INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 Reichenbach and f-geneaed implicaions in fuzzy daabase elaions Nedžad Dukić Dženan Gušić and Nemana Kajmoić Absac Applying a definiion of aibue confomance based on a similaiy elaion we inoduce an inepeaion as a funcion associaed o some fuzzy elaion insance and defined on he uniesal se of aibues As a consequence he aibues become fuzzy fomulas Conjuncions disjuncions and implicaions beween he aibues become fuzzy fomulas as well in iew of he equiemen ha he inepeaion has o agee wih he minimum - nom he maximum -conom and appopiaely chosen fuzzy implicaion The pupose of his pape is o deie a numbe of esuls elaed o hese fuzzy fomulas if he fuzzy implicaion is seleced so o be eihe Reichenbach o some f-geneaed fuzzy implicaion Keywods Confomance uzzy implicaions Inepeaions Similaiy elaions I I INTRODUCTION N his pape we elae fuzzy dependencies and fuzzy logic heoies by joining fuzzy fomulas o fuzzy funcional and fuzzy mulialued dependencies We eseach he concep of fuzzy elaion insance ha aciely saisfies some fuzzy mulialued dependency We deemine he necessay and sufficien condiions needed o gien wo-elemen fuzzy elaion insance aciely saisfies some fuzzy mulialued dependency In paicula fo Reichenbach and some f-geneaed fuzzy implicaion opeaos we poe ha a wo-elemen fuzzy elaion insance aciely saisfies gien fuzzy mulialued dependency if and only if: uples of he insance ae confoman on ceain well known se of aibues wih degee of confomance geae han o equal o some explicily known consan elaed fuzzy fomula is saisfiable in appopiae inepeaions inally fo Reichenbach and some f-geneaed fuzzy implicaion opeaos we poe ha any wo-elemen fuzzy elaion insance which saisfies all dependencies fom he se saisfies he dependency f if and only if saisfiabiliy of all N Dukić is wih he Depamen of Mahemaics Uniesiy of Saajeo Zmaja od Bosne 35 7000 Saajeo Bosnia and Hezegoina (e-mail: ndukic@pmfunsaba Dž Gušić is also wih he Depamen of Mahemaics Uniesiy of Saajeo Zmaja od Bosne 35 7000 Saajeo Bosnia and Hezegoina (phone: +387693444; e-mail: dzenang@pmfunsaba N Kajmoić is wih he Elemenay School Hasno Poodice Riba 7000 Saajeo Bosnia and Hezegoina (e-mail: nemanakajmoic@gmailcom fomulas fom he se implies saisfiabiliy of he fomula f Hee f is a fuzzy funcional o a fuzzy mulialued dependency is a se of fuzzy funcional and fuzzy mulialued dependencies esp f denoe he se of fuzzy fomulas esp he fuzzy fomula elaed o esp f II PRELIMINARIES We inoduce he minimum -nom (see eg [] [9] [] he maximum -conom (see eg [] [7] as follows ( pq = min( ( p ( q T T T ( pq = max( ( p ( q T T T T is he uh alue od m An inepeaion is said o saisfy esp falsify fomula f if T ( f esp T ( f unde (see eg [3] We inoduce he noaion following similaiy-based fuzzy elaional daabase appoach [6] (see also [3]-[5] whee 0 T ( p T ( q Hee ( m A similaiy elaion on D is a mapping sd : D [ 0] such ha (see [] ( s xx = ( xy = ( yx s s ( ( xz max min( ( x y ( yz s s s yd whee D is a se and xyzd Le R ( U = R ( n be a scheme on domains D D D n whee U is he se od all aibues n on D D D n (we say ha U is he uniesal se of aibues Hee we assume ha he domain of i is he finie se D i = n i A fuzzy elaion insance on R ( U is defined as a subse of he coss poduc of he powe ses D n D D of he ISSN: 998-4464 85
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 domains of he aibues A membe of a fuzzy elaion insance coesponding o a hoizonal ow of he able is called a uple Moe pecisely a uple is an elemen of of he fom ( d d d n whee d i D i d i (see also [8] Hee we conside d i as he alue of i on Recall ha he similaiy based daabase appoach allows each domain o be equipped wih a similaiy elaion The confomance of aibue defined on domain D fo any wo uples and pesen in elaion insance and denoed by is defined by = min min max{ s( x y } min max{ s( x y } xd yd xd yd whee d denoe he alue od aibue fo uple i i = sd : D 0 is a similaiy elaion on D and [ ] If q whee 0 q han he uples and ae said o be confoman on aibue wih q The confomance of aibue se fo any wo uples and pesen in fuzzy elaion insance and denoed by is defined by = min { } Obiously: = fo any in If hen fo any and in = and q k m 3 If ( m fo all { } k hen q fo any and in Le be any fuzzy elaion insance on scheme R ( n U be he uniesal se of aibues n and be subses of U uzzy elaion insance is said o saisfy he fuzzy funcional dependency if fo eey pai of uples and in ( min uzzy elaion insance is said o saisfy he fuzzy mulialued dependency if fo eey pai of uples and in hee exiss a uple 3 in such ha: ( ( ( 3 3 3 whee = U Hee U Moeoe 0 descibes he linguisic sengh of he dependency Namely some dependencies ae pecise some of hem ae no some dependencies ae moe pecise han he ohe ones Theefoe he linguisic sengh of he ( U means ( dependency gies us a mehod fo descibing impecise dependencies as well as pecise ones uzzy elaion insance is said o saisfy he fuzzy mulialued dependency -aciely if saisfies ha dependency and if fo all and all I follows immediaely ha he insance saisfies he dependency -aciely if and only if saisfies and fo all = be any wo-elemen fuzzy elaion insance Le { } on scheme ( n A mapping :{ } [ 0] R and 0 such ha n ( k > if ( k if < k = n is called a aluaion (o an inepeaion joined o and Le III RESULTS ( be some fuzzy funcional dependency (fuzzy mulialued dependency on U whee U is he uniesal se of aibues nand R ( n is a scheme In his pape we associae he fuzzy fomula o ( and he fuzzy fomula ( ( o whee = U Though he es of he secion we assume ha he fuzzy implicaion opeao is gien eihe by T ( p q = ( q ( p T T if T ( p 0 o T ( q 0 T ( p q = if T ( p = 0and ( o by ( p q = ( p + ( p ( q T T T T T q = 0 Noe ha he fis fuzzy implicaion opeao is known as Yage's (Y opeao (see [9] I is a ypical example of f- geneaed fuzzy implicaion opeao (see [5] [0] The second fuzzy implicaion opeao is widely-known as Kleene- ISSN: 998-4464 86
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 Dienes-Lukasiewicz opeao o Reichenbach (R opeao (see [4] I epesens a classical example of song (S and quanum logic (QL implicaion (see [5] [7] [8] We efe o [6] [0] and [] as well In geneal classes of fuzzy implicaion opeaos ae ey nicely descibed in [] and [5] Theoem Le = { } insance on scheme ( be any wo-elemen fuzzy elaion of aibues n and R n U be he uniesal se be subses of U Le = U Then saisfies he fuzzy mulialued dependency -aciely if and only if and ( > whee denoes he fuzzy fomula associaed o ( (( Y ( Poof: (fo Y is we poe ha saisfies -aciely if and only if o Suppose ha he insance saisfies he dependency -aciely Now and hee is a uple 3 such ha he condiions gien by ( hold ue ie ha 3 3 3 Hence if 3= hen Else if 3 = hen Le min = Now Hence ( hee is 3 3 3 3= such ha 3 = ie ( holds ue Analogously if min = Moeoe hee is 3 hen ( 3 3 3 3= such ha = = Theefoe ( holds ue Now since saisfies he dependency and i follows ha he insance saisfies he dependency -aciely Now we poe he main asseion -aciely We ( Suppose ha saisies hae o Suppose ha Now min min { } = { } = Hence fo all and fo all Theefoe ( > fo ( > fo Now We obain ( ( = min{ } > min{ ( } = > ( = ( ( = ( ( = max Denoe a = b = max ( Since > and > we hae ha a > b> Now a ( > if and only if b > a If b= hen b > holds ue and hence ( > Le b < < Now a b > if and only if a < logb The las inequaliy is ue since logb > Theefoe ( > Similaly if hen > and > Now easoning as in he peious case we conclude ha a > b> and hence ( > ( Suppose ha and ( > We hae a > and hen b a > If b= 0 hen 0 > ie a conadicion Hence 0< b a If b= hen b > holds ue a Le 0< b< We hae b > if and only if a < logb The las inequaliy is saisfied fo b < < We conclude b > fo all Hence fo Now Theefoe and yield he esul If b = hen ( > ISSN: 998-4464 87
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 Analogously if b = hen Now yield he esul This complees he poof Poof: (fo R ( Assume ha saisfies he dependency aciely Now o Le and hold ue We hae and min min { } = { } = - Theefoe and Consequenly ( > and ( > We obain ( min{ ( } = > and min{ ( } = > We hae ( = ( ( ( ( ( ( ( ( = + = + max Pu a = b = max ( Now > ( > yield a > b> Hence < a and hen a We obain Since a a> > 0 we hae ha ( > if and only if a + ab > if and only if ab a + > if and only if + b> which is a ue Hence ( > Similaly if and hen > and > Now a> > 0b> and hence b a + > holds ue Theefoe ( > ( Le ( > Now 0< < a hence ( > if and only if b a + > Theefoe a implies ha b > Consequenly Now Hence yield he esul Similaly if we assume ha b = we obain ha Since he heoem follows This complees he poof If b = hen ( > Theoem Le f be a fuzzy funcional o a fuzzy mulialued dependency on a se of aibues U whee is a se of fuzzy funcional and fuzzy mulialued dependencies on U Le esp f be he se of fuzzy fomulas esp he fuzzy fomula elaed o esp f The following wo condiions ae equialen: (a Any wo-elemen fuzzy elaion insance on scheme R ( U which saisfies all dependencies fom he se saisfies also he dependency f (b ( f > fo eey suc ha ( > fo all Poof: (fo Y We denoe f by funcional dependency and by when f is a fuzzy when f is a fuzzy mulialued dependency Theefoe ( ( and ( ( will denoe f in he fis and he second case especiely whee = U We may assume ha he se { pq } is he domain of each of he aibues in U ix some [ 0 whee is he minimum of he senghs of all dependencies ha appea in { f } Suppose ha < Namely if = hen eey dependency f { f} is of he sengh This case is no ineesing howee s pq = s qp = o be a similaiy elaion on Define ( ( { pq } ISSN: 998-4464 88
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 ( a ( b Suppose ha ( Now hee is some and ( f Hee b is no alid such ha ( elaion insance = { } on ( Define W U ( Assume ha W = In his case ( Hence ( > fo all is joined o some wo-elemen fuzzy R U and some 0 = > fo all U < fo any U 0 0 0 If ( = ( = esp ( ( ( 0 hen ( f = 0 = yields ie a conadicion Hence ( o ( esp ( 0 o ( ( ( We may assume ha 0 max ( 0 Now ( f esp ie esp 0 max 0 esp implies ( ( max (3 ( log (4 ( log max ( ( ( (5 Theefoe 0 < ( esp 0 < max ( yields ( = This is conadicion Hence W Assume ha W =U In his case ( > fo all U Consequenly ( > fo all U Now ( esp (3 holds ue If ( = esp max ( = hen ie a conadicion Hence (4 esp (5 holds ue Theefoe < ( < esp <max ( ( ( < yields > Define = { } by Table below is a wo-elemen fuzzy elaion insance on ( This is a conadicon We obain W U R U We shall poe ha his insance saisfies all dependencies fom he se bu iolaes he dependency f Le be any fuzzy funcional dependency fom e se Table : aibues of W ohe aibues p p p p p p p p p qqq Assume ha 0 Then hee exiss such ha ( 0 = min { ( } = ( ie 0 W Since s ( pq aibues 0 We hae = and hence { } min = = = we know ha U Theefoe fo any se of We obain ( = min ie saisies Assume ha ( > Now ( ( ( ( = > The las inequaliy is saisfied if ( ( = 0 hen Le ( If = 0 > ie a conadicion 0 < < We hae ( < log ( > > fo all and hen W fo We obain Hence = We hae Theefoe ( Now ( ISSN: 998-4464 89
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 ( = min ie saisfies he dependency Le by any fuzzy mulialued dependency fom e se Suppose ha ( Then easoning as in he peious case we obain ha = Hence hee is = such ha ( = ( = ( = whee = U Theefoe saisfies Le ( > Now max ( ( ( ( = ( ( ( = > ( ( = hen This inequaliy is saisfied if ( ( If ( ( max 0 (6 max conadicion 0<max hen ( If ( ( < ( <log max ( ( ( = ( Theefoe max 0 > ie a > Hence > o ( > If > hen and hence = Similaly since > we conclude ha = Now hee is = such ha ( ( ( = = = min Hence saisfies If ( > hen = In his case hee is = such ha (7 holds ue In ohe wods saisfies he dependency I emains o poe ha he insance iolaes esp Le ( = ( f If ( = 0 and ( = 0 hen conadicion Hence ( 0 o ( may assume ha ( 0 Now If ( ( hen = (7 ie a 0 We ie a conadicion Theefoe 0 < < We obain ( log > ie a conadicion = Now as befoe we conclude ha = and = Theefoe If ( > hen ( Hence 0 < ( and hen ( ( = < = min This means ha iolaes Now le ( ( ( f = Reasoning as in he peious case we conclude ha 0 max 0 We hae ( o ( ( ( ISSN: 998-4464 90
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 ( max Then 0<max ( and = ie = We obain = = = If = hen If = hen ( = ( = ( = < = ( = ( min ( = < = = min In ohe wods he insance iolaes ( b ( a Suppose ha (a is no alid Now hee is a wo-elemen fuzzy elaion insance R U such ha saisfies all = { } on scheme ( dependencies in and does no saisfy f Theefoe does no saisfy esp Define W = { = } U Assume ha W = Now = fo all U Theefoe = fo all U In he case when does no saisfy we obain ie min ( ( < min < = This is a conadicion Similaly in he case when does no saisfy we hae ha he condiions min ( ( ( min (8 (9 (0 don hold simulaneously Since he fis and he second condiion in (0 hold obiously ue we obain ( ( = < min = min = which is a conadicion Theefoe W Assume ha W =U Now = fo eey U Theefoe = fo eey U In he case when does no saisfy we hae ha ( ( = < min = min = This is a conadicion In he case when does no saisfy he condiions gien by (0 don hold simulaneously The fis and he second condiion in (0 ae always saisfied hence ( ( = < min = min = This is a conadicion We conclude W U Now we define in he following way Le We shall poe ha ( Suppose ha < ( if W 0 ( if U W > fo eey is of he fom ( ( and ( f This fuzzy fomula coesponds o some fuzzy funcional dependency fom he se Suppose ha ( Then as ealie i follows ha 0 o ( 0 Assume ha 0 We hae Then ( ( < We obain ( log Theefoe 0 < and ( = and = = ie ISSN: 998-4464 9
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 We obain ( ( = < = min = min which conadics he fac ha saisfies Theefoe ( > Suppose ha is of he fom ( ( ( whee = U- This fuzzy fomula coesponds o some fuzzy mulialued dependency fom he se Assume ha ( As befoe we hae ha 0 o max 0 ( ( ( Suppose ha ( ( ( max 0 We hae ( ( max ( Then ( ( max We obain < ( log ( ( ( max ( < and = ie ( ( = Hence = = = In he case he hid condiion of he condiions Theefoe 0 max ( ( ( min ( does no hold uhemoe he second condiion of he condiions ( ( ( min ( does no hold This conadics he fac ha saisfies he dependency Hence ( > I emains o poe ha ( f Suppose ha he insance does no saisfies he dependency Assume ha ( f > If hen = Hence ( ( = min = min This conadics he fac ha iolaes If > hen > This inequaliy is saisied if ( If ( = = 0 hen 0 > ie a conadicion If ( 0 < < hen Theefoe ( <log ( > ie = ( min Now which is a conadicion We conclude ( f Suppose ha does no saisfy Now he hid condiion of he condiions gien by (0 does no hold ie ( < min (3 Moeoe he fis and he second condiion of he condiions don hold simulaneously Assume ha ( f > ( ( ( min (4 ISSN: 998-4464 9
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 hen If = Hence ( ( = min = min which conadics (3 If > hen = and ( ( max > The las inequaliy is saisfied if max ( ( ( = ( If ( ( max 0 hen = conadicion 0<max hen ( If ( ( < ( log ( ( ( max ( Theefoe max > o ( = 0 > ie a > ie > Hence = o In he fis case he condiions gien by (4 ae saisfied simulaneously while in he second case he condiion (3 does no hold Hence a conadicion We conclude ( f This complees he poof Poof: (fo R We wie esp insead of f if f is a fuzzy funcional esp a fuzzy mulialued dependency Consequenly we wie ( esp ( ( insead of f whee = U- As in he case (Y choose he se { pq } o be he domain of each of he aibues in U 0 whee is he minimum of he We fix some [ senghs of all dependencies ha appea in { f } Assume ha < and pu ( ( elaion on { pq } ( a ( b s= pq = s qp = o be a similaiy Assume ha (b does no hold Then hee exiss some such ha ( > and ( f As in he case (Y is joined o some wo- = R U and some elemen fuzzy elaion insance { } on ( 0 Denoe W ( Suppose ha W = Then ( Since ( esp = U > U f we hae ha ( ( + ( ( f = = ( ( ( + max ( ( ( = + ( ( = = ( f Hence + esp ( ( ( max ( ( + Theefoe he fac ha ( 0 fo all U yields ha holds always ue Now hee exis some 0 U such ha { } ( ( 0 =min ( = This is a conadicion We conclude W ISSN: 998-4464 93
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 Suppoes ha W=U Then ( Theefoe we hae ha esp > > U fo any U Since ( f + ( ( + max Hence ( esp max This means ha we always hae ha This is a conadicion We conclude W U = be he wo elemen fuzzy Now yields As in he case (Y le { } elaion insance on R ( U gien by Table We shall poe ha saisfies all dependencies in and iolaes he dependency f Le be any fuzzy funcional dependency fom he se We hae ie ( ( ( + ( ( = > ( ( ( + > Suppose ha Reasoning as in he case (Y we conclude ha hee is 0 such ha ( 0 ie ha 0 W Hence 0 = and hen = As befoe he fac ha s ( pq = yields ha fo eey U Now and hen ( = min This means ha saisfies he dependency Now suppose ha > We obain ( ( + > Hence Since we hae ha Now ( > > and hen W In ohe wods W Theefoe = We obain = min ( This means ha saisfies Le be any fuzzy mulialued dependency fom he se We hae ie ( ( ( ( + max ( ( ( = > ( max ( ( ( + > whee =U- Assume ha Reasoning as in he peious case we conclude ha = Hence hee exiss = such ha (6 holds ue This means ha he fuzzy elaion insance saisfies he dependency Suppose ha > We obain ( ( ( + max > Since we conclude ha max ( M > Theefoe > o M > If > hen easoning as in he case (Y we obain ha = = Theefoe hee exiss = such ha (7 holds ue In ohe wods saisfies he dependency Similaly if M > we hae ha = Now hee exiss = such ha (7 is alid ie saisfies In emans o poe ha iolaes he dependency esp Le ISSN: 998-4464 94
Hence ( ( + ( ( f = = ( ( + Since ( 0 fo U we conclude ha Theefoe W and hen = Now + As befoe we obain ha ie = We hae = < = min ( Hence does no saisfy he dependency suppose ha Now ( ( ( + max ; We hae INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 ( ( ( f = = ( max ( ( + ; and hen W ie Moeoe Now = ; ( ( + max ( Theefoe max ; and hence ( Consequenly = = If and = ha easpning in he same way as in he cas (Y we conclude ha (8 holds ue Similaly if and = we obain ha (9 holds ue Theefoe he insance does no saisfy he dependency ( b ( a Assume ha (a does no hold Now as in he case (Y hee exiss a wo-elemen fuzzy elaion insance ={ } on R ( U such ha saisfies all dependencies fom he se bu iolaes he dependency f Hence iolaes esp Reasoning in exacly he same way as in he case (Y we W = U = conclude ha W and W U whee { } Now we poe ha ( whee > is defined by ( W ( 0 W U be of he fom Le ( ( and ( f The fuzzy fomula coesponds o some fuzzy funcional dependency fom he se If ( hen ( + ( ( ie + ( ( ( and hen Theefoe = ( ( + Now as befoe we conclude ha = We hae Moeoe ( ( = < = min = min and hence This howee conadicions he fac ha he insance saisfies he dependency Hence ( > Now le be of he fom ( ( ( ISSN: 998-4464 95
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 whee = U- This fuzzy fomula coesponds o some fuzzy mulialued dependency fom he se If ( hen ( ( ( ( + max ie ( max ( ( ( + and hen Now = uemoe ( ( ( + max ( We obain max ; and hen ( Theefoe = and = Consequenly he hid condiion of he condiions ( is no alid and he second condiion of he condiions ( is no alid as well This conadics he fac ha he insance insance saisfies Hence ( > I emains o poe ha ( f Assume ha iolaes Suppose ha ( f > We hae If ( ( + > hen = and hence easoning in he same way as in (Y we obain ha min ( This conadics he fac ha does no saisfy Le > Now + > We obain ( > and hen case (Y min ( Theefoe ( f = Now as in he This is a conadicion Assume ha iolaes he dependency Reasoning in exacly he same way as in he case (Y we obain ha (3 holds ue and ha fis and he second condiion of he condiions (4 don hold a he same ime Le ( f > We hae ( max ( ( + ; > hen = Hence as in he case (Y min ( This conadics (3 Le > Now = and If ( ( + max > ( We obain max ; > Hence > o ( > ie = o = In he fis case he condiions (4 ae saisfied In he second case he condiion (3 is no alid Hence a conadicion We conclude ( f This complees he poof IV CONCLUSION The esuls pesened in his pape can be similaly eified fo many ohe indiidual fuzzy implicaion opeaos Such opeaos may be esiduaed (R as well One could y o ay -noms as well as -conoms In paicula i would be nice o deemine he degee of genealiy o which ou esuls may be applied REERENCES [] M Baczynski and B Jayaam On he chaaceizaions of (SN- implicaions geneaed fom coninuous negaions in Poc Of he h Conf Infomaion Pocessing and Managemen of Unceainy in Knowledge-based Sysems Pais 006 pp 436-443 [] M Baczynski and B Jayaam uzzy Implicaions Belin-Heidelbeg: Spinge-Velag 008 [3] R P Buckles and E Pey A fuzzy epesenaion of daa fo elaion daabases uzzy Ses and Sysems ol 7 pp 3-6 98 [4] R P Buckles and E Pey uzzy daabases and hei applicaions uzzy Infom Decision Pocesses ol pp 36-37 98 [5] R P Buckles and E Pey Unceainy models in infomaion and daabase sysems J Infom Sci ol pp 77-87 985 [6] H Busince P Buillo and Soia Auomophisms negaions and implicaion opeaos uzzy Ses and Sysems ol 34 pp 09-9 003 [7] H Busince M Pagolaa and E Baenchea Consucion of fuzzy indices fom fuzzy disubsehood measues: Applicaion o he global compaison of images Infom Sci ol 77 pp 906-99 007 ISSN: 998-4464 96
INTERNATIONAL JOURNAL O CIRCUITS SYSTEMS AND SIGNAL PROCESSING Volume 08 [8] G Chen uzzy Logic in Daa Modeling Semanics Consains and Daabase Design MA: Kluwe Academic Publishes 998 [9] J odo Conaposiie symmey of fuzzy implicaions uzzy Ses and Sysems ol 69 pp 4-56 995 [0] J odo and M Roubens uzzy Pefeence Modelling and Mulicieia Decision Suppo Kluwe Academic Publishes 994 [] E Klemen R Mesa and E Pap Tiangula Noms Kluwe Academic Publishes 000 [] G Kli and B Yuan uzzy Ses and uzzy Logic Theoy and Applicaions Penice Hall 995 [3] R C T Lee uzzy Logic and he Resoluion Pinciple J Assoc Compu Mach ol 9 pp 09-9 97 [4] H Reichenbach Wahscheinlichkeislogic Ekennnis ol 5 pp 37-43 935 [5] Y Shi A Deep Sudy of uzzy Implicaions PhD disseaion aculy of Science Ghen Uniesiy Ghen 009 [6] M I Soza and A Yazici A complee axiomaizaion fo fuzzy funcional and mulialued dependencies in fuzzy daabase elaions uzzy Ses and Sysems ol 7 pp 6-8 00 [7] E Tillas and L Valede On some funcionally expessable implicaions fo fuzzy se heoy in Poc 3 d Ine Semina on uzzy Se Theoy Linz 98 pp 73-90 [8] E Tillas and L Valede On implicaion and indisinguishabiliy in he seing of fuzzy logic in Managemen decision suppo sysems using fuzzy ses and possibiliy heoy Cologne 985 pp 98- [9] R R Yage An appoach o infeence in appoximae easoning Inena J Man-Machine Sudies ol 3 pp 33-338 980 [0] R R Yage On some new classes of implicaion opeaos and hei ole in appoximae easoning Infom Sci ol 67 pp 93-6 004 [] L A Zadeh Similaiy elaions and fuzzy odeings Infom Sci ol 3 pp 77-00 970 ISSN: 998-4464 97