Neutrosophic Relational Data Model

Transcription

1 Neurosophc elaonal Daa Model Habn wang Bosascs esearch and Informacs Core Wnshp Cancer Insue Emory Unversy Alana GA 0 habn.wang@emoryhealhcare.org ajshekhar underraman Deparmen of Compuer cence Georga ae Unversy Alana GA 00 raj@cs.gsu.edu Florenn marandache Deparmen of Mahemacs Unversy of New Mexco Gallup NM 870 smarand@unm.edu André ogako Bosascs esearch and Informacs Core Wnshp Cancer Insue Emory Unversy Alana GA 0 andre.rogako@emoryhealhcare.org Absrac In hs paper we presen a generalzaon of he relaonal daa model based on nerval neurosophc se []. Our daa model s capable of manpulang ncomplee as well as nconssen nformaon. Fuzzy relaon or nuonsc fuzzy relaon can only handle ncomplee nformaon. Assocaed wh each relaon are wo membershp funcons one s called ruh-membershp funcon T whch keeps rack of he exen o whch we beleve he uple s n he relaon anoher s called falsy-membershp funcon F whch keeps rack of he

2 exen o whch we beleve ha s no n he relaon. A neurosophc relaon s nconssen f here exss one uple α such ha Tα + Fα >. In order o handle nconssen suaon we propose an operaor called spl o ransform nconssen neurosophc relaons no pseudo-conssen neurosophc relaons and do he se-heorec and relaon-heorec operaons on hem and fnally use anoher operaor called combne o ransform he resul back o neurosophc relaon. For hs daa model we defne algebrac operaors ha are generalzaons of he usual operaors such as nersecon unon selecon jon on fuzzy relaons. Our daa model can underle any daabase and knowledge-base managemen sysem ha deals wh ncomplee and nconssen nformaon. Keyword: Inerval neurosophc se fuzzy relaon nconssen nformaon ncomplee nformaon neurosophc relaon.. Inroducon elaonal daa model was proposed by Ted Codd s poneerng paper *+. nce hen relaonal daabase sysems have been exensvely suded and a lo of commercal relaonal daabase sysems are currenly avalable [ 4]. Ths daa model usually akes care of only welldefned and unambguous daa. However mperfec nformaon s ubquous almos all he nformaon ha we have abou he real world s no ceran complee and precse [5]. Imperfec nformaon can be classfed as: ncompleeness mprecson uncerany and nconssency. Incompleeness arses from he absence of a value mprecson from he exsence of a value whch canno be measured wh suable precson uncerany from he fac ha a person has gven a subjecve opnon abou he ruh of a fac whch he/she does no know for ceran and nconssency from he fac ha here are wo or more conflcng values for a varable. In order o represen and manpulae varous forms of ncomplee nformaon n relaonal daabases several exensons of he classcal relaonal model have been proposed [ In some of hese exensons a varey of null values have been nroduced o model unknown or no-applcable daa values. Aemps have also been made o generalze operaors of relaonal algebra o manpulae such exended daa models [6 8 ]. The fuzzy se heory and fuzzy logc proposed by Zadeh [4] provde a requse mahemacal framework for dealng wh ncomplee and mprecse nformaon. Laer on he concep of nerval-valued fuzzy ses was proposed o capure he fuzzness of grade of membershp self [5]. In 986 Aanassov nroduced he nuonsc fuzzy se [6] whch s a generalzaon of fuzzy se and provably equvalen o nerval-valued fuzzy se. The nuonsc fuzzy ses consder boh ruh-membershp T and falsy-membershp F wh T a F a [0 ] and T a F a. Because of he resrcon he fuzzy se nerval-valued fuzzy se and nuonsc fuzzy se

3 canno handle nconssen nformaon. ome auhors [ ] have suded relaonal daabases n he lgh of fuzzy se heory wh an objecve o accommodae a wder range of real-world requremens and o provde closer man-machne neracons. Probably possbly and Dempser-hafer heory have been proposed o deal wh uncerany. Possbly heory [4] s bul upon he dea of a fuzzy resrcon. Tha means a varable could only ake s value from some fuzzy se of values and any value whn ha se s a possble value for he varable. Because values have dfferen degrees of membershp n he se hey are possble o dfferen degrees. Prade and Tesemale [5] nally suggesed usng possbly heory o deal wh ncomplee and unceran nformaon n daabase. Ther work s exended n [6] o cover mulvalued arbues. Wong [7] proposes a mehod ha quanfes he uncerany n a daabase usng probables. Hs mehod maybe s he smples one whch aached a probably o every member of a relaon and o use hese values o provde he probably ha a parcular value s he correc answer o a parcular query. Carvallo and Parell [8] also use probably heory o model uncerany n relaonal daabases sysems. Ther mehod augmened projecon and jon operaons wh probably measures. However unlke ncomplee mprecse and unceran nformaon nconssen nformaon has no enjoyed enough research aenon. In fac nconssen nformaon exss n a lo of applcaons. For example n daa warehousng applcaon nconssency wll appear when ryng o negrae he daa from many dfferen sources. Anoher example s ha n he exper sysem here exs facs whch are nconssen wh each oher. Generally wo basc approaches have been followed n solvng he nconssency problem n knowledge base: belef revson and paraconssen logc. The goal of he frs approach s o make an nconssen heory conssen eher by revsng or by represenng by a conssen semancs. On he oher hand he paraconssen approach allows reasonng n he presence of nconssency and conradcory nformaon can be derved or nroduced whou rvalzaon [9]. Baga and underraman [0 ] proposed a paraconssen realaonal daa model o deal wh ncomplee and nconssen nformaon. The daa model has been appled o compue he well-founded and fng model of logc programmng [ ]. Ths daa model s based on paraconssen logcs whch were suded n deal by de Cosa [4] and Belnap [5]. In hs paper we presen a new relaonal daa model neurosophc relaonal daa model NDM. Our model s based on he neurosophc se heory whch s an exenson of nuonsc fuzzy se heory [6] and s capable of manpulang ncomplee as well as nconssen nformaon. We use boh ruh-membershp funcon grade α and falsymembershp funcon grade β o denoe he saus of a uple of a ceran relaon wh [0] and. NDM s he generalzaon of fuzzy relaonal daa model FDM. Tha s when α + β = neuroshophc relaon s he ordnary fuzzy relaon. Ths model s

4 dsnc wh paraconssen relaonal daa model PDM n fac can be easly shown ha PDM s a specal case of NDM. Tha s when α β = 0 or neurosophc relaon s jus paraconssen relaon. We can use Fgure o express he relaonshp among FDM PDM and NDM. Neurosophc elaonal Daa Model Fuzzy elaonal Daa Model Paraconssen elaonal Daa Model Classcal elaonal Daa Model Fgure. elaonshp among DM FDM PDM and NDM We nroduce neurosophc relaons whch are he fundamenal mahemacal srucures underlyng our model. These srucures are srcly more general han classcal fuzzy relaons and nuonsc fuzzy relaons nerval-valued fuzzy relaons n ha for any fuzzy relaon or nuonsc fuzzy relaon here s a neurosophc relaon wh he same nformaon conen bu no vce versa. The clam s also rue for he relaonshp beween neurosophc relaons and paraconssen relaons. We defne algebrac operaors over neurosophc relaons ha exend he sandard operaors such as selecon jon unon over fuzzy relaons. There are many poenal applcaons of our new daa model. Here are some examples: a Web mnng. Essenally he daa and documens on he Web are heerogeneous nconssency s unavodable. Usng he presenaon and reasonng mehod of our daa model s easer o capure mperfec nformaon on he Web whch wll provde more poenally valued-added nformaon. b Bonformacs. There s a prolferaon of daa sources. Each research group and each new expermenal echnque seems o generae ye anoher source of valuable daa. Bu hese daa can be ncomplee and mprecse and even nconssen. We could no smply hrow away one daa n favor of oher daa. o how o represen and exrac useful nformaon from hese daa wll be a challenge problem. 4

5 c Decson uppor ysem. In decson suppor sysem we need o combne he daabase wh he knowledge base. There wll be a lo of unceran and nconssen nformaon so we need an effcen daa model o capure hese nformaon and reasonng wh hese nformaon. The paper s organzed as follow. econ deals wh some of he basc defnons and conceps of fuzzy relaons and operaons. econ nroduces neurosophc relaons and wo noons of generalzng he fuzzy relaonal operaors such as unon jon projecon for hese relaons. econ 4 presens some acual generalzed algebrac operaors for he neurosophc relaons. These operaors can be used for specfyng queres for daabase sysems bul on such relaons. econ 5 gves an llusrave applcaon of hese operaors. Fnally secon 6 conans some concludng remarks and drecons for fuure work.. Fuzzy elaons and Operaons In hs secon we presen he essenal conceps of a fuzzy relaonal daabase. Fuzzy relaons assocae a value beween 0 and wh every uple represenng he degree of membershp of he uple n he relaon. We also presen several useful query operaors on fuzzy relaons. Le a relaon scheme or jus scheme be a fne se of arbue names where for any arbue name A A doma s a non-empy doman of values for A. A uple on s any map : dom A such ha A dom A for each A. Le denoe he se of all uples on. Defnon A fuzzy relaon on scheme s any map : [0 ]. We le F be he se of all fuzzy relaons on. If and are relaon schemes such ha denoe he se { A A hen for any uple we le for all A } of all exensons of. We exend hs noon for any T by defnng T T.. e-heorec operaons on Fuzzy relaons Defnon Unon: Le and be fuzzy relaons on scheme. Then relaon on scheme gven by s a fuzzy max{ } for any. 5

6 Defnon Complemen: Le be a fuzzy relaon on scheme. Then on scheme gven by s a fuzzy relaon for any. Defnon 4 Inersecon: Le and be fuzzy relaons on scheme. Then relaon on scheme gven by s a fuzzy mn{ } for any. Defnon 5 Dfference: Le and be fuzzy relaons on scheme. Then s a fuzzy relaon on scheme gven by mn{ } for any.. elaon-heorec operaons on Fuzzy relaons Defnon 6 Le and be fuzzy relaons on schemes and respecvely. Then he naural jon or jus jon of and denoed s a fuzzy relaon on scheme gven by mn{ } for any. Defnon 7 Le be a fuzzy relaon on scheme and le. Then he projecon of ono denoed by s a fuzzy relaon on scheme gven by Defnon 8 max{ u u } for any. Le be a fuzzy relaon on scheme and le F be any logc formula nvolvng arbue names n consan symbols denong values n he arbue domans equaly symbol negaon symbol and connecves and. Then he selecon of by F denoed F s a fuzzy relaon on scheme gven by f F F 0 oherwse where s he usual selecon of uples sasfyng F. F. Neurosophc elaons 6

7 In hs secon we generalze fuzzy relaons n such a manner ha we are now able o assgn a measure of belef and a measure of doub o each uple. We shall refer o hese generalzed fuzzy relaons as neurosophc relaons. o a uple n a neurosophc relaon s assgned a measure 0. wll be referred o as he belef facor and wll be referred o as he doub facor. The nerpreaon of hs measure s ha we beleve wh confdence and doub wh confdence ha he uple s n he relaon. The belef and doub confdence facors for a uple need no add o exacly. Ths allows for ncompleeness and nconssency o be represened. If he belef and doub facors add up o less han we have ncomplee nformaon regardng he uple s saus n he relaon and f he belef and doub facors add up o more han we have nconssen nformaon regardng he uple s saus n he relaon. In conras o fuzzy relaons where he grade of membershp of a uple s fxed neurosophc relaons bound he grade of membershp of a uple o a subnerval for he case. The operaors on fuzzy relaons can also be generalzed for neurosophc relaons. However any such generalzaon of operaors should manan he belef sysem nuon behnd neurosophc relaons. Ths secon also develops wo dfferen noons of operaor generalzaons. We now formalze he noon of a neurosophc relaon. ecall ha denoes he se of all uples on any scheme. Defnon 9 A neurosophc relaon on scheme s any subse of 0 [0 ] For any we shall denoe an elemen of as where s he belef facor assgned o by and s he doub facor assgned o by. Le V be he se of all neurosophc relaons on. Defnon 0 A neurosophc relaon on scheme s conssen f for all. Le C be he se of all conssen neurosophc relaons on. s sad o be complee f for all. If s boh conssen and complee.e. 7

8 for all hen s a oal neurosophc relaon and le T be he se of all oal neurosophc relaons on. Defnon s sad o be pseudo-conssen f max{ b d b d } max{ d 8 b b d } where for hese b d b d. Le P be he se of all pseudo-conssen neurosophc relaons on. Example Neurosophc relaon { a0.0.7 a b0.0.5 c0.40. } s pseudo-conssen. Because for a max{0.0.4} max{0.70.6}.. I should be observed ha oal neurosophc relaons are essenally fuzzy relaons where he uncerany n he grade of membershp s elmnaed. We make hs relaonshp explc by defnng a one-one correspondence : T F gven by. Ths correspondence s used frequenly n he followng dscusson.. Operaor Generalzaons for all I s easly seen ha neurosophc relaons are a generalzaon of fuzzy relaons n ha for each fuzzy relaon here s a neurosophc relaon wh he same nformaon conen bu no vce versa. I s hus naural o hnk of generalzng he operaons on fuzzy relaons such as unon jon and projecon ec. o neurosophc relaons. However any such generalzaon should be nuve wh respec o he belef sysem model of neurosophc relaons. We now consruc a framework for operaors on boh knds of relaons and nroduce wo dfferen noons of he generalzaon relaonshp among her operaors. An n -ary operaor on fuzzy relaons wh sgnaure... n s a funcon F F n F n where... n are any schemes. mlarly an n - ary : operaor on neurosophc relaons wh sgnaure... n s a funcon : n n V V V. Defnon An operaor on neurosophc relaons wh sgnaure... n s oaly preservng f for any oal neurosophc relaons... s also oal. n Defnon... n on schemes n... respecvely A oaly preservng operaor on neurosophc relaons wh sgnaure... n s a weak generalzaon of an operaor on fuzzy relaons wh he same

9 sgnaure f for any oal neurosophc relaons we have... n on scheme n... respecvely n n n n The above defnon essenally requres o concde wh on oal neurosophc realons whch are n one-one correspondence wh he fuzzy relaons. In general here may be many operaors on neurosophc relaons ha are weak generalzaons of a gven operaor on fuzzy relaons. The behavor of he weak generalzaons of on even jus he conssen neurosophc relaons may n general vary. We requre a sronger noon of operaor generalzaon under whch a leas when resrced o conssen neurosophc relaons he behavor of all he generalzed operaors s he same. Before we can develop such a noon we need ha of represenaon of a neurosophc relaon. We assocae wh a conssen neurosophc relaon he se of all fuzzy relaons correspondng o oal neurosophc relaons obanable from by fllng he gaps beween he belef and doub facors for each uple. Le he map F reps : C be gven by reps { Q F Q }. The se reps conans all fuzzy relaons ha are compleons of he conssen neurosophc relaon. Observe ha reps s defned only for conssen neurosophc relaons and produces ses of fuzzy relaons. Then we have followng observaon. Proposon For any conssen neurosophc relaon on scheme reps s he sngleon { } ff s oal. Proof I s clear from he defnon of conssen and oal neurosophc relaons and from he defnon of reps operaon. We now need o exend operaors on fuzzy relaons o ses of fuzzy relaons. For any operaor F F F on fuzzy relaons we le : n n F F n F n : be a map on ses of fuzzy relaons defned as follows. For any ses M... M n of fuzzy relaons on schemes...n respecvely M... M {... M for all n}. n n 9

10 In oher words M... M s he se of - mages of all uples n he Caresan produc n M M n. We are now ready o lead up o a sronger noon of operaor generalzaon. Defnon 4 An operaor on neurosophc relaons wh sgnaure... n s conssency preservng f for any conssen neurosophc relaons... n respecvely... s also conssen. Defnon 5 n... n on schemes A conssency preservng operaor on neurosophc relaons wh sgnaure... n s a srong generalzaon of an operaor on fuzzy relaons wh he same sgnaure f for any conssen neurosophc relaons respecvely we have reps n n reps reps n n. n on schemes...n Gven an operaor on fuzzy relaons he behavor of a weak generalzaon of s conrolled only over he oal neurosophc relaons. On he oher hand he behavor of a srong generalzaon s conrolled over all conssen neurosophc relaons. Ths self suggess ha srong generalzaon s a sronger noon han weak generalzaon. The followng proposon makes hs precse. Proposon. If s a srong generalzaon of hen s also a weak generalzaon of Proof Le... n be he sgnaure of and and le... n be any oal neurosophc relaons on schemes conssen and s a srong generalzaon of we have ha reps... n reps... reps...n respecvely. nce all oal relaons are 0 n n n Proposon gves us ha for each n reps s he sngleon se { }. Therefore reps... reps s jus he sngleon se:... }. n n { n n Here... s oal and e. s a weak n generalzaon of. n n n n Though here may be many srong generalzaons of an operaor on fuzzy relaons hey all behave he same when resrced o conssen neurosophc relaons. In he nex secon we propose srong generalzaons for he usual operaors on fuzzy relaons. The proposed

11 generalzed operaors on neurosophc relaons correspond o he belef sysem nuon behnd neurosophc relaons. Frs we wll nroduce wo specal operaors on neurosophc relaons called spl and combne o ransform nconssen neurosophc relaons no pseudo-conssen neurosophc relaons and ransform pseudo-conssen neurosophc relaons no nconssen neurosophc relaons. Defnon 6 pl Operaor { b d b d and Le be a neurosophc relaon on scheme. Then b d } { b d b d and b d and b b and d b} { b d b d and b d b d and d d}. I s obvous ha s pseudo-conssen f s nconssen. Defnon 7 Combne Operaor { b d b d b d Le be a neurosophc relaon on scheme. Then and b d b d b b and b d and b d b d d d }. I s obvous ha s nconssen f s pseudo-conssen. Noe ha srong generalzaon defned above only holds for conssen or pseudo-conssen neurosophc relaons. For any arbrary neurosophc relaons we should frs use spl operaon o ransform hem no non-nconssen neurosophc relaons and apply he seheorec and relaon-heorec operaons on hem and fnally use combne operaon o ransform he resul no arbrary neurosophc relaon. For he smplfcaon of noaon he followng generalzed algebra s defned under such assumpon. 4. Generalzed Algebra on Neurosophc elaons In hs secon we presen one srong generalzaon each for he fuzzy relaon operaors such as unon jon and projecon. To reflec generalzaon a ha s placed over a fuzzy relaon operaor o oban he correspondng neurosophc relaon operaor. For example denoes he naural jon mong fuzzy relaons and denoes naural jon on neurosophc relaons. These generalzed operaors manan he belef sysem nuon behnd neurosophc relaons.

12 4. e-theorec Operaors We frs generalze he wo fundamenal se-heorec operaors unon and complemen. Defnon 8 Le and be neurosophc relaons on scheme. Then a he unon of and denoed s a neurosophc relaon on scheme gven by max{ }mn{ } for any ; b he complemen of denoed s a neurosophc relaon on scheme gven by for any. An nuve apprecaon of he unon operaor can be obaned as follows: Gven a uple snce we beleved ha s presen n he relaon wh confdence presen n he relaon wh confdence and ha s we can now beleve ha he uple s presen n he eher - - or - relaon wh confdence whch s equal o he larger of and. Usng he same logc we can now beleve n he absence of he uple from he eher - - or - relaon wh confdence whch s equal o he smaller because mus be absen from boh and for o be absen from he unon of and. The defnon of complemen and of all he oher operaors on neurosophc relaons defned laer can and should be undersood n he same way. Proposon The operaors and unary on neurosophc relaons are srong generalzaons of he operaors and unary on fuzzy relaons. Proof Le and be conssen neurosophc relaons on scheme. Then reps s he se { Q Ths se s he same as he se max{ } Q mn{ }} { r s r s }

13 whch s. reps reps uch a resul for unary can also be shown smlarly. For sake of compleeness we defne he followng wo relaed se-heorec operaors: Defnon 9 Le and be neurosophc relaons on scheme. Then a he nersecon of and denoed s a neurosophc relaon on scheme gven by } }max{ mn{ for any. b he dfference of and denoed s a neurosophc relaon on scheme gven by } }max{ mn{ for any. The followng proposon relaes he nersecon and dfference operaors n erms of he more fundamenal se-heorec operaors unon and complemen. Proposon 4 For any neurosophc relaons and on he same scheme we have. and Proof By defnon. max mn mn max so and The second par of he resul can be shown smlarly. 4. elaon-theorec Operaors

14 We now defne some relaon-heorec algebrac operaors on neurosophc relaons. Defnon 0 Le and be neurosophc relaons on schemes and respecvely. Then he naural jon furher for shor called jon of and denoed s a neurosophc relaon on scheme gven by mn{ }max{ } where s he usual projecon of a uple. I s nsrucve o observe ha smlar o he nersecon operaor he mnmum of he belef facors and he maxmum of he doub facors are used n he defnon of he jon operaon. Proposon 5 s a srong generalzaon of. Proof Le and be conssen neurosophc relaons on schemes and respecvely. Then reps s he se { Q F mn{ } Q max{ }} and reps reps { r r reps s reps }. Le Q reps. Then Q reps where s he usual projecon over of fuzzy relaons. mlarly Q reps Therefore Q reps reps. Le Q reps reps. Then Q mn{ } and Q mn{ } max{ } for any because and are conssen. Therefore Q reps. We now presen he projecon operaor. Defnon Le be a neurosophc relaon on scheme and. Then he projecon of ono denoed s a neurosophc relaon on scheme gven by 4

15 max{ u u }mn{ u u }. The belef facor of a uple n he projecon s he maxmum of he belef facors of all of he uple s exensons ono he scheme of he npu neurosophc relaon. Moreover he doub facor of a uple n he projecon s he mnmum of he doub facors of all of he uple s exensons ono he scheme of he npu neurosophc relaon. We presen he selecon operaor nex. Defnon Le be a neurosophc relaon on scheme and le F be any logc formula nvolvng arbue names n consan symbols denong values n he arbue domans equaly symbol negaon symbol and connecves and. Then he selecon of by F denoed F s a neurosophc relaon on scheme gven by F f F 0 oherwse where and f F oherwse where F s he usual selecon of uples sasfyng F from ordnary relaons. If a uple sasfes he selecon creron s belef and doub facors are he same n he selecon as n he npu neurosophc relaon. In he case where he uple does no sasfy he selecon creron s belef facor s se o 0 and he doub facor s se o n he selecon. Proposon 6 The operaors and are srong generalzaons of and respecvely. Proof mlar o ha of Proposon 5. Example elaon schemes are ses of arbue names bu n hs example we rea hem as ordered sequence of arbue names whch can be obaned hrough permuaon of arbue names so uples can be vewed as he usual lss of values. Le { a b c} be a common doman for all arbue names and le and be he followng neurosophc relaons on schemes X Y and Y Z respecvely. 5

16 aa <0> ab <0> ac <0> bb <0> bc <0> cb <> ac <0> ba <> cb <0> For oher uples whch are no n he neurosophc relaons and her 0 0 whch means no any nformaon avalable. Because and are nconssen we frs use spl operaon o ransform hem no pseudo-conssen and apply he relaon-heorec operaons on hem and ransform he resul back o arbrary neurosophc se usng combne operaon. Then T s a neurosophc relaon on scheme T T T X Y Z and X Z and T XZ are neurosophc relaons on scheme Z. T and T are shown below: T aaa <0> aab <0> aac <0> T X 6

17 aba <0> abb <0> abc <0> aca <0> acb <0> acc <0> bba <> bcb <0> cba <> cbb <0> cbc <0> ccb <0> T aa <0> ab <0> ac <0> ba <0> ca <0> T aa <0> ab <0> ac <0> 7

18 ba <0> bb <0> ca <0> cc <0> 5. An Applcaon Consder he arge recognon example presened n [6]. Here an auonomous vehcle needs o denfy objecs n a hosle envronmen such as a mlary balefeld. The auonomous vehcle s equpped wh a number of sensors whch are used o collec daa such as speed and sze of he objecs anks n he balefeld. Assocaed wh each sensor we have a se of rules ha descrbe he ype of he objec based on he properes deeced by he sensor. Le us assume ha he auonomous vehcle s equpped wh hree sensors resulng n daa colleced abou radar readngs of he anks her gun characerscs and her speeds. Wha follows s a se of rules ha assocae he ype of objec wh varous observaons. adar eadngs: eadng r ndcaes ha he objec s a T-7 ank wh belef facor 0.80 and doub facor 0.5. eadng r ndcaes ha he objec s a T-60 ank wh belef facor 0.70 and doub facor 0.0. eadng r ndcaes ha he objec s no a T-7 ank wh belef facor 0.95 and doub facor eadng r 4 ndcaes ha he objec s a T-80 ank wh belef facor 0.85 and doub facor 0.0. Gun Characerscs: Characersc c ndcaes ha he objec s a T-60 ank wh belef facor 0.80 and doub facor

19 Characersc c ndcaes ha he objec s no a T-80 ank wh belef facor 0.90 and doub facor Characersc c ndcaes ha he objec s a T-7 ank wh belef facor 0.85 and doub facor 0.0. peed Characerscs: Low speed ndcaes ha he objec s a T-60 ank wh belef facor 0.80 and doub facor 0.5. Hgh speed ndcaes ha he objec s no a T-7 ank wh belef facor 0.85 and doub facor 0.5. Hgh speed ndcaes ha he objec s no a T-80 ank wh belef facor 0.95 and doub facor Medum speed ndcaes ha he objec s no a T-80 ank wh belef facor 0.80 and doub facor 0.0. These rules can be capured n he followng hree neurosophc relaons: adar ules eadng Objec Confdence Facors r T-7 < > r T-60 < > r T-7 < > r 4 T-80 < > Gun ules eadng Objec Confdence Facors c T-60 < > 9

20 c T-80 < > c T-7 < > peed ules eadng Objec Confdence Facors low T-60 < > hgh T-7 < > hgh T-80 < > medum T-80 < > The auonomous vehcle uses he sensors o make observaons abou he dfferen objecs and hen uses he rules o deermne he ype of each objec n he balefeld. I s que possble ha wo dfferen sensors may denfy he same objec as of dfferen ypes hereby nroducng nconssences. Le us now consder hree objecs o o and o whch need o be denfed by he auonomous vehcle. Le us assume he followng observaons made by he hree sensors abou he hree objecs. Once agan we assume cerany facors maybe derved from he accuracy of he sensors are assocaed wh each observaon. adar Daa Objec-d eadng Confdence Facors o r < > o r < > o r 4 < > 0

21 Gun Daa Objec-d eadng Confdence Facors o c < > o c < > o c < > peed Daa Objec-d eadng Confdence Facors o hgh < > o low < > o medum < > Gven hese observaons and he rules we can use he followng algebrac expresson o denfy he hree objecs: Objecd Ojbec Objecd Objec Objecd Objec adar Gun peed Daa Daa Daa Gun adar peed ules ules ules The nuon behnd he nersecon s ha we would lke o capure he common nersecng nformaon among he hree sensor daa. Evaluang hs expresson we ge he followng neurosophc relaon: Objec-d eadng Confdence Facors o T-7 < > o T-80 < >

22 o T-80 < > I s clear from he resul ha by he gven nformaon we could no nfer any useful nformaon ha s we could no decde he saus of objecs o o and o. 6. Conclusons and Fuure Work We have presened a generalzaon of fuzzy relaons nuonsc fuzzy relaons nervalvalued fuzzy relaons and paraconssen relaons called neurosophc relaons n whch we allow he represenaon of confdence belef and doub facors wh each uple. The algebra on fuzzy relaons s appropraely generalzed o manpulae neurosophc relaons. Varous possbles exs for furher sudy n hs area. ecenly here has been some work n exendng logc programs o nvolve quanave paraconssency. Paraconssen logc programs were nroduced n [7] and probablsc logc programs n [8]. Paraconssen logc programs allow negave aoms o appear n he head of clauses hereby resulng n he possbly of dealng wh nconssency and probablsc logc programs assocae confdence measures wh lerals and wh enre clauses. The semancs of hese exensons of logc programs have already been presened bu mplemenaon sraeges o answer queres have no been dscussed. We propose o use he model nroduced n hs paper n compung he semancs of hese exensons of logc programs. Explorng applcaon areas s anoher mporan hrus of our research. We developed wo noons of generalzng operaors on fuzzy relaons for neurosophc relaons. Of hese he sronger noon guaranees ha any generalzed operaor s wellbehaved for neurosophc relaon operands ha conan conssen nformaon. For some well-known operaors on fuzzy relaons such as unon jon and projecon we nroduced generalzed operaors on neurosophc relaons. These generalzed operaors manan he belef sysem nuon behnd neurosophc relaons and are shown o be wellbehaved n he sense menoned above. Our daa model can be used o represen relaonal nformaon ha may be ncomplee and nconssen. As usual he algebrac operaors can be used o consruc queres o any daabase sysems for rerevng vague nformaon.

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