A Fuzzy Logic Based Resolution Principal for Approximate Reasoning

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1 Journal of Software Engneerng and pplcatons, 07, 0, ISSN Onlne: ISSN Prnt: Fuzzy Logc ased Resoluton Prncpal for pproxmate Reasonng lex Tserkovny Dassault Systemes, oston, M, S How to cte ths paper: Tserkovny,. (07 Fuzzy Logc ased Resoluton Prncpal for pproxmate Reasonng. Journal of Software Engneerng and pplcatons, 0, Receved: ugust 6, 07 ccepted: September 5, 07 Publshed: September 8, 07 Copyrght 07 by author and Scentfc Research Publshng Inc. Ths work s lcensed under the Creatve Commons ttrbuton Internatonal Lcense (CC Y Open ccess bstract In ths artcle, we present a systemc approach toward a fuzzy logc based formalzaton of an approxmate reasonng methodology n a fuzzy resoluton, where we derve a truth value of from both values of and by some mechansm. For ths purpose, we utlze a t-norm fuzzy logc, n whch an mplcaton operator s a root of both graduated conjuncton and dsjuncton operators. Furthermore by usng an nverse approxmate reasonng, we conclude the truth value of from both values of and, applyng an altogether dfferent mechansm. current research s utlzng an approxmate reasonng methodology, whch s based on a smlarty relaton for a fuzzfcaton, whle smlarty measure s utlzed n fuzzy nference mechansm. Ths approach s appled to both generalzed modus-ponens/modus-tollens syllogsms and s well-llustrated wth artfcal examples. Keywords Fuzzy Logc, Deducton, Fuzzy Resolvent, Implcaton, Dsjuncton, Conjuncton, ntecedent, Consequent, Modus-Ponens, Modus-Tollens, Fuzzy Condtonal Inference Rule. Introducton Ths study s a contnuaton of a research, whch s based on a proposed t-norm fuzzy logc, presented n []. Here we also use an automated theorem provng, where a resoluton prncpal s a rule of an nference, leadng to a refutaton theorem-provng technque. pplyng the resoluton rule n a sutable way, t s possble to check whether a propostonal formula s nversally Vald (V and construct a proof of a fact that relatve consequent s frst-order formula s V or non V. In 965, J.. Robnson [] ntroduced the resoluton prncple for frst-order logc. fuzzy resoluton prncpal, n ts part, was ntroduced by M. DOI: 0.436/jsea Sep. 8, Journal of Software Engneerng and pplcatons

2 . Tserkovny Mukadono [3]. Takng nto account the above mentoned, we present the followng. Defnton [3]. fuzzy resolvent of two fuzzy clauses C and C, contanng the complementary lterals x and x respectvely, s defned as R C, C = L L (. x where C = x L and C = x L. L, L are fuzzy clauses, whch don t contan x and respectvely. The operator s understood as the dsjuncton of the lterals present n them. It s also a logcal consequence of C C. resoluton deducton of a clause C from a set S of clauses s a fnte sequence of clauses C C,, Cn = C such that each C s ether a member of or s a resolvent of two clauses taken from the resoluton prncple n propostonal logc we deduce that, f S s true under some truth valuaton v, then vc ( = TRE for all, and n partcular, v( C = TRE []. Example : Here s a dervaton of a clause from a set of clauses presented by means of a resoluton Tree n Fgure. In frst order logc, resoluton condenses the tradtonal syllogsm of logcal nference down to sngle rule. smple resoluton scheme s: ntecedent : a b ntecedent : b (. Consequent: a. The entre hstorcal analyss of ths approach toward applyng of a resoluton prncpal to a logcal nference s presented n [4].. asc Theoretcal spects Frst, Let us consder that, ', and ' are fuzzy concepts represented by fuzzy sets n unverse of dscourse,, V and V, respectvely and correspondent fuzzy sets be represented as such µ : [ 0,], V µ : V [ 0,], where = µ ( u u, = µ ( v v (. V Fgure. Resoluton Tree. DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

3 . Tserkovny Gven (. let us formulate the argument form of smple Fuzzy Resoluton as follows (. Gven (. the scheme for Generalzed Fuzzy Resoluton looks lke that ntecedent : If x s OR y s ntecedent : y s ' (.3 Consequent: x s '. In case (.3, we can say that the Dsjunctve Syllogsm holds f ' s close to not, whereas ' s close to. The second approach s called Inverse pproxmate Reasonng. Its scheme looks lke that: ntecedent : If x s, then y s ntecedent : y s ' (.4 Consequent: x s '. We shall transform the dsjuncton form of rule nto fuzzy mplcaton from fuzzy logc, ntroduced n [], or fuzzy relaton and apply the method of nverse approxmate reasonng to get the requred resolvent. However, n the case of complex set of clauses the method s not sutable. Hence, we nvestgate for another method of approxmate reasonng based on smlarty to get the fuzzy resolvent. Let us consder Generalzed Fuzzy Resoluton frst. The key operaton used n ths method s dsjuncton. The dsjuncton operaton s presented n Table S and, beng appled to above ntroduced fuzzy sets and, looks lke that [], + < = + (.5 Whereas correspondent conjuncton operaton s also presented n Table S and looks lke that [], + > = 0, + (.6 Takng nto account (.5 and (.6 and the fact that =, let us formulate the followng Lemma. If there are two fuzzy clauses C C and R( C C s a fuzzy resolvent of them wth keyword x, then the followng nequalty holds T( C C T R C C (.7 ( where T(x s a truth value of an x. Proof: Snce C = x L and C = x L, where L L 0, then DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

4 . Tserkovny whereas from (. R( C C = L L. From (.8 let defne the followng values of truth: ( ( = ( x x ( L x ( x L ( L L C C = x L x L (.8 x x, x + x > T = ( x x = 0, x + x Snce x + x then from (.9 we are gettng that In a meantme from the same (.8 we have From (. let s take a note that (.9 T 0 (.0 L x, L > x, T = ( L x = 0, L x Sup T L x (. = = 0 (. lso from (. let x L, x + L > T3 = ( x L = 0, x + L (.3 Contnung from (. let L L, L+ L > T4 = ( L L = 0, L+ L (.4 nd fnally from (. we have L L, L+ L < L L = L+ L (.5 Let s rewrte (.8 n the followng way T( C C = T T T3 T4 (.6 From (.6 gven both (.0 and (. we have 0, T < 0, T < T = T T = 0 T = = (.7 T T = Takng nto account (. fnally we are gettng T 0, T < = L = x = 0. (.8 Furthermore from the same (.6 let T3 T4, T3 + T4 < T34 = T3 T4 = T3 + T4 Note that from (.8 x = 0, whch means that from (.3 T3 0 from (.9 we have (.9, therefore DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

5 . Tserkovny T 34 0, T4 < = T4 (.0 ut from (.4 and (.0 T 4 = when L L = or equally L L = = (. Fnally from (.6, gven (.8 and (.0, we are gettng the followng ( T T34, T + T34 < 0, T + T34 < T C C = T T34 = = (. T + T34 T + T34 Takng nto account (. from (.5 and (. R( C C = L L (.3 From (. and (.3 we are gettng T( C ( C T R C C (Q. E. D.. Corollary Let C, C are two fuzzy clauses. If s true: Proof: Frst note that for T ( ( R C C < 0.5, then the followng ( T( C C < T R C C (.4 [ ] L, L 0, L + L < L L < 0.5 (.5 From (. and (.5, gven (.5 we are gettng the followng ( L L, L+ L < 0.5, L+ L < T( R C C = L L = = L+ L L+ L Frst from (.4 we have the followng 0, T3 < T4 0 L + L T34 = T3 0= T3 = (.6 (.7 ut from (.3 we have the followng T 3 = x = L = L 0, but L + L < T 0, therefore 3 T34 0 T T34 0, e.. < 0.5, n other words the followng s true. T( C C < T R C C (Q. E. D.. ( Corollary Let C, C are two fuzzy clauses. If T ( R( C C 0.5, then the followng s true: T( C C = T( R( C C (.8 Proof: From (. we have 0, T + T34 < T( ( C C =, whch means that T + T34 T( C C 0.5. From (.4, (.7 and (.9 we are gettng DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

6 . Tserkovny T( C C L = L =, but from (.6 T R( C C L = L =, whch means that T C C = T R C, C (Q. E. D.. ( ( ased on above presented results let formulate the followng Theorem (Deducton: Let,, n and are fuzzy concepts. fuzzy concept s a logcal consequent of,,, n f and only f the followng nequalty holds f Or equally T ( < T. (.9 { and and and } n (.30 Whch means a fuzzy formula (.30 s V. Note, that a fuzzy formula F s called V, f T( f 0.5. efore provdng a proof of ths theorem let us gve the followng Defnton fuzzy concept s a logcal consequent of,, n,.e.,, n =, f and only when V of,, n has caused V of a fuzzy concept, n other words the followng s true. T 0.5 ( T 0.5, where = mn { } Proof: Let fuzzy concept s a logcal consequent of fuzzy concepts,, n. If fuzzy concepts,, n are V,.e. T( 0.5, = n, then n accordance wth Defnton a fuzzy concept s also V,.e. T. n 0.5 mplcaton operator n a fuzzy logc, used n ths artcle s defned as the followng (see Table S and Table S T ( T, [ ] ( n T T = T (, T T > T Let us consder a set of cases. (.3 If ( T T, then T T, therefore V of a fuzzy formula (.30 s apparent. If ( T > T, then T T T T(, but snce T( 0.5, = n and T( 0.5, then T( < 0.5, = n and T ( < T. Therefore t s obvous then DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

7 . Tserkovny T ( T < 0.5,.e. a fuzzy formula (.30 s not V, a logcal contradcton takes place. If * ( = T * < 0.5, then T = mn{ T( } 0.5 < a fuzzy sub = n formula (antecedent from (.30 and and and n s not V,.e. contradctve, but n a meantme T, therefore 0.5 ( T ( T T T V. If,.e. a fuzzy formula (.30 s T 0.5, ( < T < 0.5 (.3 and f ( T ( T T T, therefore formula (.30 s V, whereas f T ( > T, then agan ( T ( T T T and gven condtons (.3 we have T 0.5 ( T T 0.5, whch means that a fuzzy formula (.30 s not V or s contradctve. t last let a fuzzy formula (.30 be V and also let T ( T. Then f a fuzzy sub formula (antecedent from (.30 and and and n s also V,.e. T 0.5, then from (.3 we have T ( ( T T 0.5. In other words a fuzzy concept s V. Therefore from Defnton a fuzzy concept s a logcal consequent of fuzzy concepts,, n (Q. E. D.. ased on these results we formulate the followng Theorem If there are two fuzzy clauses C C and R( C C s a fuzzy resolvent of them wth keyword x, then R( C C s logcal consequent of both C and C.e. C, C = R C, C (.33 DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

8 . Tserkovny Proof: Let C = x L and C = x L, where L L 0, whereas R C, C = L L. y Defnton and n accordance wth (.5 ( L L, L+ L < T( R C C = L+ L ( x L x + L < T C = x + L ( x L, x L T C = + < x + L (.34 (.35 (.36 Let C, C are both V,.e. T C 0.5 and T C 0.5. Snce from (.35 and (.36 the followng s takng place x L < 0.5 x + L < and x L < 0.5 x + L < then V of C, C s n realty means that x + L x + L Takng nto account that T( x T( x (.37 = let sum both nequaltes (.37 together and get the followng L+ L. From (.34 T ( ( R C C = L+ L T( R( C C 0.5.e. R( C C s V. Therefore by Defnton we are gettng a fact that f C C are both V, and then R C, C s also V. (Q. E. D.. ( Let us present some consderatons about usng a noton of smlarty, whch plays a fundamental role n theores of knowledge and behavor and has been dealt wth extensvely n psychology and phlosophy. careful analyss of the dfferent smlarty measures reveals that t s mpossble to sngle out one partcular smlarty measure that works well for all purposes. We wll utlze a consstent approach toward defnton of a smlarty measure, based on the same fuzzy logc we used above []. ut ths tme we wll use the operaton Equvalence (see Table S. Suppose be an arbtrary fnte set, and I ( be the collecton of all fuzzy subsets of. For, I (, a smlarty ndex between the par {, } s denoted as S(, ; or smply S(, whch can also be consdered as a functon S: I ( I( [ 0,]. In order to provde a defnton for smlarty ndex, a number of factors must be consdered. Defnton 3 functon S(, defnes a smlarty between fuzzy concepts, f t satsfes the followng axoms: P. S(, = S(,, S(, = S(,, where =, P. 0 S(, P3. S(, = ff =, P4. For two fuzzy concepts,,, S(, = 0 mn ( µ ( u, µ ( u = 0, u,.e. =, C C S, C mn S,, S, C. P5. ( DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

9 . Tserkovny Lemma If a functon S(, s defned as operaton equvalence from Table S then t could be consdered as a smlarty measure. Proof: From Table S we have P. From (.38 b aa, < b, S(, = a= b, ( a ba, > b, b a, a< b, a ba, > b, S(, = a= b, = a= b, S, = S, a b, a b, > ( b aa, < b, whereas b a, b< a, S(, = b= a, = S, ( b ab, > a, (.38 (.39 (.40 xoms P and P3 are trvally satsfed by (.38. P4. From (.38 S, 0 b= a= 0 or b= 0, a= = (.4 P5. From (.38 Case: a< b< c From (.38 and (.4 we have c bb, < c, S( C, = b= c, ( b cb, > c, (, ( ; (, ( ; (, ( (.4 S = b a S C = c b S C = c a (.43 From (.43 ( c a< ( c b S( C, < S( C,, whereas ( c a< ( b a S(, C < S(,. Snce S( C, S( C, S(, C < S(,, then the followng s also true: S(, C mn S(,, S(, C Case: a> b> c From (.38 and (.4 we have < and (.44 (, ( ; (, ( ; (, ( S = a b S C = b c S C = a c (.45 From (.45 ( a c< ( a b S(, C < S(,, whereas ( a c< ( b c S(, C < S(,. Snce S( C, S( C, S(, C < S(,, then the followng s also true: S(, C mn S(,, S(, C (Q. E. D.. < and DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

10 . Tserkovny To llustrate our further research before gvng the defnton of smlarty ndex, we wll present couple examples. Let and be two normal fuzzy sets defned over the same unverse of dscourse and presented by unmodal lnear monotonc membershp functons and supp{ } Card ; supp{ } Card. Correspondent lngustc scale could consst of the terms lke { SMLL, MEDIM, LRGE }. Let us consder the followng cases. labeled SMLLER THN LRGE = 0. u u u u u u u u +.0 u u u Whereas labeled LRGE = 0.0 u + 0. u + 0. u u u u u u u u +.0 u From (.38 the smlarty matrx (, labeled MEDIM nd labeled LRGE = 0.0 u + 0. u + 0. u u u u u u u u +.0 u S would look lke that = 0.5 u u u u u +.0 u u u u u u From (.38 the smlarty matrx S(, would look lke that (for smplcty sake we show elements of a matrx wth sngles only = 0.5 u u u u u +.0 u 3 labeled MEDIM u u u u u nd labeled SMLL =.0 u u u u u u u u + 0. u + 0. u u From (.38 the smlarty matrx (, 4 labeled LRGE nd labeled SMLL =.0 u u u u u u S would look lke that = 0.0 u + 0. u + 0. u u u u u u + 0. u + 0. u u From (.38 the smlarty matrx (, 5 labeled MEDIM nd labeled MEDIM = 0.5 u u u u u +.0 u u u u u u u u u u +.0 u S would look lke that = 0.5 u u u u u +.0 u u u u u u From (.38 the smlarty matrx S(, would look lke that Let consder smlarty measure as a matrx (, ; ; ; S = s = n j = n n = Card. We are presentng the followng j Proposton Snce and are two normal fuzzy sets, then [ ] = =. Then the followng functon could be cons-, j n s ; n Card j DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

11 . Tserkovny dered as a smlarty ndex sj = + n; j= j + n SI (, = max ; n = Card. n In Table there are three sets of pars of ndces, j + n ; j j + n ; s = ; n = Card. j * = j * = 3, Gr, j =, 4;3,5; 4,6;5,7;6,8;7,9;8,0;9, 8 SI ( Gr {, j} = = = 0, j = 0, Gr, j = SI Gr, j = 0 For { } { } For { } ( { } 3 For j Gr3{ j} SI ( Gr3{ j} (.46 = = 9,, =, = 0. From (.46 we are gettng SI (, = max SI ( Grk {, j } = 0.8. Ths value k= 3 perfectly matches our ntuton and percepton of a closeness of terms SMLLER THN LRGE and LRGE and membershp functons of correspondent fuzzy sets. In Table there are sx sets of pars of ndces, j + n ; j j + n ; s = ; n = Card. j For * = j * = 6, Gr {, j} = {, 7;3,8; 4, 9;5,0; 6,} 5 SI ( Gr {, j} = = = 7, j = 0, Gr, j = SI Gr, j = 0 For { } ( { } 3 For j Gr3{ j} SI ( Gr3{ j} 4 For j Gr4{ j} SI ( Gr4{ j} 5 For j Gr5{ j} SI ( Gr5{ j} 6 For j Gr6{ j} SI ( Gr6{ j} = 8, = 9,, =, = 0 = 9, = 8,, =, = 0 = 0, = 7,, =, = 0 = = 6,, =, = 0 Table. Smaller Than Large Large. S(, DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

12 . Tserkovny Table. Medum Large. S(, Table 3. Medum Small. S(, From (.46 SI ( SI ( Grk { j }, = max, = 0.5. Ths value s n a mddle k= 6 of a scale [0, ] and also perfectly matches our ntuton and percepton of an average closeness of terms LRGE and MEDIM and membershp functons of correspondent fuzzy sets. Smlarly n Table 3 there are sx sets of pars of ndces j + n j j + n s = n = Card., ; ; ; j = j = Gr j = For 6, {, } { 7, ;8, 3;9, 4;0, 5; 6} 5 SI ( Gr {, j} = = For j Gr{ j} SI ( Gr{ j} 3 For j Gr3{ j} SI ( Gr3{ j} = 5, =,, =, = 0 = 4, = 3,, =, = 0 DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

13 4 For j Gr4{ j} SI ( Gr4{ j} 5 For j Gr5{ j} SI ( Gr5{ j} 6 For j Gr6{ j} SI ( Gr6{ j} = 3, = 4,, =, = 0 =, = 5,, =, = 0. Tserkovny = = 6,, =, = 0. From (.46 we are gettng SI (, = max SI ( Grk {, j } = 0.5. Ths value k= 6 s also n a mddle of a scale [ 0, ] and also perfectly matches our ntuton and percepton of an average closeness of terms SMLL and MEDIM and membershp functons of correspondent fuzzy sets. In Table 4 there are eleven sets of pars of ndces, j + n ; j j + n ; s = ; n = Card. j = j = Gr, j = SI Gr, j = 0 For { } ( { } = 0, j =, Gr{, j} = SI ( Gr{, j} = 0 3 = 9, j = 3, Gr3{, j} = SI ( Gr3{, j} = 0 4 = 8, j = 4, Gr4{, j} = SI ( Gr4{, j} = 0 5 = 7, j = 5, Gr5{, j} = SI ( Gr5{, j} = 0 6 = 6, j = 6, Gr6{, j} = SI ( Gr6{, j} = 0 7 For = 5, j = 7, Gr7{, j} = SI ( Gr7{, j} = 0 8 For = 4, j = 8, Gr8{, j} = SI ( Gr8{, j} = 0 9 For = 3, j = 9, Gr9{, j} = SI ( Gr9{, j} = 0 0 For j Gr0 { j} SI ( Gr0 { j} For j Gr{ j} SI ( Gr{ j} =, = 0,, =, = 0 = =, =, = 0. From (.46 we are gettng SI ( SI ( Grk { j }, = max, = 0. Ths value k= also perfectly matches our ntuton and percepton of a fact that terms SMLL and LRGE has nothng n common. Table 4. Large Small. S(, DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

14 . Tserkovny Table 5. Medum Medum. S(, In Table 5 there are twelve sets of pars of ndces j n j j n s n Card, + ; + ; = ; =. j j Gr j SI Gr j For = = {, } = ( {, } = 0 = 0, j =, Gr{, j} = SI ( Gr{, j} = 0 3 = 9, j = 3, Gr3{, j} = SI ( Gr3{, j} = 0 4 = 8, j = 4, Gr4{, j} = SI ( Gr4{, j} = 0 5 = 7, j = 5, Gr5{, j} = SI ( Gr5{, j} = 0 6 = 6, j = 6, Gr6{, j} = SI ( Gr6{, j} = 0 7 For = 5, j = 7, Gr7{, j} = SI ( Gr7{, j} = 0 8 For = 4, j = 8, Gr8{, j} = SI ( Gr8{, j} = 0 9 For = 3, j = 9, Gr9{, j} = SI ( Gr9{, j} = 0 0 For j Gr0 { j} SI ( Gr0 { j} For j Gr{ j} SI ( Gr{ j} =, = 0,, =, = 0 = =, =, = 0. For * = j * = Gr {, j} = {, ;3,3; 4, 4;5,5;6,6;7,7;8,8;9,9;0,0;} 0 SI ( Gr {, j} = =. 0 From (.46 we are gettng SI (, = max SI ( Grk {, j } =. Ths value s k= a confrmaton of a fact that both fuzzy sets are dentcal. 3. Generalzed Fuzzy Resoluton ased pproxmate Reasonng Let us remnd that the scheme for Generalzed Fuzzy Resoluton (.3 looks lke that DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

15 . Tserkovny ntecedent: If x s OR y s ntecedent: y s (3. Consequent: x s. Frst consder the followng classcal logc equvalence a b b a b a (3. The classcal logc equvalence (3. can be extended n fuzzy logc wth mplcaton and negaton functons. We use the same fuzzy logc, whch operatons are presented n Table S. Let us frst proof that (3. holds. Snce nd because a ba, + b< b ab, + a< a b= b a= a b b a a+ b b+ a a ba, > b, a b= therefore a b b a, b> a, b ab, + a< b a = = a b b a b+ a (3.3 (3.4 oth (3.3 and (3.4 proofs that classcal logc equvalence (3. holds. It s very mportant to show that we transform Generalzed Fuzzy Resoluton rule (3. nto ts equvalent form ntecedent: If y s then x s ntecedent: y s ' (3.5 Consequent: x s '. Let us formulze an nference method for a rule (3.5. Followng a well-known pattern, establshed a couple of decades ago and the standard approaches toward such formalzaton, presented and extensvely used n [5] [6] [7] [8] [9], let and V be two unverses of dscourses and correspondent fuzzy sets be µ : 0, V µ : V 0,, where represented as such [ ], [ ] µ, µ (3.6 = u u = v v Whereas gven (3.6 a bnary relatonshp for the fuzzy condtonal proposton of the type: If y s then x s for a fuzzy logc s defned as ( R D y D x = V ( µ ( v ( vu, µ ( u ( vu, = V V (( µ ( v µ ( u ( vu, = V Gven an mplcaton operator from Table S expresson (3.7 looks lke V (3.7 DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

16 . Tserkovny µ ( v µ ( u µ ( v > µ ( u ( µ ( v µ ( u = µ ( v µ ( u. µ ( v µ ( u, µ ( u + µ ( v < = µ ( u + µ ( v.,, R D y = one can ob- It s well known that gven a unary relatonshp ( tan the consequence ( (CRI to R( D ( y and R( D( y, D( x of type (3.7: R( D( x = R( D( y R( D( y, D( x = µ ( µ µ (3.8 R D x by applyng compostonal rule of nference V vv V ( (, v v v u vu ( = µ v µ v µ u v In order that Crteron I (see ppendx s satsfed, that s ( from (3.9 the equalty ( ( (3.9 R D x = µ v µ v µ u = µ u (3.0 vv must be satsfed for arbtrary u and n order that the equalty (3.0 s satsfed, t s necessary that the nequalty ( ( µ v µ v µ u µ u (3. holds for arbtrary u and v V. Let us defne new methods of fuzzy condtonal nference of the type (3.5, whch requres the satsfacton of Crtera I-IV from ppendx. Theorem 3 If fuzzy sets µ : [ 0,], V µ : V [ 0,] and R( D( y, D( x s defned by (3.7, where µ ( v µ ( u µ ( v < µ ( u ( µ ( v µ ( u = µ ( v µ ( u. then Crtera I, II, III and IV- are satsfed. Proof: are defned as (3.6,, For Crtera I-III let R( D ( y = ( > 0 then R( D( x = R( D( y, D( x = µ ( µ µ V vv V ( (, v v v u vu ( = µ v µ v µ u u V, V V V V = V; V V = v V µ v < µ u ; v V µ v µ u From (3.3 and gven subsets from (3.4 we have R D x v v u u v u (3. (3.3 (3.4 ( = µ ( µ µ µ. (3.5 vv vv DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

17 . Tserkovny Let us ntroduce the followng functon (as a part of mplcaton operaton Then the followng s takng place: (, µ µ µ µ f vu = v u v < u. (3.6 ( v, ( v f( vu,, µ > µ µ v V µ ( v f( vu, = f vu,, v f vu,, (3.7 Snce from (3.6 µ ( u + µ ( v < f( vu, < 0.5, but ( v [ 0,] therefore from (3.7 we have ( ( x R D From (3.6-(3.9 we have µ, v V µ v f vu, = f vu, < 0.5 (3.8 v V µ v = µ v (3.9 (3.5 = µ ( v u = µ ( u u =. (Q. E. D.. vv For Crtera IV- let R( D ( y = then R( D( x = R( D( y, D( x = ( µ ( µ µ V v v v u vu, vv V ( µ ( µ µ = v v u u From (3.0 and gven subsets from (3.4 we have (3.0 = µ ( v ( µ ( v µ ( u u µ ( v u (Q. E. D. (3. v V v V = µ ( u u µ ( v u = µ ( u u = v V v V To llustrate these results we wll present couple examples. Example Let and V be two unverses of dscourses and correspondent fuzzy sets µ : 0, V µ : V 0, ; related are represented as n (3.6 [ ], [ ] lngustc scale could consst of the terms lke { SMLL, MEDIM, LRGE }. Let us consder the followng cases. = 0.0 u+ 0. u + 0. u u u u6 labeled LRGE u u u u +.0 u lke nd labeled SMLL The negaton of a fuzzy set would look lke =.0 v v v v v v v v + 0. v + 0. v v = 0.0 v + 0. v + 0. v v v v v v v v +.0 v The bnary relatonshp matrx ( R D y D x of a type (3.7 would look DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

18 . Tserkovny \ Let labeled very SMLL =.0 v v v v v v v v v v v pplyng (3.3 ( = ( = µ ( µ µ R D x R D y D x V V ( (, v v v u uv =.0 u u u u u u u u u u u = Example labeled LRGE = 0.0 u + 0. u + 0. u u u u u u u u +.0 u nd also labeled LRGE = 0.0 v + 0. v + 0. v v v v v v v v +.0 v lke The negaton of a fuzzy set would look lke =.0 v v v v v v v v + 0. v + 0. v v The bnary relatonshp matrx ( R D y D x of a type (3.7 would look \ DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

19 . Tserkovny Contnued pplyng (3.3 ( = ( = µ ( µ µ R D x R D y D x V v v v u uv, V = 0.0 u + 0. u + 0. u u u u u u u u0 +.0 u = Let us revst the fuzzy condtonal nference rule (3.5. It wll be shown that when the membershp functon of the observaton s contnuous, then the concluson depends contnuously on the observaton; and when the membershp functon of the relaton R(, s contnuous then the observaton has a contnuous membershp functon. We start wth some defntons. fuzzy set wth membershp functon µ : R [ 0,] = I, s called a fuzzy number f s normal, contnuous, and convex. The fuzzy numbers represent the contnuous possblty dstrbutons of fuzzy terms of the followng type = µ x x R Let be a fuzzy number, then for any θ 0 we defne ϖ ( θ the modulus of contnuty of by max ( x ( x ϖ θ = µ µ. (3. x x θ n -level set of a fuzzy nterval s a non-fuzzy set denoted by [ ] s defned by [ ] = { t R µ ( t } for ( 0,] and [ ] = cl ( suppµ for = 0. Here we use a metrc of the followng type ([ ] [ ] and D, = sup d,, (3.3 [ 0,] where d denotes the classcal Hausdorff metrc expressed n the famly of compact subsets of R,.e. ([ ],[ ] max ( (, ( ( { } d = a b a b. whereas [ ] = a(, a(, [ ] b, b and both have fnte support { x x } s defned as = when the fuzzy sets,, n then ther Hammng dstance DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

20 . Tserkovny n. = (, = µ µ H x x In the sequel we wll use the followng lemma. Lemma 3 [0] Let δ 0 be a real number and let, be fuzzy ntervals. If D(, δ, then sup µ ( t µ ( t max ϖ ( δ, ϖ ( δ. t R { } Consder the fuzzy condtonal nference rule (3.5 wth dfferent observatons and : ntecedent : If y s then x s ntecedent : y s Consequent: x s ntecedent : If y s then x s ntecedent : y s Consequent: x s ccordng to the fuzzy condtonal nference rule (3.5, the membershp functons of the conclusons are computed as Or ( u = ( v ( ( v ( u µ µ µ µ ; v R ( u = ( v ( ( v ( u µ µ µ µ v R ( u ( v ( ( v ( u µ = sup µ µ µ ; v R ( u ( v ( ( v ( u, (3.4 µ = sup µ µ µ v R The followng theorem shows the fact that when the observatons are closed to each other n the metrc D (. of (3.3 type, then there can be only a small devaton n the membershp functons of the conclusons. Theorem 4 (Stablty theorem Let δ 0 and let, be fuzzy ntervals and an mplcaton operaton n D, δ then the fuzzy condtonal nference rule (3.5 s from Table S. If sup µ ( u µ ( u max { ϖ ( δ, ϖ ( δ } Proof: u R Gven an mplcaton operaton n the fuzzy condtonal nference rule (3.5 s from Table S for the observaton we have V, V R V V = V; V V = µ µ ; µ µ ( µ ( v v (( µ ( v µ ( u ( vu, V V µ ( v (( µ ( v µ ( u u v V v < u v V v u = R D y D x = = v V (3.5 DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

21 . Tserkovny From (3.3 and gven subsets from (3.4 we have = µ ( v ( µ ( v µ ( u u µ ( v u vv vv. (3.6 Then from (3.6 the followng s takng place: ( v, ( v f( vu,, µ > µ µ v V µ ( v f( vu, = f vu,, v f vu,, (3.7 Snce from (3.6 µ ( u + µ ( v < f( vu, < 0.5, but ( v [ 0,] therefore from (3.7 we have From (3.7-(3.9 we have,, 0.5 µ, v V µ v f vu = f vu < (3.8 v V µ v = µ v (3.9 (3.6 = µ ( v u µ ( u u = = vv From (3.8 and (3.9, we see that the dfference of the values of conclusons for both and observatons for arbtrary fxed v R µ ( v µ ( v s defned as follows v V µ v µ v = ( u ( u µ µ 0, v V µ v µ v = µ u µ u. Therefore from Lemma we have { } ( v ( v ( u ( u sup µ µ = sup µ µ max ϖ δ, ϖ δ (Q. E. D. v R u R Theorem 5 (Contnuty theorem Let bnary relatonshp R( vu, = ( µ ( v µ ( u be contnuous. Then ϖ δ ϖr δ for each δ 0. Proof: Let δ 0 be a real number and let u u R such that u u δ. From ϖ δ = max µ u µ u. Then s contnuous and (3. we have u u δ, ( u ( u = sup ( v ( ( v ( u µ µ µ µ µ v R ( v ( ( v ( u sup µ µ µ v R ( (( ( ( sup µ v µ v µ u µ v µ u v R ( v ( u u sup ( v sup µ ϖ µ ϖ δ = ϖ δ v R R R R v R (Q. E. D.. Results from Theorem 4 and Theorem 5 could be used for formulatng another smlarty measure, based on Hammng dstance between two fuzzy sets DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

22 . Tserkovny Fgure. Terms very and more or less. and, whch presented by non-lnear membershp functons,.e. n ( u ( u µ µ SI n Card n = (, = ; =. (3.30 For nstance let us apply (3.9 to fuzzy sets from Example (see Fgure : =.0 v+ 0.9 v v v v v6 Let labeled SMLL v v + 0. v + 0. v v nd labeled very SMLL =.0 v v v v v v v v v v v SI, = It s mportant to pay attenton to a fact that SI, = 0.85 from the same Example whch confrms results of both Theorem 4 and Theorem Generalzed Modus Tollens ased Inverse pproxmate Reasonng Let us remnd that the scheme for Generalzed Modus Tollens (.3 looks lke that Frst consder classcal logc equvalence ntecedent : If x s, then y s ntecedent : y s (4. Consequent: x s. a b b a (4. The classcal logc equvalence (4. can be extended n fuzzy logc wth mplcaton and negaton functons. We use the same fuzzy logc, whch operatons are presented n Table S. Let us frst proof that (4. holds. DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

23 . Tserkovny Snce a ba, > b, a b= a b (4.3 nd because b a, b> a, b aa, > b, b a= = a b b a a b (4.4 oth (4.3 and (4.4 proofs that classcal logc equvalence (4. holds. It s very mportant to show that we transform Generalzed Modus Tollens rule (4. nto ts equvalent form ntecedent: If y s then x s ntecedent: y s (4.5 Consequent: x s. Let us formulze an nference method for a rule (4.5. Followng a standard approaches toward such formalzaton, let and V be two unverses of dscourses and correspondent fuzzy sets be represented as such µ : [ 0,], V µ : V [ 0,]. Whereas gven (3.6 a bnary relatonshp for the fuzzy condtonal proposton of the type: If y s then x s for a fuzzy logc s defned as ( R D y D x = V ( µ ( v ( vu, ( µ ( u ( vu, V (( µ ( v ( µ ( u ( vu, = V = V Gven an mplcaton operator from Table S expresson (4.6 looks lke ( µ ( v ( µ ( u µ ( v ( µ ( u, µ ( v > µ ( u, = µ ( v µ ( u. µ ( v ( µ ( u, µ ( u > µ ( v, = µ ( u µ ( v. nd agan gven a unary relatonshp R( D ( y consequence ( R( D ( y and R( D( y, D( x of type (4.6: R( D( x = R( D( y R( D( y, D( x = µ ( µ ( µ (4.6 = one can obtan the R D x by applyng compostonal rule of nference (CRI to V vv V ( (, v v v u vu ( ( = µ v µ v µ u v In order that Crteron V (see ppendx s satsfed, that s ( from (3.9 the equalty (4.7 R D x = DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

24 . Tserkovny ( ( µ v µ v µ u = µ u (4.8 vv must be satsfed for arbtrary u and n order that the equalty (3.0 s satsfed, t s necessary that the nequalty ( ( µ v µ v µ u µ u (4.9 holds for arbtrary u and v V. Let us defne new methods of fuzzy condtonal nference of the type (4.6, whch requres the satsfacton of Crtera V-VIII from ppendx. Theorem 6 If fuzzy sets µ : [ 0,], V µ : V [ 0,] and R( D( y, D( x s defned by (4.6, where µ ( v ( µ ( u µ ( u > µ ( v ( µ ( v ( µ ( u = µ ( u µ ( v. are defned as (3.6,, (4.0 then Crtera V,VI,VII and VIII- are satsfed. Proof: For Crtera V-VII let R( D ( y = ( > 0 then R( D( x = R( D( y, D( x = µ ( v v (( µ ( v ( µ ( u ( vu, (4. V V = µ ( v ( ( µ ( v ( µ ( u u vv V V V V V = V; V V = (4. v V µ ( u > µ ( v ; v V µ ( u µ ( v From (4. and gven subsets from (4. we have R( D ( x. (4.3 = µ ( v ( µ ( v ( µ ( u u µ ( v u v V v V Let us ntroduce the followng functon (as a part of mplcaton operaton (, µ ( µ µ µ f v u = v u u > v. (4.4 Then the followng s takng place: ( v, ( v f( vu,, µ > µ µ v V µ ( v f( vu, = f vu,, v f vu,, From (4.5, (4.6 we have DOI: 0.436/jsea Journal of Software Engneerng and pplcatons (4.5 v V µ v = µ v, (4.6 (4.3 = µ ( v u = µ ( u u = vv For Crtera VIII- let ( R D y = then. (Q. E. D..

25 ( = ( = ( µ ( µ ( µ R D x R D y D x V v v v u vu, vv V ( µ ( µ ( µ = v v u u From (4.7 and gven subsets from (4.5 and (4.6 we have R( D ( x = µ ( v f( vu, u µ ( v u vv vv the followng s takng place. Tserkovny (4.7. Snce ( µ ( v f( vu ( µ ( v,. Therefore (4.7 = ( µ ( v u = µ ( u u =. (Q. E. D. vv To llustrate these results we wll present couple examples. We use smlar fuzzy sets as n Examples and. Example 3 Let and V be two unverses of dscourses and correspondent fuzzy sets µ : 0, V µ : V 0, ; related are represented as n (3.6 [ ], [ ] lngustc scale could consst of the terms lke { SMLL, MEDIM, LRGE }. Let us consder the followng cases. = 0.0 u+ 0. u + 0. u u u u6 labeled LRGE u u u u +.0 u lke nd labeled SMLL =.0 v v v v v v v v + 0. v + 0. v v The negatons of fuzzy sets would look lke =.0 u u u u u u u u + 0. u + 0. u u = 0.0 v + 0. v + 0. v v v v v v v v +.0 v The bnary relatonshp matrx ( R D y D x of a type (4.6 would look \ DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

26 . Tserkovny Contnued Let labeled very SMLL =.0 v v v v v v v v v v v pplyng (4.7 ( = ( µ ( µ ( µ R D x R D y D x V V ( (, = v v v u vu = 0.0 u u u u u u u u u u0 +.0 u = ( very LRGE. Example 4 labeled LRGE = 0.0 u + 0. u + 0. u u u u u u u u +.0 u nd also labeled LRGE = 0.0 v + 0. v + 0. v v v v v v v v +.0 v The negatons of fuzzy sets would look lke =.0 u u u u u u u u + 0. u + 0. u u =.0 v v v v v v v v + 0. v + 0. v v The bnary relatonshp matrx ( R D y D x of a type (4.6 would look lke \ DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

27 . Tserkovny pplyng (4.7 ( = ( = µ ( µ ( µ R D x R D y D x 5. Concludng Remarks V v v v u vu, V =.0 u u u u u u u u u u u = In ths artcle, we presented a systemc approach toward a fuzzy logc based formalzaton of an approxmate reasonng methodology n a fuzzy resoluton. We derved a truth value of from both values of and by some mechansm. We used a t-norm fuzzy logc, n whch an mplcaton operator s a root of both graduated conjuncton and dsjuncton operators. We nvestgated features of correspondent fuzzy resolvent, whch was based on ntroduced operators. We proposed two types of Smlarty Measures for both lnear and non-lnear membershp functons. We appled ths approach to both generalzed modus-ponens/modus-tollens syllogsms, for whch we formulated a set of Crteron. The content of ths nvestgaton s well-llustrated wth artfcal examples. References [] Tserkovny,. (07 t-norm Fuzzy Logc for pproxmate Reasonng. Journal of Software Engneerng and pplcatons, 0, [] Robnson, J.. (965 Machne Orented Logc ased on the Resoluton Prncple. Journal of the CM, [3] Mukadono, M. (988 Fuzzy Inference of Resoluton Style. In: Yager, R.R., Ed., Fuzzy Set and Possblty Theory, Pergamon Press, New York, 4-3. [4] Mondal,. and Raha, S. (03 pproxmate Reasonng n Fuzzy Resoluton. Internatonal Journal of Intellgence Scence, 3, [5] lev, R.. and Tserkovny,. (988 The Knowledge Representaton n Intellgent Robots ased on Fuzzy Sets. Sovet Mathematcs Doklady, 37, [6] lev, R.., Mamedova, G.. and Tserkovny,.E. (99 Fuzzy Control Systems, Energoatomzdat, Moscow. [7] lev, R.., Fazlollah,. and lev, R.R., Eds. (004 Soft Computng and Its pplcaton n usness and Economcs. In: Studes n Fuzzness and Soft Computng, Vol. 57, Physca-Verlag, Sprger-Verlag Company, erln Hedelberg. [8] lev, R.. and Tserkovny,. (0 Systemc pproach to Fuzzy Logc Formalzaton for pproxmate Reasonng. Informaton Scences, [9] Tserkovny,. (06 Fuzzy Logc for Incdence Geometry. The Scentfc World Journal, 06, rtcle ID: [0] Fuller, R. and Zmmermann, H.-J. (993 On Zadeh s Compostonal Rule of Infe- DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

28 . Tserkovny rence. In: Lowen, R. and Roubens, M., Eds., Fuzzy Logc: State of the rt, Theory and Decson Lbrary, Seres D, Kluwer cademc Publsher, Dordrecht, DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

29 . Tserkovny ppendx Table S. Fuzzy Logc Operatons. Name Desgnaton Value Tautology I Controversy O 0 Negaton Dsjuncton Conjuncton Implcaton Equvalence Perce rrow Shaffer Stroke a ba, + b< a+ b a ba, + b> 0, a+ b a ba, > b, a b b aa, < b, a = b, ( a ba, > b, a b, a+ b< 0, a+ b a b, a+ b> a+ b Table S. Fuzzy Logc Implcaton Operaton. a b Crteron I ntecedent : If y s then x s ntecedent : y s Consequent: x s. DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

30 . Tserkovny Crteron II ntecedent : If y s then x s ntecedent : y s very Consequent: x s (very. Crteron III ntecedent : If y s then x s ntecedent : y s more or less Consequent: x s (more or less. Crteron IV- ntecedent : If y s then x s ntecedent : y s Consequent: x s unknown Crteron IV- ntecedent : If y s then x s ntecedent : y s Consequent: x s. Crteron V ntecedent : If y s then x s ntecedent : y s Consequent: x s. Crteron VI ntecedent : If y s then x s ntecedent : y s very Consequent: x s very Crteron VII ntecedent : If y s then x s ntecedent : y s more or less Consequent: x s more or less Crteron VIII- ntecedent : If y s then x s ntecedent : y s Consequent: x s unknown DOI: 0.436/jsea Journal of Software Engneerng and pplcatons

31 . Tserkovny Crteron VIII- ntecedent : If y s then x s ntecedent : y s Consequent: x s. Submt or recommend next manuscrpt to SCIRP and we wll provde best servce for you: cceptng pre-submsson nqures through Emal, Facebook, LnkedIn, Twtter, etc. wde selecton of journals (nclusve of 9 subjects, more than 00 journals Provdng 4-hour hgh-qualty servce ser-frendly onlne submsson system Far and swft peer-revew system Effcent typesettng and proofreadng procedure Dsplay of the result of downloads and vsts, as well as the number of cted artcles Maxmum dssemnaton of your research work Submt your manuscrpt at: Or contact jsea@scrp.org DOI: 0.436/jsea Journal of Software Engneerng and pplcatons