Attribute reduction theory and approach to concept lattice

Transcription

1 Scence n Chn Ser F Informton Scences 2005 Vol48 No Attrbute reducton theory nd pproch to concept lttce ZHANG Wenxu 1, WEI Lng 1,2 & QI Jnun 3 1 Insttute for Informton nd System Scences, Fculty of Scence, X n Jotong Unversty, X n , Chn; 2 eprtment of Mthemtcs, Northwest Unversty, X n , Chn; 3 Insttute of Computer Archtecture & Network, X n Jotong Unversty, X n , Chn Correspondence should be ddressed to Zhng Wenxu (eml: wxzhng@mlxtueducn) Receved October 14, 2004; revsed My 1, 2005 Abstrct The theory of the concept lttce s n effcent tool for knowledge representton nd knowledge dscovery, nd s ppled to mny felds successfully One focus of knowledge dscovery s knowledge reducton Ths pper proposes the theory of ttrbute reducton n the concept lttce, whch extends the theory of the concept lttce In ths pper, the udgment theorems of consstent sets re exmned, nd the dscernblty mtrx of forml context s ntroduced, by whch we present n pproch to ttrbute reducton n the concept lttce The chrcterstcs of three types of ttrbutes re nlyzed Keyword: forml context, concept lttce, ttrbute reducton, dscernblty mtrx OI: / The concept lttce, lso clled Glos lttce, ws proposed by Wlle n 1982 [1] A concept lttce s n ordered herrchy tht s defned by bnry reltonshp between obects nd ttrbutes n dt set As n effcent tool of dt nlyss nd knowledge processng, the concept lttce hs been ppled n mny felds, such s knowledge engneerng, dt mnng, nformton serches, nd softwre engneerng [2 7] Most of the reserches on the concept lttce concentrte on such topcs s [8] : Constructon of the concept lttce [9 11], prunng of the concept lttce [2], cquston of rules [10,11], reltonshp between the concept lttce nd rough set [12 14], nd pplctons [2,15 17] Ref [18] proposed reducble ttrbute nd reducble obect from the vewpont of shortenng lne or row In ths pper, we propose the ttrbute reducton theory n the concept lttce Attrbute reducton n the concept lttce s to serch for mnml subset of ttrbutes so tht the new concept lttce s somorphsm wth the rw concept lttce The ttrbute reducton mkes the dscovery of ntl knowledge n the forml context Copyrght by Scence n Chn Press 2005

2 714 Scence n Chn Ser F Informton Scences 2005 Vol48 No eser, nd the representton of the knowledge smpler Ths theory extends the concept lttce theory, nd hs very mportnt sgnfcnce n reserches nd pplctons of the concept lttce The contents of ths pper re s follows: Secton 1 gves the defntons nd propertes bout ttrbute reducton n the concept lttce Secton 2 gves the udgment theorems of reducton Secton 3 ntroduces the dscernblty mtrx nd functon of forml context nd obtns the pproch to reducton Secton 4 studes the chrcterstcs of ttrbutes wth dfferent types Secton 5 concludes the pper 1 efntons nd propertes of ttrbute reducton n the concept lttce Generlly, the dt nlyzed by the concept lttce s represented by forml context, whch s defned s follows efnton 1 [18] We sy s forml context, n whch, U = { x1,, x n } s n obect set, nd ech set, nd ech x ( n) s clled n obect; A = { 1,, m } s n ttrbute ( m) s clled n ttrbute; I s bnry relton between U nd A, I U A If ( x, ) I, then we sy x hs the ttrbute, nd denoted by xi In ths pper, let 1 denote ( x, ) cn be represented by tble whch hs only 0 nd 1 Two opertons re defned on the obect set respectvely n : I, nd 0 denote ( x, ) I Then, forml context X U nd the ttrbute set B A X = { A, x X, xi}, B = { x x U, B, xi} (1) (2) For the ske of smplcty, we wrte x nsted of {} x, x U; nd wrte nsted of {}, A We cll context s cnoncl f x U, x, x A, nd A,, U We ssume tht the contexts we study n the sequel re ll cnoncl contexts before reducton efnton 2 [18] Let ( U, A, I ) be forml context ( X, B) s clled forml concept (or concept), f nd only f X = B nd X = B, where extenson nd B s the ntenson of (X, B) X s clled the The concepts of gven context re nturlly ordered by the subconcept-superconcept relton defned by ( X, B ) ( X, B ) X X ( B B ) (3) Copyrght by Scence n Chn Press 2005

3 Attrbute reducton theory nd pproch to concept lttce 715 The ordered set of ll concepts n ( U, A, I ) s denoted by LU (, AI, ) nd clled the concept lttce of ( U, A, I ) In fct, LU (, AI, ) s complete lttce n whch nfmum nd supremum re gven by [18] ( X, B ) ( X, B ) = ( X X,( B B ) ), = ( X, B ) ( X, B ) (( X X ), B B ) (4) (5) For forml context, ( X1, X2, X U nd B1, B2, B A) [18] : the followng propertes cn be esly proved X X X X, B B B B 2 X X, B B 3 X = X, B = B 4 X B B X 1 2 = ( X X ) X X, ( B B ) = B B ( X X ) X X, ( B B ) B B 7 ( X, X ) nd ( B, B ) re ll concepts efnton 3 Let LU (, A1, I1) nd LU (, A2, I2) be two concept lttces If for ny ( X, B) L( U, A, I ), there exsts ( X ', B') L( U, A, I ) such tht X = X, then LU (, A1, I1) s sd to be fner thn LU (, A2, I2), whch s denoted by LU (, A, I) LU (, A, I ) (6) If LU (, A, I) LU (, A, I ) nd LU (, A, I ) LU (, A, I), then these two concept lttces re sd to be somorphc to ech other, nd denoted by LU (, A, I) LU (, A, I ) (7) In forml context, A, let I = I ( U ) Then ( U,, I ) s lso context For ny X U, X s represented by X n nd X n A A ( U,, I ) It s evdent tht I A = I, X = X X = X = X, nd X X, Theorem 1 Let ( U, A, I ) be forml context, A, Then LU (, AI, ) LU (, I, ) holds Proof ( X, B) L( U,, I ), we hve ( X, X ) L( U, A, I) from the property 7 Then we only need to prove X = X From the property 2, we hve X X ; wwwscchncom

4 716 Scence n Chn Ser F Informton Scences 2005 Vol48 No Furthermore, X X = B X B = X So, X = X The proof s over efnton 4 Let be forml context If there exsts n ttrbute set A such tht LU (, I, ) LU (, AI, ), then s clled consstent set of ( U, A, I ) And further, f d, LU (, { d}, I { }) LU (, AI, ), then s clled reduct of It s esy to prove the followng theorem d The ntersecton of ll the reducts s clled the core of Theorem 2 Let ( U, A, I ) be forml context, A, Then, s consstent set LU (, I, ) LU (, AI, ) efnton 5 Let ( U, A, I ) be forml context The set { s reduct, τ} ( τ s n ndex set) ncludes ll of the reducts n The ttrbute set s dvded nto 3 prts 1 Absolute necessry ttrbute (core ttrbute) 2 Reltve necessry ttrbute d : d A, τ e: e A τ c: c 3 Absolute unnecessry ttrbute n whch, the ttrbute not n the core s clled unnecessry ttrbute It s ether reltve necessry ttrbute or n bsolute unnecessry ttrbute For bsolute necessry ttrbute b, reltve necessry ttrbute c, nd bsolute unnecessry ttrbute d, t s cler tht b c, c d, b d τ τ A b: b τ Theorem 3 The reduct exsts for ny forml context Proof Let ( U, A, I ) be forml context If A, LU (, A { }, I ) A { } LU (, AI, ), then A tself s ts reduct If there s n ttrbute A such tht LU (, A { }, I ) LU (, AI, ), then we study B1 = A {} Further, f b 1 B 1 A { } such tht LU (, B { b}, I ) LU (, AI, ), then B s reduct; otherwse, we study 1 1 B b { 1} B { b} Repetng the bove process, we cn fnd t lest one reduct becuse A s fnte set So, the reduct of must exst Generlly spekng, t s possble tht forml context hs multple reducts Exmple 1 Tble 1 shows forml context A = {,,, bcde,} ( U, A, I ), n whch, U = {1,2,3,4}, There re 6 concepts n ths context They re: (1, bde ), (24, bc ), (13, d), Copyrght by Scence n Chn Press 2005

5 Attrbute reducton theory nd pproch to concept lttce 717 Tble 1 A context of Exmple 1 b c d e (124, b ), ( U, ), (, A), nd lbeled s FC ( = 1,2,,6) respectvely The concept lttce s shown n Fg 1 hs two reducts: 1 = {,, cd}, 2 = {,, bcd} In ths exmple, cd, re bsolute necessry ttrbutes, b, re reltve necessry ttrbutes, nd e s bsolute unnecessry ttrbute The concept lttce of forml context (,, ) U 1 I 1 s shown n Fg 2 It s evdent tht LU (,, I ) LU (, AI, ) 1 1 Fg 1 L( U, A, I ) n Exmple 1 Fg 2 L( U, 1, I ) n Exmple 1 1 Let be forml context Clerly, we hve the followng results: Corollry 1 The core of the forml context s reduct There s only one reduct n the forml context Corollry 2 A s n unnecessry ttrbute A {} s consstent set Corollry 3 A s n element of the core A {} s not consstent set 2 Judgment theorems of consstent sets n concept lttce As we hve known, A s reduct of forml context f nd only f: s consstent set, nd { d} s not consstent set for ny d So we only need to fnd the udgment theorem of consstent set Theorem 4 (Judgment Theorem 1 of consstent sets) Let context, A,, E = A Then, s consstent set ( F E) = ( F ) = F be forml F E, F, wwwscchncom

6 718 Scence n Chn Ser F Informton Scences 2005 Vol48 No Proof Necessty Becuse s consstent set, t s esy to see tht LU (,, I ) LU (, AI, ) For ny F E, F, there must be ( F, F ) L, so C, ( F, C) L( U,, I ), nd then C = F Furthermore, we hve C = F then X = F, so ( F E) = ( F ) = C = F Suffcency If we cn prove tht LU (,, I ) LU (, AI, ), nd = B, ( B ) = X Frst, t s cler tht ( X, B) L, ( X, B ) L( U,, I ), s consstent set So, we need to prove: X = X = B ( B ) = X It s evdent tht B = ( B ) ( B E) If B E =, then B E B ( B E) B = X = B ( B E) = (( B E) ) ( B ) Thus, X = B = ( B ) ( B E) = ( B ) Combnng two fcts bout B E, Second, t should be proved tht X = B = ( B ) If B E, then (( B E) ) = ( B E) holds becuse of B E E So, we cn conclude ( B ) = X Corollry 4 Let ( U, A, I ) be forml context, A, If s consstent set, then ( A ) holds Proof If s consstent set, we get (( A ) ) = ( A ) from Theorem 4 On the other hnd, becuse ( A ), (( A ) ) = ( A ) holds Theorem 5 (Judgment Theorem 2 of consstent sets) Let context, C, C, be forml A,, E = A Then, s consstent set F E, F, C = F Proof Necessty Ths cn be proved by Theorem 4 mmedtely Suffcency It cn be obtned tht C C = F by C F And we know C, holds, so so C F, ( F ) C = F Further, = ( F ) = F estblsh Thus, s consstent set from Theorem 4 Theorem 6 (Judgment Theorem 3 of consstent sets) Let context, ( F ) F F be forml A,, E = A Then, s consstent set e E, Copyrght by Scence n Chn Press 2005

7 Attrbute reducton theory nd pproch to concept lttce 719 C, C, C = e Proof Necessty Ths follows mmedtely from Theorem 5 Suffcency F E, F, let F = { e k τ} From the known condtons, we hve ek F E, Ck, Ck, C k = e k Thus F = ek = C k = C k k τ k τ k τ k Let C = C, then C, C, C = F So, s consstent set from Theorem 5 k τ k Theorem 7 (Judgment Theorem 4 of consstent sets) Let context, be forml A,, E = A Then, s consstent set e E, ( e E) = ( e ) = e Proof Necessty Ths follows mmedtely from Theorem 4 Suffcency As we known, e E, ( e ) = e Let C = e, then C, Theorem 6 C, nd C = e So, t cn be derved tht s consstent set from Theorem 8 (Judgment Theorem 5 of consstent sets) Let context, LU (, E, I E ) A,, E = A Then, s consstent set Proof Necessty Suppose s consstent set, then be forml LU (,, I ) On the other hnd, LU (, AI, ) LU (, E, I E ) becuse of E A, so LU (,, I ) LU (, E, I E ) holds Suffcency Becuse LU (,, I ) LU (, E, I ), nd lso E ( F, F ) L( U, E, I E ) holds, there exsts C, C LU (,, I ), thus C E LU (,, I ) LU (, AI, ) such tht F E, F, = F Fnlly, we get s consstent set from Theorem 5 Exmple 2 For the forml context n Exmple 1, we exmne Theorem 7 Suppose 1 = {, cd}, then E1 = {,,} be It s esy to clculte tht ( F, C) E1 ( ) = =U, wheres = {1,2,4}, they re not equl So 1 = {, cd} s not consstent set In fct, t s the core of the forml context Suppose 2 = {,, cde,}, then E2 = {} b It s esy to clculte tht 2 ( b E ) = b = {1,2,4} So, 2 = {,, cde,} s consstent set Ths result hs been concluded n Exmple 1 wwwscchncom

8 720 Scence n Chn Ser F Informton Scences 2005 Vol48 No Theorem 9 (Judgment Theorem of reduct) Let A,, E = A Then, s reduct = e e E,, nd d, ( d ( E { d})) = ( d ( { d})) d be forml context, ( e E) = ( e ) Proof Necessty s reduct s consstent set ( e E) = ( e ) = e e E, In ddton, suppose be proved tht d,( d ( E { d})) = ( d ( { d})) = d Then t cn { d} s consstent set In fct, we cn study ech element n E { d} usng Theorem 7 e E { d}, f e = d, we hve from ssumpton; f e E nd d e, then e = ( e E) = ( e ( E { d} )) holds; f e E nd d e, then d e holds Thus, e = ( e ) = (( e ( { d})) { d}) In summry, = ( e ( { d})) d = ( e ( { d})) ( d ( { d})) e = ( e ( E { d}) ) = (( e ( { d})) ( d ( { d}))) = (( e d ) ( { d})) = ( e ( { d})) = ( e ( E { d})) e E { d}, e = ( e ( E { d}) ) lwys hold Thus A ( E { d}) = { d} s consstent set nsted of reduct, whch s contrdctory to the known condton So, d, ( d ( E { d})) = ( d ( { d})) d Suffcency s consstent set becuse of s not reduct, then d = ( d ( E { d})) = ( d ( { d})) holds from Theorem 7 Ths s contrdctory to the ssumpton So ( e E) = ( e ) = e Further, f d such tht { d} s consstent set Thus, s reduct 3 Approch to ttrbute reducton n the concept lttce Ths secton ntroduces the dscernblty mtrx nd dscernblty functon of forml context Bsed on these notons, we wll obtn the pproch to ttrbute reducton n the concept lttce efnton 6 Let ( U, A, I ) be forml context, ( X, B ), ( X, B ) L We cll IS (( X, B ),( X, B )) = B B B B (8) FC the dscernblty ttrbutes set between ( X, B ) nd ( X, B ) And Copyrght by Scence n Chn Press 2005

9 Attrbute reducton theory nd pproch to concept lttce 721 Λ = ( IS (( X, B ),( X, B )), ( X, B ),( X, B ) L) FC FC s clled the dscernblty mtrx of the context In the dscernblty mtrx of forml context, only those non-empty elements re useful So, we lso denote the set of non-empty elements n the mtrx by The Λ FC menng of Λ FC cn be decded by the content Exmple 3 Tble 2 gves the dscernblty mtrx of Exmple 1 Tble 2 The dscernblty mtrx of Exmple 1 FC1 FC2 FC3 FC4 FC5 FC6 FC1 FC2 {c,d,e} FC3 {,b,e} {,b,c,d} FC4 {d,e} {c} {,b,d} FC5 {,b,d,e} {,b,c} {d} {,b} FC6 {c} {d,e} {,b,c,e} {c,d,e} A Theorem 10 Let ( U, A, I ) be forml context ( X, B ),( X, B ),( X, B ) LU (, AI, ) The followng propertes hold: 1 IS (( X, B ),( X, B )) = FC 2 IS (( X, B ),( X, B )) = IS (( X, B ),( X, B )) FC FC k k 3 IS (( X, B ),( X, B )) IS (( X, B ),( X, B )) IS (( X, B ),( X, B )) FC FC k k FC k k Proof It s evdent tht 1 nd 2 re rght, we only need to prove 3 If IS (( X, B ),( X, B )), then, ether B or B, nd B B FC Wthout loss of generlty, we suppose B, B If B k, then ISFC (( X, B ),( X k, B k )); f B k, then ISFC (( X, B ),( Xk, Bk )), but ISFC (( X k, Bk ),( X, B )) So, IS (( X, B ),( X, B )) IS (( X, B ),( X, B )) FC k k FC k k Theorem 11 (Judgment Theorem 6 of consstent sets) Let be forml context A,, ( X, B ),( X, B ) L, ( X, B ) ( X, B ), the followng propostons re equvlent: 1 s consstent set 2 B B wwwscchncom

10 722 Scence n Chn Ser F Informton Scences 2005 Vol48 No IS (( X, B ),( X, B )) 4 B A, f B =, then B Λ Proof FC 1 2 LU (,, I ) LU (, AI, ) cn be obtned becuse FC s consstent set So, ( X, B ), ( X, B ) L, ( X, B ) ( X, B ), C, C, such tht ( X, C ), ( X, C ) L( U,, I ), ( X, C ) ( X, C ) Thus C C, C = X = X = B, C = X = X = B estblsh As result, B B holds 2 1 We need only to prove tht ( X, B) L, ( X, B ) L( U,, I ), thus s consstent set So, we need to prove: we hve LU (, AI, ) X = X = B Second, f ( B ) X, ( B ) = X Frst,, then ( X, B) (( B ),( B ) ) Bsed on the ssumpton, t should be B ( B ) But, on the one hnd, X = B, (( B ),( B ) ) B B ( B ) B = X ( B ) X = B ( B ) B On the other hnd, B ( B ) B = B ( B ) Thus B = ( B ), whch s contrdctory to B ( B ) So ( B ) = X 2 3 Suppose ( X, B ),( X, B ) L, nd B B Then, ether B B = B B, or B B = B B So, IS (( X, B ),( X, B )) = ( B B B B ) FC = ( B B ) ( B B ) = ( B B ) ( B B ) 3 2 Suppose ( X, B ), ( X, B ) L, nd IS (( X, B ), ( X, B )) Then there exst t lest one ttrbute FC, nd IS (( X, B),( X, B)), e, ether B or B, nd B B Thus, f B, B, then B, B ; f B, B, then B, B So B B 3 4 It s obvous tht the equvlence s rght Copyrght by Scence n Chn Press 2005 FC

11 Attrbute reducton theory nd pproch to concept lttce 723 Theorem 11 shows tht to fnd reduct of forml context s to fnd the mnml subset of ttrbutes, whch stsfes the condton H ( H Λ ) Exmple 4 We gve the reducts of Exmple 1 bsed on ts dscernblty mtrx gotten n Exmple 3 Frstly, we cn obtn tht Λ FC = {{ c},{ d},{ b, },{ de, },{ cde,, },{ be,, },{ bc,, },{ bd,, },{ bde,,, },{ bcd,,, }, {,,,}, bce A} Secondly, we exmne ts reduct For exmple, the subset 1 = {,, cd} stsfes H H Λ ); n ddton, for ts subset: { c, },{ d, },{ cd, }, there exsts 1 ( FC correspondng element { d},{ c},{, b} n ΛFC such tht the ntersecton between them s empty Smlrly, for 2 = {,, bcd}, on the one hnd, t stsfes 2 H ( H Λ FC ); on the other hnd, for ts subset: { bc, },{ bd, },{ cd, }, there exsts correspondng element { d},{ c},{, b} n ΛFC such tht the ntersecton between them s empty So, 1 nd 2 re ll reducts of the forml context FC efnton 7 Let be forml context, ts dscernblty functon s defned s the followng formul: f ( ) ( Λ = (9) FC h) H ΛFC h H By the bsorpton lw nd the dstrbutve lw, f ( Λ FC ) cn be trnslted to the mnml dsunctve norml form [19], whose components (ll of ts conunctve norml form) re ll of the reducts of Exmple 5 We gve the dscernblty functon of Exmple 1, nd clculte ll of ts reducts f( Λ ) = ( h) FC H ΛFC h H = ( d e) c ( c d e) ( b c e) ( b c d e) ( b c d) ( b c) ( b e) ( b d e) ( b d) ( b) d = c d ( b) = ( c d) ( b c d) So, there re 2 reducts: Exmple 4 { cd,, },{ bcd,, } 4 Attrbute chrcterstcs n the concept lttce, whch re the sme s our concluson n In efnton 5, we clssfed the ttrbutes n forml context nto 3 types: bsolute wwwscchncom

12 724 Scence n Chn Ser F Informton Scences 2005 Vol48 No necessry ttrbute, reltve necessry ttrbute, nd bsolute unnecessry ttrbute Attrbutes wth dfferent types ply dfferent role n the reducton of the concept lttce Ths secton wll gve ther dfferent chrcterstcs Theorem 12 Let be forml context A, s core ttrbute ( X, B ),( X, B ) L, IS (( X, B ),( X, B )) = { } FC Proof Usng Corollry 3 nd Theorem 11 cn prove ths theorem Theorem 13 Let ( U, A, I ) be forml context A, put G = { g g A, g } Then 1 s core ttrbute ( { }) 2 s n bsolute unnecessry ttrbute ( { }) =, nd G = 3 s reltve necessry ttrbute ( { }) =, nd G Proof 1 Becuse, s n unnecessry ttrbute A {} s consstent set ( { }) =, we obtn tht s n bsolute necessry ttrbute s not n unnecessry ttrbute ( { }) 2 Necessty Assume s n bsolute unnecessry ttrbute It s esy to see ( { }) = On the other hnd, for ny reduct, thus ( ) = b, f b, then b ; f b, then b nd b s not n bsolute unnecessry ttrbute, thus b So b From whch G, nd G ( ) = Suffcency Suppose tht reduct hold In ddton, we know G So G = s not n bsolute unnecessry ttrbute, there exsts such tht So ( ( { })) And nother reson ( ( { })) ( { }), we get ( ( { })) G, we get G = ( G ( { })) ( G ( A ) ), thus {})) ( G ( A )) ( { } ) holds It cn be obtned tht Becuse from G, thus, we hve ( G ( {})) ( ( {})) Assume e G ( A ) On the one hnd, e cn be concluded from e G, G = ( G ( G ( { }) Copyrght by Scence n Chn Press 2005

13 Attrbute reducton theory nd pproch to concept lttce 725 then e, e On the other hnd, e = ( e ) = ( e ( { })) ( ( { }) ) cn be obtned from e A nd s consstent set Thus e G ( A ) ( G ( A )) = e ( ( { }) ) So G ( ( { })), G Ths s contrdcton to ssumpton, so s n bsolute unnecessry ttrbute 3 It follows mmedtely from 1 nd 2 Exmple 6 We exmne the type of ech ttrbute n Exmple 1 usng Theorem 13 = {1,2,4}, = {, b},( { }) = {1,2,4},( { }) = ; G =, G = {1,2,3,4}, G b = {1,2,4}, b = {, b},( b { b}) = {1,2,4},( b { b}) =b ; b b b G =, G = {1,2,3,4}, G b c = {2,4}, c = {, b, c},( c { c}) = {1,2,4},( c { c}) c d = {1,3}, d = { d},( d { d}) = {1,2,3,4},( d { d}) d e = {1}, e = {, b, d, e},( e { e}) = {1},( e { e}) = e ; e e = G = {,, b d}, G = {1}, G e e From the bove results we know,, b re reltve necessry ttrbutes, c, d re core ttrbutes, nd e s n bsolute unnecessry ttrbute, whch s the sme s the results of Exmple 1 5 Concluson As useful tool for dt nlyss nd knowledge processng, the concept lttce hs been ppled to mny felds Ths pper hs proposed the theory of reducton n the concept lttce Reducton theory n the concept lttce s to fnd mnml subset of the ttrbute set, whch cn decde ll of the concepts nd herrchy entrely n the rw context The reducton extends the theory of the concept lttce nd hs mportnt sgnfcnce n ts study nd pplcton Ths pper hs gven some udgment theorems of concept lttce reducton, ntroduced dscernblty mtrx of context, nd obtned pproch to reducton bsed on t At the sme tme, ths pper hs lso studed the chrcterstcs of dfferent types of ttrbutes The context studed n ths pper hs no decson ttrbute We wll study the context wth decson ttrbute usng reducton theory n our wwwscchncom

14 726 Scence n Chn Ser F Informton Scences 2005 Vol48 No future work Acknowledgements Ths work ws supported by the Ntonl 973 Progrm of Chn (Grnt No 2002CB312200), the Ntonl Nturl Scence Foundton of Chn (Grnt No ), nd the Nturl Scentfc Reserch Proect of the Educton eprtment of Shnx Provnce n Chn (Grnt No 04JK131) References 1 Wlle, R, Restructurng lttce theory: n pproch bsed on herrches of concepts, n Ordered Sets (ed Rvl, I), ordrecht Boston: Redel, 1982, Oosthuzen, G, The Applcton of Concept Lttce to Mchne Lernng, Techncl Report, Unversty of Pretor, South Afrc, Ho, T B, Incrementl conceptul clusterng n the frmework of Glos lttce, n K: Technques nd Applctons (eds Lu, H, Lu, H, Motod, H,), Sngpore: World Scentfc, 1997, Kent, R E, Bowmn, C M, gtl Lbrres, Conceptul Knowledge Systems nd the Nebul Interfce, Techncl Report, Unversty of Arknss, Corbett,, Burrow, A L, Knowledge reuse n SEE explotng conceptul grphs, Interntonl Conference on Conceptul Grphs (ICCS 96), Sydney, Unversty of New South Wles, 1996, Schmtt, I, Ske, G, Mergng Inhertnce herrches for scheme ntegrton bsed on concept lttces [EB/OL] http: //wwwmthemtctu-drm stdtde/gs/g1 7 Sff, M, Reps, T, Identfyng modules v concept nlyss, n Interntonl Conference on Softwre Mntennce (eds Hrrold, M J, Vsggo, G), Br, Itly, Wshngton, C: IEEE Computer Socety, 1997, Hu, K Y, Lu, Y C, Sh, C Y, Advnces n concept lttce nd ts pplcton, Journl of Tsnghu Unversty (Scence & Technology), 2000, 40(9): Ho, T B, An pproch to concept formton bsed on forml concept nlyss, IEICE Trns Informton nd Systems, 1995, E782 (5): Crpneto, C, Romno, G, Glos: n order-theoretc pproch to conceptul clusterng, n Proceedngs of ICML 293 (ed Utgoff, P), Amherst: Elsever, 1993, Godn, R, Incrementl concept formton lgorthm bsed on Glos (concept) lttces, Computtonl Intellgence, 1995, 11 (2): Yo, Y Y, Concept lttces n rough set theory, In Proceedngs of 2004 Annul Meetng of the North Amercn Fuzzy Informton Processng Socety (NAFIPS 2004) (eds ck, S, Kurgn, L, Pedrycz, W et l), IEEE Ctlog Number: 04TH8736, June 27-30, 2004, Oosthuzen, G, Rough sets nd concept lttces, n Rough Sets, nd Fuzzy Sets nd Knowledge scovery (RSK 93) (ed Zrko, W P), London: Sprnger-Verlg, 1994, eogun, J S, Squer, J, Concept pproxmtons for forml concept nlyss, n Workng Wth Conceptul Structures, Contrbutons to ICCS 2000, (ed Stumme, G), Achen: Shker-Verlg, Tonell, P, Usng Concept Lttce of ecomposton Slces for Progrm Understndng nd Impct Anlyss, IEEE Trnsctons on Softwre Engneerng, 2003, 29(6): [OI] 16 Grgorev, P A, Yevtushenko, S A, Elements of n Agle scovery Envronment, In Proc 6th Interntonl Conference on scovery Scence (S 2003) (eds Greser, G, Tnk, Y, Ymmoto, A), Lecture Notes n Artfcl Intellgence, 2843, 2003, Gtzemeer, F H, Meyer, O, Text schem mnng usng grphs nd forml concept nlyss, n Conceptul Structures-Integrton nd Interfces, ICCS 2002 (eds Prss, U, Corbett,, Angelov, G), LNAI 2393, Hedelberg: Sprnger, 2002, Gnter, B, Wlle, R, Forml Concept Anlyss, Mthemtcl Foundtons, Berln: Sprnger, Zhng, W X, Leung, Y, Wu, W Z, Informton System nd Knowledge scovery, Beng: Scence Press, 2003 Copyrght by Scence n Chn Press 2005