainprobabilitydegree.Thisdefinesamethodthatcanbeused inmanyapplicationsofneutrosophicsets,particularlyindecisionmakingproblems,solutionspace searching,andmanymore.Weprovedsomeresultsabouttherelationshipbetweentheenumeration setsofagivenneutrosophicsubsetofnaturalnumbersandthecriterionfunction.Afutureworkof thisstudyistoinvestigatethepropertiesofequivalenceclassesinducedbytheoperator ≡N .Wemay callthisequivalenceclass,neutrosophicdegreeofcomputability.Itwouldbeinterestingtostudy therelationshipbetweenneutrosophicdegreesofcomputabilityandclassicalTuringdegrees.The resultsalsoarisefurtherdevelopmentsinachievingofnewgenerationofcomputingmachinessuchas
fuzzycellularnonlinearnetworksparadigmorthememristor-basedcellularnonlinearnetworks[9]. Thelatterofcoursehaspracticalbenefits.
AuthorContributions: Conceptualization,A.Ç.andS.T.;Methodology,A.Ç.;Validation,A.Ç.,S.T.,andF.S.; Investigation,A.Ç.andS.T.;Resources,A.Ç.,S.T.,andF.S.;Writing-originaldraftpreparation,A.ÇandS.T.; Writing-review&editing,A.Ç.,S.T.,andF.S.;Supervision,A.Ç.,S.T.andF.S.;FundingAcquisition,F.S.
Funding: Thisresearchreceivednoexternalfunding
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest
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