GENERATE FUZZY CONCEPTS BASED ON JOIN-IRREDUCIBLE ELEMENTS

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1 GENERATE FUZZY CONCEPTS BASED ON JOIN-IRREDUCIBLE ELEMENTS Hua Mao ad *Zhe Zheg Departmet of Mathematcs ad Iformato Scece Hebe Uversty Baodg Cha *Author for Correspodece: ABSTRACT The costructo of fuzzy cocept lattces s oe of the maor matters of fuzzy formal cocept aalyss. However the costructo of effcecy of fuzzy cocept lattces s stll epoetal. Irreducble elemets push forward a mmese fluece o mprovg costructo effcecy of fuzzy cocept lattces. Hece ths paper frstly proposes a weakeed obect topology graph for fdg o-rreducble elemets. After that t provdes a ovel way to compute o-rreducble elemets ad the geerate all crsp-fuzzy cocepts. Keywords: fuzzy cocept lattces rreducble elemets o-rreducble elemets INTRODUCTION Cocept lattce s a brach of lattce theory ad also a mmedate outcome of formal cocept aalyss (FCA) whch was put forward by Gater ad Wlle (1982) the early 1980s. Cocept lattce s a cocept herarchcal structure establshed by aalyzg ad dsposg the bary relatoshp betwee obects ad attrbutes of a data set. At preset cocept lattce has bee successfully appled to mathematcs ad computer scece ad stll ows a great deal of potetal applcato value. A formal cocept s a composto of eteso ad teso. However the real world the publc's cogto s fuzzy rather tha crsp for the great maorty of thgs. Wth the rapd developmet of techology some scholars proposed a fuzzy formal cocept whch dcates the vague relatoshp amog the obects ad attrbutes. Such as Burusco ad Fuetes (1994) frstly defed fuzzy cocepts usg mplcato operators based o fuzzy closure operators ad Krac (2003) preseted oe-sded fuzzy cocepts. Researchers have employed fuzzy cocepts to fuzzy classfcato ad fuzzy decso though the approaches of costructg fuzzy cocept lattces are less productve. Cosequetly the costructo of fuzzy cocept lattces s cout for research. It s geerally kow that every elemet ca be depcted by the o (meet) of o-rreducble (meet-rreducble) elemets. Thus a lattce all the o-rreducble elemets ad meet-rreducble elemets are the most fudametal elemets. Smlarly a cocept lattce obect cocepts ad attrbute cocepts are the most fudametal elemets where each formal cocept s ether the tersecto of attrbute cocepts or the uo of obect cocepts. I other words each obect cocept ad attrbute cocept correspods to a meet-rreducble elemet ad a o-rreducble elemet a cocept lattce. So t s very sgfcat to study rreducble elemets a cocept lattce. O the oe had Cetre for Ifo Bo Techology (CIBTech) 1

2 usually rreducble elemets ca be used to eplore the attrbute characterstcs attrbute reducto such as Shao ad Leug (2016) proposed a graular reduct methods based o o-rreducble elemets a fuzzy formal cotet. O the other had rreducble elemets also make cotrbutos to the costructo of cocept lattces. For stace L ad Shao (2017) bult the lattce from meet-rreducble attrbute cocepts by usg geerators drectly. I fuzzy formal cotets Zhag (2018) rased a batch-mode algorthm for drectly costructg fuzzy cocept lattce based o uo ad tersecto operatos yet the costructo of effcecy of fuzzy cocept lattces s stll epoetal. Accordgly a fuzzy formal cotet usg rreducble elemets to geerate fuzzy cocepts ca greatly mprove the costructo of effcecy of fuzzy cocept lattces. Ispred by attrbute topology proposed by Zhag (2014) ths paper presets a weakeed obect topology graph order to fd o-rreducble elemets a fuzzy formal cotet. Furtherly we ca obta all fuzzy cocepts by the o of the o-rreducble elemets. The remader of ths paper s orgazed as follows. Frstly we brefly revew some basc otos of fuzzy formal cotets ad graph theory. Secodly we wll defe the oto of a weakeed obect topology graph ad dscuss how to fd o-rreducble elemets from a weakeed obect topology graph ad the geerate all fuzzy cocepts. Fally we coclude the paper ad outle the future work. Prelmares I ths secto we revew some otos ad propertes about fuzzy formal cotets ad graph theory. For more detals please refer to Krac (2003) Body ad Murty (2008). Defto 1 (1) (Krac 2003) Let U ad B as the set of obects ad attrbutes respectvely R s a fuzzy relato o ther Cartesa product.e. R: A B [0 1]. A table wth rows ad colums correspodg to obects ad attrbutes s used for represetg ths fuzzy relato. The the value R (a b) epresses the grade whch the obect b have the attrbute a. (2) (Krac 2003) Let (U A R) be a fuzzy formal cotet. Defe a mappg: f : ( U ) ( A) whch assgs to every crsp set X of obects a fuzzy set f( X) of attrbutes a value a pot a Aof whch s f ( X )( a) R( a) a A (1) X.e. ths fucto assgs to every attrbute the greatest value so that all obects from X have ths attrbute at least such grade. Coversely defe a mappg g : ( A) ( U ) whch assgs to every fucto : A [01] a set g( ) { U a A ( a) R( a)} (2).e. such obects whch have all attrbutes at least the grade set by fucto ( other words the fucto of ther fuzzy-membershp to obects domates over ). It s easy to see that operators f ad g form a Galos coecto betwee ( U ) ad ( A) propertes ca be obtaed. ad the followg Property 1 (Krac 2003) Let (U A R) be a formal cotet X X1 X 2 X ( U) B B1 B2 B ( A) J ( J s a de set). The X X f ( X ) f ( X ) B B g( B ) g( B ); (ⅰ) ad Cetre for Ifo Bo Techology (CIBTech) 2

3 (ⅱ) X g of ( X ) B f o g( B); () f ( X ) f og o f ( X ) g( B) g o f o g( B); (v) f ( U ) I f ( X ) g( U B ) I g( B). J J J J Defto 2 (1) (Krac 2003) Let (U A R) be a formal fuzzy cotet a par ( X B) ( U) ( A) satsfyg X g( B) ad B f ( X ) s called crsp-fuzzy cocept of ( U A R ). (2) (Krac 2003) For a set of obects X ( U) ad a fuzzy set of attrbutes B f ( A) from Property1 () both ( g o f ( X ) f ( X )) ad ( g( B) f og( B)) are crsp-fuzzy cocepts. I partcular ( g o f ( X ) f ( X )) s a crsp-fuzzy cocept for each U ad s called a obect cocept. For two crsp-fuzzy cocept ( X1 B1) ad ( X2 B2) to make ( X1 B1 ) ( X 2 B2 ) f ad oly f X1 X2(or equvaletly B2 B1). Eample 1 A fuzzy formal cotet K= (U A R) wth U { } A { a b c d e} ad fuzzy relato s descrbed Table 1. Accordg to Defto 1(2) all the correspodg crsp-fuzzy cocepts are showed Table 2. I addto Fg.1 s the crsp-fuzzy cocept lattce L (U A R). Table 1 A fuzzy formal cotet (U A R) Table 2 All crsp-fuzzy cocepts geerated R a b c d e from Table 1 Crsp-fuzzy cocepts C1 ( a b c d e ) C2 ( a b c d e ) C3 ( 23 5 a b c d e ) C4 ( a b c d e ) C a b c d e ( ) C6 ( 1 a b c d e ) C7 ( 5 a b c d e ) C8 ( a b c d e ) C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 8 Fg.1 crsp-fuzzy cocept lattce L(UAR) Defto 3 (1) (Body ad Murty 2008) A graph G s a ordered trple ( V ( G) E( G) G ) cosstg of oempty set VGof ( ) a vertees a set EG ( ) dsot from VG ( ) of edges ad a fucto G that assocates wth each edge of G a uordered par of (ot ecessarly dstct) vertees of G. If e s a edge ad u ad v are vertees such that G () e uv the e s sad to o u ad v ; The vertees u ad v are called the eds of e. (3) (Body ad Murty 2008) A drected graph D s a ordered trple (V(D)A(D)d(v)) cosstg of a Cetre for Ifo Bo Techology (CIBTech) 3

4 oempty set V(D) of vertces a set A(D) dsot from V(D) of arcs ad ordered par of vertces of D. If a s a arc ad u ad v are vertces such that d(v)(a)=(u v) the a s sad to o u to v; u s the tal of a ad v s ts head. Defto 4 (B.Gater ad R.Wlle 1999) Let L be fte lattce adv L.We deote v* { L v} v* s sad to be o-rreducble f v v*. Lemma 1 (Gater ad Wlle 1999) Let L be fte lattce. Every elemet L s a o of some o-rreducble elemets. It should be oted that every crsp-fuzzy cocept (X B) the cocept lattce L (U A R) ca be represeted as a o of obect cocepts of ts eteso that s ( X B ) g( o f ( f) (. )) X Geeratg all crsp-fuzzy cocepts by the o-rreducble elemets I ths secto we propose a method to fd the o-rreducble elemets by a weakeed obect topology graph ad further we ca geerate all crsp-fuzzy cocepts. Weakeed obect topology graph a fuzzy formal cotet Defto 5 Let K= (U A R) be a fuzzy formal cotet. A weakeed obect topology graph deoted as G s defed as follows. The verte set V (G) s U ad the edges set E (G) s ( U ) where s defed as: (ⅰ)For a A R( a) R( a) the there s a double-arrow to that s ; (ⅱ)For a A R( a) R( a) the there s a sgle-arrow to that s. Defto 6 I a weakeed obect topology graph G for ay V ( G) ts graule of kowledge ca be derve from the arrow relato deoted as T ( ) { V ( G) } T( ) { V ( G) }. Eample 2 Cotug from Eample 1 by the Deftos 5 ad 6 we ca obta a weakeed obect topology graph G (U A R) fuzzy formal cotet K= (U A R) whch s show Fg Fg 2 A weakeed obect topology graph G (U A R) The graule of kowledge for each obect V ( G) s epressed as follows: T ( 1 ) T ( 2)={ 3 5} T ( 3)={ 5} T ( 4)={ } T ( 5 ). T ( 1) { 1} T ( 2)={ 2} T( 3)={ 3} T ( 4)={ 4} T( 5) { 5} How to fd the o-rreducble elemets Irreducble elemets play a crucal role the costructo of fuzzy formal cocept lattces. I ths secto we dscuss the property of the o-rreducble elemets obtaed from a weakeed obect topology graph a fuzzy formal cotet (see Theorems 1 ad 2). I addto we debate how to determe whether a crsp-fuzzy cocept s a o-rreducble elemets a fuzzy cocept lattce (see Theorems 3 ad 4). Cetre for Ifo Bo Techology (CIBTech) 4

5 Theorem 1 Let K= (U A R) be a fuzzy formal cotet G s the correspodg weakeed obect topology graph for a A U the f ( T ( ))( a) f ( )( a) f ( T ( ))( a ) f ( a )( ) Proof: I the lght of Deftos 4 ad 5 we kow that T ( ) { V ( G) } the a A R( a) R( a) we ca obta T ( ) f ( T ( ))( a) R( a) f ( )( a) a A by Defto 1 ad Property1. Smlarly we ca get f ( T( ))( a) f ( )( a) Theorem 2 Let K= (U A R) be a fuzzy formal cotet U the ( T ( ) UT( ) f ( T ( ))) s a crsp-fuzzy cocept ad T ( ) UT( ) g o f ( ). Proof: Frstly t ca be deduced from theorem 1 that g o f ( T ( ) UT( )) g o ( f ( T ( )) f ( T( ))) g o f ( ). Accordg to Property1 (ⅱ) we elct T ( ) U T( ) og f( T( U ) T( ))= o g f( ) (ⅰ) Secodly for ay g of ( T ( ) U T( )) we have f ( ) f og o f ( T ( ) U T( )) f( ) o the bass of Property1 (ⅰ) ad (). I other words ths meas fora A f ( ) R( a) R( a) f ( ).Therefore T ( ) ad we ca get T ( ) UT( ) g o f ( T ( ) U T( )) (ⅱ) I summary based o (ⅰ) ad (ⅱ) we ca cofrm that g o ( f T ( ) UT( )) g o f ( )= T ( ) UT( ) ad the ( T ( ) UT( ) f ( T ( ))) s a crsp-fuzzy cocept. It s kow from Theorems 1 ad 2 that the relatoshp s troduced betwee a graule of kowledge ad a par of operators f ad g. For the sake of determg whether a crsp-fuzzy cocept s a o-rreducble elemet we deduce the Theorems 3 ad 4 as follows. Theorem 3 Let K= (U A R) be a fuzzy formal cotet U f T ( ) 1 ad the the obect cocept ( T ( ) UT( ) f ( )) s a o-rreducble elemet the fuzzy formal cocept lattce. Proof: Suppose that T ( ) 1 the T ( ) =0 or T ( ) =1. If T ( ) =0 t meas that there s o obect yu T ( ) hece ( T ( ) UT( ) f ( )) s a o-rreducble elemet accordg to Lemma 1. If T ( ) =1 t meas that there s oly oe obect yu T ( ) ad t ca be epressed as( T ( y) U T( y) f ( y)). By the Deftos 5 ad 6 we have f ( ) f ( y) that s ( T ( ) UT( ) f ( )) ( T ( y) U T( y) f ( y)). Thus t ca be affrm that ( T ( ) UT( ) f ( )) s a o-rreducble elemet by the Defto 6. Theorem 4 Let K= (U A R) be a fuzzy formal cotet U f T ( ) = { y y L y } 1ad f ( y ) f ( ) the the obect cocept Cetre for Ifo Bo Techology (CIBTech) 5

6 ( T ( ) UT( ) f ( )) s a o-rreducble elemet the fuzzy formal cocept lattce. Proof: Suppose that T ( ) = { y1 y2 L y } 1 ad f ( y ) f ( ). Owg to T ( ) = { y1 y2 L y} 1 we have T ( )={ y1 y2 L y} ad f( y1) f ( ) f ( y ) f ( ) L f ( y ) f ( ). So 2 1 L ( T ( ) UT( ) f ( )) ( T ( y ) UT( y ) f ( y )) ca be obtaed. 1 I other words f ( y ) f ( a) t meas 1 1 ( T ( ) UT( ) f ( )) ( T ( y ) U T( y ) f ( y )). Accordg to the Deftos 5 ad 6 t s evdet that the obect cocept ( T ( ) UT( ) f ( )) s a o-rreducble elemet. The Theorems 3 ad 4 provde a way for fdg the o-rreducble elemets from a weakeed obect topology graph. The by Lemma 1 we ca geerate all crsp-fuzzy cocepts. The procedure for computg the o-rreducble elemets s show Algorthm 1. Algorthm 1 Computg the o-rreducble elemets of a fuzzy formal cotet Iput: A fuzzy formal cotet K= (U A R). Output: Jo-rreducble elemets set JI. 1: Italze JI ; 2: For aa 3: If R( a) R( a ) the ed f 4: If R( a) R( a ) the ed f 5: Ed for; 6: For each U compute T ( ) T( ) ; 7: If T ( ) 1 the JI JI U( T ( ) UT( ) f ( )) ed f ; 8: If T ( ) = { y1 y2 L y} 1ad 1 JI JI U( T ( ) UT( ) f ( )) ed f; 9: Retur JI. f ( y ) f ( ) the Eample 3 We use the cotet Table 1 to eame Algorthm 1. Cotug from Eample 2 we ca fd the o-rreducble elemets accordg to Algorthm 1. Frstly we calculate T ( 1 ) = =0 T ( ) = { } = T ( ) = { } =1 3 5 T ( ) = { } 4 T ( 5 ) =0. Owg to T ( 1 ) = T ( ) =1 3 T 5 ( ) =0 we ca fer that ( T ( 1 ) U T( 1 ) f ( 1 )) Cetre for Ifo Bo Techology (CIBTech) 6

7 ( T ( 5 ) U T( 5 ) f ( 5 )) ( T ( 3) UT( 3) f ( 3)) are the o-rreducble elemets due to T ( 2) =2>1 T 4 ( ) 4 1ad f ( 3) f ( 5 ) f ( 2) f ( 1 ) f ( 2) f ( 3) f ( 5) f ( 4) we ca coclude that ( T ( 2) U T( 2) f ( 2)) ( T ( 4 ) UT( 4 ) f ( 4 )) are the o-rreducble elemets. The et we ca geerate all crsp-fuzzy cocepts the lght of Lemma 1 all crsp-fuzzy cocepts are show Table 2 ad the crsp-fuzzy cocept lattce s preseted Fg.2. Cocluso I ths paper we put forward a dea of costructg a fuzzy cocept lattce by usg o-rreducble elemets ad preset how to fd the o-rreducble elemets from a weakeed obects-topology graph. Smlarly rreducble elemets have also a vtal posto attrbute reducto ad the we wll devote to attrbute reducto of fuzzy formal cocept aalyss the future. ACKNOWLEDGMENT Ths work s grated by the Natoal Nature Scece Foudato of Cha [ ]; Nature Scece Foudato of Hebe Provce [A ]; Post-graduate's Iovato Fud Proect of Hebe Uversty [hbu2018ss45]. REFERENCES Body JA ad Murty USR (2008). Graph Theory. Sprger Lodo. Burusco A ad Fuetes R (1994). The study of the L-fuzzy cocept lattces. Math-ware ad Soft Computg Gater B ad Wlle R (1999). Formal cocept aalyss: Mathematcal foudatos Sprger-Verlag. New York. Krac S (2003). Cluster based effcet geerato of fuzzy cocepts. Neural Network World 13(5) L X ad Mgwe Shao (2017). Costructg lattce based o rreducble cocepts. Iteratoal Joural of Mache Learg & Cyberetcs 8(1) Shao MW ad Leug Y (2016). Graular reducts of formal fuzzy cotets. Kowledge-Based Systems T. Zhag (2014). Deep frst formal cocept search. The Scetfc World Joural Z. Zhag (2018). Costructg L-fuzzy cocept lattces wthout fuzzy Galos closure operato. Fuzzy Sets ad Systems Cetre for Ifo Bo Techology (CIBTech) 7