A New Functional Dependency in a Vague Relational Database Model

Transcription

1 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 New Fucioal Depedecy i a ague Relaioal Daabase Model Jaydev Mishra College of Egieerig ad Maageme, Kolagha Wes egal, Idia Sharmisha Ghosh ellore Isiue of Techology Uiversiy, ellore, Tamiladu, Idia STRCT I order o model he real world wih imprecise ad ucerai iformaio, various exesios of he classical relaioal daa model have bee sudied i lieraure usig fuzzy se heory However, vague se, as a geeralized fuzzy se, has more powerful abiliy o process fuzzy iformaio ha fuzzy se I his paper, we have proposed a vague relaioal daabase model ad have defied a ew kid of vague fucioal depedecy (called - based o he oio of -equaliy of uples ad he idea of similariy measure of vague ses Nex, we prese a se of soud vague iferece rules which are similar o rmsrog s axioms for he classical case Fially, parial - ad vague key have bee sudied wih he ew oio of - ad also esed wih examples Geeral Terms ague Daabase Desig Keywords ague se, similariy measure of vague ses, -, parial -, vague key 1 INTRODUCTION Iformaio i he real world is very ofe imprecise or ucerai i aure Fuzzy se heory, iroduced by adeh i 1965 [1] has bee widely used i lieraure [, 3, 4, 5, 6, 7, 8, 9] o icorporae such imprecise daa io classical relaioal daabases Exesive research has bee carried ou i his direcio ad several fuzzy relaioal daabase models have bee proposed o model vague iformaio i relaioal daabases lso, based o such fuzzy relaioal models, here have bee may sudies o differe daa iegriy cosrais [, 3, 4, 8, 9], fuzzy relaioal algebra [5], fuzzy query laguages [10] ad so o However, vague se heory was pu forward by Gau ad uehrer [11] i 1993 as a more efficie ool o deal wih imperfec or ambiguous daa vague se, coceived as a geeralizaio of he cocep of fuzzy se, is a se of objecs each of which has a grade of membership whose value is a coiuous sub-ierval of [0,1] O he corary, i is well kow ha a radiioal fuzzy se F i he verse of discourse U is characerized by a sigle membership fucio F ha assigs o each objec uu a sigle membership value F u which is a real umber lyig bewee 0 ad 1 F u is called he grade of membership of he eleme u i he se F However, i real life i is hard o make sure of he precisio degree ha a eleme belogs o a fuzzy se Furher, i was poied ou by Gau e al i [11] ha he sigle membership value i he fuzzy se heory combies he evidece for uu ad he evidece agais uu wihou idicaig how much here is of each To resolve his problem, hey had iroduced he cocep of vague se which is characerized by a ruh membership fucio ad a false membership fucio f Thus, a vague se separaes he posiive ad egaive evidece for membership of a eleme i he se ad provides lower ad upper bouds o he grade of membership of a eleme i he se These lower ad upper bouds are used o creae a sub-ierval o [0, 1], amely,,1 f, o geeralize he membership fucio of fuzzy ses, where 1 f Sice vague ses have bee iroduced o deal wih imprecise iformaio i a more efficie maer ha radiioal fuzzy ses, classical relaioal daabases may also be exeded o represe ad deal wih ucerai daa wih he cocep of vague se heory The exeded daabase model is called a vague relaioal daabase model However, compared o fuzzy relaioal daabases, much less research has bee repored so far i he area of vague relaioal daabase This is, i paricular, rue for he sudy of vague fucioal depedecy u, i is well kow ha daa depedecies play a impora role i ay daabase desig ad implemeaio of fucioal depedecy of oe se of aribues upo aoher is oe of he mos vial coceps i relaioal daabases Thus, similar o he heory of classical relaioal daabases, vague fucioal depedecies ca also be used as a gdelie for he desig of a vague relaioal schema hao e al [1] have proposed a vague relaioal model based o vague se heory ad a ew similariy measure (SE bewee vague ses They have also exeded ad sudied he vague relaioal algebra i ref [13] I paricular, hao e al have focused o he issues of vague fucioal depedecy ad vague rmsrog s axioms i [1] ccordig o hao e al, a vague relaio r o a relaioal schema R saisfies he vague fucioal depedecy : Y, where Y R if ( r( s r( SE( [ ], s[ ] ( SE( [ Y], s[ Y] I may be clearly observed ha he above defiiio of fails if he SE( [ Y], s[ Y] is slighly less ha SE ( [ ], s[ ], which should o be he case i realiy However, such siuaios have bee ake care by hao e al [1] i he saisfacio degree of s Such vague fucioal depedecies have bee uilized by Lu & Ng [14] o maiai he cosisecy of a vague daabase 9

2 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 I his paper, we propose a ew kid of vague fucioal depedecy (called - based o he cocep of - equaliy of uples ad he oio of similariy measure bewee vague ses which ca resolve he problem i such siuaios a defiiio level Nex, we prese a se of soud vague iferece rules which are similar o rmsrog s axioms for he classical case Fially, parial - ad vague key have also bee sudied wih he ew oio of - ad esed wih examples The orgaizaio of he paper is as follows: The basics of vague se heory have bee reviewed i secio The vague relaioal model has bee proposed i secio 3 Secio 4 discusses he similariy measure of vague ses used i his sudy We prese he ewly proposed vague fucioal depedecy ( - i secio 5 The vague iferece rules have also bee ivesigaed ad he defiiios of parial - ad vague key have bee iroduced i he same secio The fial coclusios are repored i secio 6 SICS OF GUE SET I his secio, we review some basic defiiios o he heory of vague ses [11] Le U be he verse of discourse where a eleme of U is deoed by u Defiiio 1 vague se i he verse of discourse U is characerized by wo membership fucios give by: (i a ruh membership fucio : U 0,1 ad, (ii a false membership fucio : U 0,1 f, where is a lower boud of he grade of membership of u derived from he evidece for u, ad f is a lower boud o he egaio of u derived from he evidece agais u, ad f 1 Thus, he grade of membership of u i he vague se is bouded by a subierval,1 f of [0,1] ie, 1 f The, he vague se is wrie as u,,1 f : u U Here, he ierval,1 f is said o be he vague value o he objec u ad is deoed by For example, i a voig process, he vague value [05, 08] ca be ierpreed as he voe for a resoluio is 5 i favour, agais ad 3 eural (abseious The precisio of kowledge abou u is clearly characerized by he differece 1 f If his is small, he kowledge abou u is relaively precise However, if i is large, we kow correspodigly lile If is equal o 1 f, he kowledge abou u is precise, ad vague se heory revers back o fuzzy se heory If ad 1 f are boh equal o 1 or 0, depedig o wheher u does or does o belog o, he kowledge abou u is very exac ad he heory goes back o ha of ordiary ses Thus ay crisp or fuzzy se may be cosidered as a special case of vague se However, i may be oed ha ierval-valued fuzzy ses (i-v fuzzy ses [15] are o vague ses I i-v fuzzy ses, a ierval based membership value is assiged o each eleme of he verse cosiderig oly he evidece for u, wihou cosiderig he evidece agais u I vague ses boh are idepedely proposed by he decisio maker This makes a major differece i judgme abou he grade of membership Nex, we prese several special vague ses ad various operaios o vague ses ha are obvious exesios of he correspodig defiiios for ordiary ses ad fuzzy ses Defiiio vague se is a empy vague se, deoed by, if ad oly if is ruh-membership fucio u 0 ad falsemembership fucio u 1 Defiiio 3 f for all u o U c The compleme of a vague se is deoed by ad is defied by he ruh-membership ad false-membership fucios c ad f c as follows: f ad f c Defiiio 4 c vague se is coaied i aoher vague se, wrie as, if ad oly if, ad f f Defiiio 5 Two vague ses ad are equal, wrie as =, iff, ad, ha is, ad f f Defiiio 6 The o of wo vague ses ad is a vague se C, deoed as C =, whose ruh-membership ad false-membership fucios are relaed o hose of ad by C max, ad f C mi f, f Defiiio 7 The iersecio of wo vague ses ad is a vague se C, wrie as C =, such ha C mi, ad f C max f, f Defiiio 8 U U U U Le 1 be he Caresia produc of verses, ad i, i 1,,, be vague ses i heir correspodig verse of discourses U i, i 1,,, 30

3 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 respecively lso, le Caresia produc u U, i 1,, U U i i, The he 1 is defied o be a U U vague se of 1 wih he ruh ad false membership fucios defied as follows: u u,, u mi u, u,, u 1, 1 1 f u u,, u max f u, f u,, f u 1, GUE RELTIONL DTSE MODEL (RD I his secio, we make a aemp o exed he classical relaioal daabase model o icorporae imprecise or vague daa by meas of vague se heory, which resuls i he ague Relaioal Daabase model (RD Table-I : vague relaioal isace r of EMP_SL relaio NME EP SL Smih {<1, [1, 1]>, <15, [9, 95]>, <0, [0, 0]>} {<45000, [8, 95]>} David {<1, [9, 95]>, <15, [95, 95]>, <0, [, 3]>} {<45000, [1, 1]>, <75000, [1, ]>} Sachi {<1, [1, 15]>, <15, [, 3]>, <0, [1, 1]>} {<75000, [1, 1]>} Joh { <0, [9, 95]>} {<45000, [, 3]>, <75000, [9, 95]>} Defiiio 31 Le i, i 1,,, be aribues defied o he verses of discourse ses U i, respecively The, a vague relaio r o he relaio schema R,,, 1 is defied as a subse of he Caresia produc of a collecio of vague subses: r (U 1 (U (U, where (U i deoes he collecio of all vague subses o a verse of discourse U i Each uple of r cosiss of a Caresia produc of vague subses o he respecive U i s, ie, [ i ] = ( i, where ( i is a vague subse of he aribue i defied o U i for all i The relaio r ca hus be represeed by a able wih colums I may be observed ha he vague relaio ca be cosidered as a exesio of classical relaios (all vague values are [1, 1] ad fuzzy relaios (all vague values are [a, a], 0 a 1 I is clear ha he vague relaio ca capure more iformaio abou vagueess Example 311: Cosider he vague relaioal isace r over EMP_SL (NME, EP, SL give above i Table-I I r, EP(Experiece ad SL(Salary are vague aribues The firs uple i r meas he employee wih NME = Smih has he experiece of {<1, [1, 1]>, <15, [9, 95]>, <0, [0, 0]>} ad he salary of {<45000, [8, 95]>}, which are vague ses Here he vague daa <1, [1, 1]> meas he evidece i favour of The experiece is 1 is 1 ad he evidece agais i is 0 Similarly he vague daa <45000, [8, 95]> idicaes he evidece i favour of The salary is Rs is 08 while he evidece agais i is 005 ad so o 4 SIMILRITY MESURE OF GUE DT There have bee some sudies i lieraure which discuss he opic cocerig how o measure he degree of similariy bewee vague ses [16, 17, 18] I [18], i was poied ou by Lu e al ha he similariy measure i [16, 17] did o fi well i some cases They have proposed a ew similariy measure bewee vague ses which ured ou o be more reasoable i more geeral cases The same has bee used i he prese work which is defied as follows: Defiiio 41: Similariy Measure bewee wo vague values Le x ad y be wo vague values such ha x = [ x, 1-f x ], y = [ y, 1-f y ], where 0 x 1-f x 1, ad 0 y 1-f y 1 Le SE(x, y deoe he similariy measure bewee x ad y The, x y f x f y SE( x, y 1 1 f f Defiiio 4: Similariy Measure bewee wo vague ses Le U = { u 1, u, u 3, u } be he verse of discourse Le ad be wo vague ses o U, where: {,[ (,1 f ( ], U}, where (u i ( u i 1- f (u i ad 1 i {,[,1 f ], U}, where (u i ( u i 1- f (u i ad 1 i Now, he similariy measure bewee ad, deoed by SE (, is defied as: 1 SE(, i1 SE([ ( u,1 f i i x y ( u ],[ ( u,1 f i x y ( u ] 1 ( ( ( f ( f 1 1 ( ( ( f ( f i 1 5 DESIGN OF GUE RELTIONL DTSE MODEL Fucioal Depedecy (fd is oe of he mos impora daa depedecies which play a crucial role i desig of ay logical daabase Here we proceed o exed he cocep of fucioal depedecy i he ligh of vague se heory This is ermed as ague Fucioal Depedecy ( I he prese work, we have proposed a ew, called -, which is based o he idea of - equaliy of wo vague uples Here, [0, 1] is a choice parameer ad he cocep of - equaliy has bee iroduced usig he oio of similariy measure of vague daa as follows: i 31

4 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February ague Fucioal Depedecy ( Defiiio 511: - equaliy of wo vague uples Le r(r be a vague relaio o he relaioal schema R( 1,,, Le 1 ad be ay wo vague uples i r Le [0, 1] be a hreshold or choice parameer, predefied by he daabase desiger, ad = { 1,,, k } R The he vague uples 1 ad are said o be -equal o if SE( 1 [ i ], [ i ] i=1,,3 k We deoe his equaliy by he oaio ]( E [ ] 1[ The, he followig proposiio is sraighforward from he above defiiio 1 1, Proposiio 511: If 0 1[ ]( E [ ] [ ]( [ ] 1 1 E Nex, we defie our - as follows: Defiiio 51: ague Fucioal Depedecy ( - Le, Y R = { 1, } Choose a hreshold value [0, 1] The a vague fucioal depedecy (, deoed by Y, is said o exis if, ]( E wheever 1[ [ ], i is also he case ha 1[ Y]( E [ Y] We may read his as follows: he se of aribues vague fucioally deermies he se of aribues Y a -level of choice I aoher ermiology, he se of aribues Y is vague fucioally deermied by he se of aribues a -level of choice Durig he course of aalysis, differe values may be se by he daabase desiger so, ha he above may be ermed as - lso, we have he followig sraighforward proposiio for - Proposiio 51 : If 0 1 1, Y Y 1 Example 511: Cosider he vague relaioal isace r preseed i Table-I Le us ow check wheher EP SL holds o a cerai -level of choice or o Firs, we obai SE( p [EP], q [EP] ad SE( p [SL], q [SL] for every pair of uples p & q i r, respecively, usig Defiiio 4 The resuls are show below by he marices E ad S respecively i Table-II Table-II : Marices E & S showig Similariy Measure E For = 08 (give by decisio maker, we ca see from he above wo marices, ha for ay pair of uples p & q if SE( p [EP], q [EP], i is also he case ha SE( p [SL], q [SL] So, we ca say ha he vague relaioal isace r S EP SL 8 holds for he lso, from above marices E ad S, we ca say ha he EP SL holds rue However he 85 EP SL does o hold because for uples 1 ad, 9 SE( 1 [EP], [EP] = , bu SE( p [SL], q [SL] = Iferece rules for - I is well kow ha i classical relaioal daabases, fucioal depedecies saisfy a se of iferece rules called rmsrog s axioms I his secio, we have derived a se of iferece rules for our proposed - These vague iferece rules are similar o rmsrog s axioms for fd We call hem vague rmsrog s axioms ad are give as follows: (1 - reflexive rule: If Y R, Y ( - augmeaio rule: If Y ad R, Y 3

5 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 (3 - rasiive rule: If Y ady mi( 1, 1, Theorem 51 : ague rmsrog s axioms (1- (3 are soud Proof: (1 - reflexive rule: [ i ] Le 1[ ]( E [ ] is rue, ie, SE( 1 [ i ], i i Y holds (Y ie, ] This implies ( Le The, SE( 1 [ i ], [ i ] 1[ Y]( E [ Y is also rue Y holds Hece proved - augmeaio rule: Y Now, from Defiiio 51, for ay wo uples 1 ad if 1[ ]( E [ ] (i is rue, 1[ Y]( E [ Y] (ii is also rue Nex, suppose 1[ ]( E [ ] (iii is rue This implies, SE( 1 [ i ], [ i ] SE( 1 [ i ], [ i ] i ]( E [ ] i 1[ (iv The from (ii ad (iv, we ge 1[ Y ]( E [ Y ] (v Thus, for ay wo uples 1 ad if 1[ ]( E [ ], i is also he case ha 1[ Y ]( E [ Y ] which implies Y proved (3 - rasiive rule: Le us assume ha boh he s hold i he relaio r(r Hece Y ad Y 1 Case I: 1 so ha mi ( 1, = Give ha Y ad So, usig Proposiio 51 we ge The, from (i we ca wrie Y (i ]( E [ ] Y]( E [ ] 1[ gai, sice Y 1[ Y ----(ii holds, so we have Y]( E [ ] ]( E [ ] 1[ Y Combiig (ii ad (iii, we ge 1[ -----(iii ]( E [ ] ]( E [ ] 1[ which implies Hece for = mi ( 1,, if 1[ Y ady 1 Case II: 1 follows similarly Hece proved, Usig he above vague rmsrog s axioms, he followig resuls are also derived for - 33

6 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 (4 - decomposiio rule: If Y Y ad Proof: Give ha Y (i, Sice Y Y, by - reflexive rule, we have Y Y (ii From (i ad (ii usig - rasiive rule, we ge Y also follows similarly Thus, if proved Y, Y ad Hece (5 - o rule: If Y ad 1 Y mi( 1, Proof: Give ha -(ii From (i we may wrie augmeaio rule Y (i ad 1 Y 1, (iii (usig - Similarly, from (ii we ca wrie Y Y (iv Thus from (iii ad (iv usig - rasiive rule, we ge mi( 1, Y Hece proved (6 - pseudo rasiive rule: If Y WY, W Proof: Give ha ad WY Y (ii mi( 1, (i 1 ad From (i, usig - augmeaio rule we ca wrie W WY (iii From (iii ad (ii usig - rasiive rule, we ge W mi( 1, Hece proved 53 Parial vague fucioal depedecy (parial - fer validaio of rmsrog s axioms i he vague evirome wih our prese oio of -, le us defie parial vague fucioal depedecy (parial - as follows: Defiiio 531: Parial vague fucioal depedecy (parial - Y is called parially vague fucioally depede o a - level of choice, ie, Y parially, if Y hold ad 34

7 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 also here exiss a o empy se, such ha, Y The cocep of parial - expresses he fac ha afer removal of a aribue i from, he depedecy sill holds ie, for a aribue i, -{ i } sill vague fucioally deermies Y a -level of choice The oio of parial - is eeded o defie vague key Example 531 Le he relaioal schema be R={,, C, D, E} ad he se of s o R be give by ={ C D ad C D observed ha he 54 ague Key C 08 D } The i may be easily is a parial - I classical relaioal daabase, key is kow o be a special case of fucioal depedecy Le us ow exed he idea of classical key i he vague evirome o defie vague key wih -level of choice where [0, 1] is a choice parameer defied by he daabase desiger formal defiiio of vague key is as follows: Defiiio 541: ague Key Le K R ad be a se of s for R The, K is called a vague key of R a -level of choice where [0, 1] iff K R ad K R is o a parial - Example 541: Le us assume a relaio schema R= (,, C, D ad a se of s = {, C, D vague key of R Soluio: Give 08 } of R Fid a (i, C ---- (ii ad D (iii pplyig - o rule o (i ad (ii, we ge C (iv gai, applyig - o rule o (iii ad (iv, we ge CD (v lso, 1 is rivial ---- (vi Thus from (v ad (vi usig - o rule, we ge CD ie, R which implies ha is a vague key of R a -level of choice 6 CONCLUSION I his paper, we prese a exesio of he classical relaioal daabase model wih he coceps of vague se heory, a geeralized versio of fuzzy ses The paper maily coceraed o he sudy of fucioal depedecy i vague relaioal daabase For his purpose, we have iroduced a ew kid of vague fucioal depedecy (called - based o he idea of -equaliy of uples ad similariy measure of vague ses We have also derived he vague iferece rules ad defied parial - ad vague key i he paper The work may be exeded o sudy Mulivalued Depedecy ad Normalizaio usig - which cosiue a impora par of a relaioal daabase desig 7 REFERENCES [1] adeh L, 1965, Fuzzy Ses, Iformaio ad Corol, ol 8, No3, pp [] l-hamouz S, iswas R, 006, Fuzzy Fucioal Depedecies i Relaioal Daabases, Ieraioal Joural of Compuaioal Cogiio, ol 4, No1, pp [3] Liao S Y, Wag H Q ad Liu WY, 1993, Fucioal Depedecies wih Null values, Fuzzy values, ad Crisp values, IEEE Trasacio o Fuzzy Sysems, ol 7, No 1, pp [4] Liu, W Y, 1997, Fuzzy Daa Depedecies ad Implicaio of Fuzzy Daa Depedecies, Fuzzy Ses ad Sysems, ol 9, No 3, pp [5] Ma M, hag WJ, Ma W Y, ad Mili F, 00, Hadlig Fuzzy Iformaio i Exeded Possibiliy- ased Fuzzy Relaioal Daabases, Ieraioal Joural of Iellige Sysems, ol 17, No 10, pp [6] Mishra J ad Ghosh S, 008, Sudy of Fuzzy Relaioal Daabase, Ieraioal Joural of Compuaioal Cogiio, ol 6, No 4, pp [7] Yazici ad Soza M I, 1998, The Iegriy Cosrais for Similariy-ased Fuzzy Relaioal Daabases, Ieraioal Joural of Iellige Sysems, ol 13, No 7, pp

8 Ieraioal Joural of Compuer pplicaios ( olume 39 No8, February 01 [8] Yazici ad Soza M I, 001, Complee xiomaizaio for Fuzzy Fucioal ad Mulivalued Depedecies i Fuzzy Daabase Relaios, Fuzzy Ses ad Sysems, ol 117, No, pp [9] Sigaraju J, 010, Fuzzy Subse Depedecies, Ieraioal Joural of Compuer pplicaios, ol 8, No 13, pp 9-36 [10] osc P ad Piver O, 1995, SQLf: Relaioal Daabase Laguage for Fuzzy Queryig, IEEE Trasacios o Fuzzy Sysems, ol 3, No 1, pp 1-17 [11] Gau W L ad uehrer D J, 1993, ague Ses, IEEE Tras Sys Ma, Cybereics, ol 3, No, pp [1] hao F ad Ma M, 006, Fucioal Depedecies i ague Relaioal Daabases, IEEE Ieraioal Coferece o Sysems, Ma ad Cybereics [13] hao F, Ma M, ad Ya L, 007, ague Relaioal Model ad lgebra, Fourh Ieraioal Coferece o Fuzzy Sysems ad Kowledge Discovery, (FSKD [14] Lu ad Ng W, 009, Maiaiig cosisecy of vague daabases usig daa depedecies, Daa ad Kowledge Egieerig, ol 68, No 7, pp [15] adeh L, 1975, The Cocep of a Ligsic ariable ad is pplicaio o pproximae Reasoig - I, Iformaio Scieces, ol 8, pp [16] Che S M, 1997, Similariy Measure bewee ague Ses ad bewee Elemes, IEEE Tras Sys Ma, Cyber, ol 7, No 1, pp [17] Hog D H ad Kim C, 1999, Noe o Similariy Measures bewee ague Ses ad bewee Elemes, Iformaio Scieces, ol 115, pp [18] Lu, ad Ng W, 004, Maagig Merged Daa by ague Fucioal Depedecies, LNCS 388, pp 59-7, Spriger-erlag erli Heidelberg 36