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K[1}= b% ) H :<; XY; OE87X>{2 !V3 !2g% ;t6HGV4F>@G>J 6HGMCE>.Q9GHABCE> HG9>*R{;@4 I> 6 H >{287:I87X87GCE>{2TH8L6HGf; :lad;Nl87CE> 2E8L:B:<; XY; G 3 % !" > " 4 2&)3T" 6 U6HGV=698L:I>{2 ; :lda ;N 87CE>{2]M= 698; Nx; OE8,Jh 4J; GV87G8F2PCE8:BA7NHOE>8F2PC5; Om; G a >@OPXY;@TH>{2aN>@O:I87CPO5;@2a87GL87:#2E87GCPAITH> 692E6x; :vTH87:vCED7OPX*ABG9> 8F2PCE>YG9>2PAI87X*NHOE88F2G98F4F8F25; OPAI> 6 =?>@O(8 JP87X*NH:I> :<;@2:I87CPO5;@2TH86HG ; :lda ;N 87CE>VN>MT\O <; G4F>@OPOE8F2PN>@G9TH87O; 8F2PCPOP6947CP6HO5;@2TH8T9; CE>{2t87G 6HG>@OETH87Gx;@TH>@O 6 U :I> :<; O SI >TH87:9:BA7HN OE> S6HG; :la ;N 87CE>2E8OE87NHOE8F2587GC5; Om ; 692P6x; :BX87GCE8X8FT\A<; GCE8:<;:I87CPO5;?{I OPAI8,I ; a:<;@2:I87CPO5;@2b{I OPAI8,I ;@2LX*ABG 92E476H:<;@2 4F>@X> TH87G9>@C5; Om; G 2 BX@N >@:I>{2bSI 87G9D7OPAI4F>{2TH87: Σ ; :lda ;N 87CE> X*AI87GMCPO5;@2&M= 698a:I>{2v2 BX@N >@:I>{234F>@G947OE87σCE>{2gM= 698":I> a >@OPX87G*2E87Om; GG9>@OPXY; :BX87GCE8 :<;@2iNHOPABX87O5;@2i:I87CPO5;@2iX*ABG 92E476H:<;@2 TH87:v; :la ;N 87CE>*692P6x; : a, b, c . . . 6 K 8(6HCPAB:BA7h ; Om ; 87:r2 BX@N >@:I>TH8(N87OPCE87G987G947A<; x9 ;HN ABCP6x; :?87GVCE8F>@O <;YTH8(4F>@G>J 6HGMCE>{2 Nx; O5;*ABG9T\AI4J; O = 698$6HGx;t:I87CPO5;Nl87OPCE87G98F4F8t;t6HG ; :la ;N 87CE> N>@Oi8 J 87X*NH:I> a ∈ Σ 6 ∈ Q G; :la ;N 87CE> 2E8TH8R9Q G98<SI 87G987O5; :BX87GMCE8LNl>@Ot87G6HX87O5;@47AIW@GTH8:I>{22 BXjlN >@:I>{2 M= 698 4F>@GMCPAI87G98 6 =v>@OV8 J 87X*NH:I> Σ = {a, b, c, d, e, . . . , z} 8F2L87:(; :lda ;N 87CE>mTH8[:<;@2:I87CPO5;@2 :<; CPABGx;@2X*ABG 92E476H:<;@2 Σ = {0, 1} 8F2$87:; :lda ;N 87CE>VTH8*:I>{2$G HX87OE>{2@HN ABGx; OPAI>{2 Σ = 8F2 87:c; :la ;N 87CE><M= 698N87OPX*ABCE8t47>@G92ECPOP6HABO 476x; : M= 6HAI87OG HX87OE> {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Gx; CP6HO5; :o87G xN ;@2E8 5 c Σ = {/, \} 8F287:c; :la ;N 87CE>.a >@OPXY;@TH>Nl>@O:I>{2TH>{2 2 BXjlN >@:I>{2 E \6 1 2 3 4 ! &" K+A7"+ >+#4 b% ) [ K 8:B:<; XY; !V3!2! Za >@OPXY;@T9;4F>@G:I>{2v2 BX@N>@:I>{2cTH86HG; :lad;Nl87CE> ;tCE> T9;*2E8F476987G947A<; 9Q GHABC5;*TH8(:<;@2:I87CPO5;@2iTH88F2E8$; :la ;N87CE> 6 1YJ 87X*NH:I>{2TH8VNx; :<;HN O5;@22E>>HN OE8-87:; :la ;N 87CE> 2E>@G cVEHW CFfQNHGQL E ^ZLD^crf 6 K >@G Nx; :<;HN O5;@22E>>HN OE8*87:"; :la ;N 87CE> Σ 0 1 01010101Σ E 1111 6 =v>@O H:BCPABX> 0 1 125 E 12357384 HTH:B8 CPAB6HXYG ;@;2:lda :I;87CPlN 2EO5>@87;@GCE2$> Nx6 X*;U AB:<G 2; 9HN 2EO5Nl47;@6H>@2tO:<;@2E8 2>>PJ HN 87THOEX*878:NH87:I; :a> :la ; ;:lN da ;87lN CE>b87CEx9> ;ΣHN ABCP8F66x21c; 6H:GmGxNx;:I; > NxO5;L; 2P69:<;OE4F87HN 8FNHO52EOE;*AB8FRS2E2E> 87>>?GHN 2EC5OE8; 8 O6H:<CP;@AB:B2$ A7h Nx; ;Om;:<;GmHN O5:<;@;@22 x = juan Σ y = 1234 8F2i6HGx;Nx; :<;HN O5;*2E>>HN OE8 Σ 6 1 2 3 1 3 i i i i i i i 5 c i K 8?x9 ;6HCPAB:BA7h ;@TH>Y87:rCED7OPX*ABG9>\" %'%+! Nx; O5;*TH8RQ9GHABOi6HGx;Nx; :<;NHO5;TH8 JP; G9TH>; : J 6HAI47AI>TH87::I8F47CE>@Oi2E6ABGMCE87OPNHOE87C5;@47AIW@G 6 =?>@O:I>{28 JP87X*NH:I>{2i;@T\6947AITH>{2 \2E8 NH698FTH8(TH8FT\6947ABO M= 698:<;tNx; :<;HN O5;258F476987G947A<;*2E8 OE8RxQ 87OE8;t:<;*4F>@:I> 4J;@47AIW@Gba I2PAI4J;>@OETH87Gx;@T9;*TH8(:<;@2:I87CPO5;@2 6HGx;[;\J 6HGMCE> ;->@CPO5;o4F>@X>[2E8Kx9 ;@4F8 9x;NHABCP6x; :BX87GCE8V4F>@G:I>{2tCE8 CE>{2*8F2547OPABCE>{2 6 KMABG 87XjxN ; O SI > "8F2PC5; ;@4F87N47AIW@GyG9> ;@47:<; O5; :I> = 698LN>MT\O <; 2E87O*6HGx;Nx; :<;HN O5; 476x; G9TH> :I>{2 2 BX@N >@:I>{2 TH87:c; :lda ;lN 87CE>G9>CE87GI ; G[6HGx;YOE87NHOE8F2E87GC5;@47AIW@GDa I2PAI4J;2587G947AB:B:<;l4F>@X>87G[87: 4J;@25>-TH8 = 6982E8YCPO5; CE8TH88F2PCPOP6947CP6HO5;@2TH8T9; CE>{2 6 U6HGVM= 698; :I> :<; O SI >[TH87:a:BA7HN OE>-258 692 ; Om; G2 BXjlN >@:I>{2ABX*NHOE8F2E>{2 E :<;@2YNx; :<;HN O5;@2L4F>@OPOE8F2PN>@G9TH87Om; G ; :<; ABGCE87OPNHOE87C5;@47AIW@G M= 698*2E8*;@4J;xN ;TH8*T9; O #4F>@GMRMAI87G98T9; O(6HGx;ABGCE87OPNHOE87C5;@47AIW@G a >@OPXY; :3TH87:34F>@G94F87NHCE>-TH8 Nx; :<;HN O5;\= 698$258J;tR>; :BAIT9;*87GV87:?4F>@GCE8 \CE>.SI 87G987O5; :?TH8$; :lda ;lN 87CE>{2 476x; :I8F2 = 6HAI87O5; 6 H ;VAITH8J;-8F2E87G947A<; :aNx; O5; TH8R9Q GHABOtNx; :<;HN O5;@2*8F2i= 698DF2PC5;@2258NH698FTH87G4F>@G92PCPOP6HABO; Nx; OPCPABOTH8*:I87CPO5;@2 X*AI87GCPO5;@2:<;@2(:I87CPO5;@22E8NH698FTH87G OE8F476HN87O5; O;Nx; OPCPABOTH8*:<;@2(Nx; :<;k HN O5;@2 6 =v>@O:I>tC5; GCE> 6HGx;tTH8R9Q GHAI47AIW@G a >@OPXY; :#TH8 Nx; :<;HN O5;Nx; OPCPABOm ; TH8 :<;t8 \AI2PCE87G947A<;tTH8 4F>@G92ECPOP6947CE>@OE8F2 Y E TH8F2PCPOP6947CE>@OE8F2TH8 Nx; :<;HN O5;@2 Nx; O5;Nl> TH87OTH8R9Q GHABONx; :<;HN O5;@22E>>HN OE8 6HG ; :lda ;N 87CE>2E>@GNHOE8F47AI25;@2CPOE8F24F>@X*Nl>@G987GMCE8F2 Q Gx;Nx; :<;HN O5;8F2PN8F47A<; : {:B:<; XY;@T9;!V3!2! ! ! M= 698i2E8iTH87G9>@C5; Om$ ; X8FT\A<; GCE8 • :<;:I87CPO5;?{I OPAI8,I ; λ 6_ GCP6HABCPABR{; X87GCE8 λ 8F2c:<;Nx; :<;HN O5;1M= 698G9>$4F>@GMCPAI87G98 GHABG{I 6HGx; :I87CPO5; 6 Q Gm4F>@G92PCPOP6947CE>@OTH8YNx; :<;HN O5;@2 :<; >@Nl87O5;@47AIW@Gm4F>@G94J; CE87Gx;@47AIW@GTH86HG 2 BXjlN >@:I> • 4F>@G 6HGx;mNx; :<;HN O5; = 698 ;mNx; OPCPABOVTH86HG 2 BXjlN >@:I> σ f E TH8[6HGx;mNx; :<;HN O5; 4F>@G92PCPOP6SE 86HGx;*Nx; :<;HN O5;YG6987R{; σ.x )= 6982E8>>HN CPAI87G98t4F>@:I>M4J; G9TH>Y:<;*:I87CPO5; σ x; :<;A7h M= 6HAI87OET9;*TH8(:<;@2:I87CPO5;@2iTH8 x 6 M >{2(; Gx; :BA7h ;@TH>@OE8F2(TH8Nx; :<;HN O5;@2 Y= 698:B:<; XY; OE87X>{2 inicial E resto TH8X> TH> • = 6982PA x 8F2"6HGx;$Nx; :<;HN O5;$G9>R@;@4 <; inicial(x) 8F236HGx;$:I87CPO5; :<;$NHOPABX87O5;$:I87CPO5; TH8 x E resto(x) 8F2i>@CPO5;Nx; :<;HN O5; 6 132G98F4F8F25; OPAI>\M= 698:I>{24F>@G92PCPOP6947CE>@OE8F2S E ; Gx; :BA7h ;@TH>@OE8F2TH8(Nx; :<;HN O5;@2ABGCPOE> T\6947AITH>{2 RS87OPAlQTM= 6987GV:<;@22PA7{I 6HAI87GCE8F2 OE87:<;@47AI>@G98F2 5$6 KMA x 8F26HGx;*Nx; :<;HN O5;; O HN ABCPO5; OPA<;T\AI2PCPABGMC5;*TH8 λ H2E8(476HX*NH:I8@M= 698 BX6 KMA x 8F2i6HGx;YNx; :<;NHO5;476x; : =6HAI87O5;KE σ 8F2 476x; : =M6HAI87O 2 BXjNl>@:I> l2E8$476HX*NH:I87G[:<;@2 TH>{2iNHOE>@NHAI8FT9;@TH8F22PA7{I 6HAI87GCE8F2 inicial(x).resto(x) = x inicial(σ.x) = σ resto(σ.x) = x i i i i i i i i U 6HVG =M698;:I>:<; O IS>YTH87:r:BA7NHOE>Y2E8(6HCPAB:BA7h ; Om;4F>@G92PC5; GCE87X87GMCE8t87:?4F>@G94F87NHCE>TH8(Nx;k :<;NHO5;4F>@X>Y2PA/2E8CPO5; C5; O5;YTH8(6HGx;*4J;@TH87Gx;YTH8(4J; O5;@47CE87OE8F2 9CE> TH>{2i:I>{2i4F>@G94F87NHCE>{2 =M698 6HCPAB:B7A h ; GNx; :<; NHO5;@2 4F>@X>87:9:I87GI{6x;JP8;@4F87NHC5;@TH>>(OE8F4F>@G9> 47AITH>Nl>@O6HG; 6HCEW@XY; C5; \87: :I87G I{6x; JP8 IS87G987O5;@TH>>(OE87NHOE8F2E87GC5;@TH>N>@Oc6HGx: ; I{O5; < X ; CPAI4J;>N>@O36HGx;(8 \NHOE8F2PAIW@GOE8,I{6Xk :<; O 87CE4 6 /258cNH698FTH87Gt8 \CE87G9TH87OvABGHX8FT\A<; C5; X87GCE8i;Nx; :<; NHO5;@2 Nx;@25;@T9;@2v87G:I>{2o; AI>@XY;@2 ; GMCE87OPAI>@OE8F2 9N>@O:I. > =M698CE>MT9;@2:<;@2; Q9OPXY;@47AI>@G98F2 ECE8F>@OE87XY;@2iC5; j X NHAID7G-2E87mO ; GV47AI87OPCE>{2 87G-8F2PCE8(4F>@GCE8 \CE>< X ;@2 ; X*NH:BAI> 6 ; EmX694 9x;@2XY; G987O5;@2YTH8VABX*NH:I87X87GC5; OL6HGf2PAI2PCE87XY; TH8Nx; :<; NHO5;@K2 =M698V25; CPAI2 k ad; I;[:I>{2; AI>@XY;@2L; GCE87OPAI>@OE8F2X8FT\A<; GCE88F2PCPOP6947CP6HO5;@2TH8 T9; CE>{287Gf6HGf>@OETH87Gx;@TH>@O 2E6HNl>@GHAI87G9THj > =698CE87G I;$X87X>@OPA<;(ABXG Q9GHABC5; 4F>@GYRS8F47CE>@OE8F2TH8i4J; O5;@47CE87OE8F2 4F>@G:BAI2PC5;@2 87GH:<; h ;@T9;@2 #87CE4 6 132PCE>Nl87OPX*ABCE8*THR8 Q9GHABO$8F2ECPOP6947CP6HO5;@2$TH8T9; CE>{2< X ;@2$4F>@X*NH:I8 JP;@2Nx; O5; OE87NHOE8F2587GC5; OL:I>{24F>@G94F87NHCE>{2 =M698VABmO ; G ; Nx; OE8F47AI87G9TH> ;@2 4F>@X> NHOE>> I{O5; XY;@2 =M698 258 4F>@X*N>@OPC5; G-4F>@X>:I>{2 ; 7: IS>@OPABCPX>{2 TH8F2E47OPABCE>{287GV8F2PCE8(:B7A NHOE> 6 b% ) K 8$:B:<; XY;Y:I>@G I{ABCP69TTH8$6HGx;YNx; :<; NHO5;; :?G HX87OE>LTH8$:I87CPO5;@:2 =M698 :<;4F>@X*Nl>@G987G 6 H ;t:I>@G I{ABCP69TTH8 :<;Nx; :<; NHO5; x 258 OE87NHOE8F2E87GMC5;*4F>@GL:<;G9>@C5;@47AIW@G |x| 6 =?>@Oi8 JP87X*NH:I> |juan| = 4 |1| = 1 |12345678| = 8 |λ| = 0 1cGL6HGL2EAI2PCE87XY;jIS87G987O5; 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Om; GL:<;@2aTH8R9Q GHAI47AI>@G98F2 E TH87X>{2PCPO5;@47AI>@G98F2OE8F476HOE2PABR{;@2 a >@OPXY; :I8F2 ; 6HGVM= 69887G; :7I HG4J;@2E>2E8aOE8F476HOPOPABOm( ; ;:<;(; O {I 6HX87GC5;@47AIW@G\a >@OPXY; : Nx; O5;87RMABC5; OcT\69T9;@2 >ABGMCE87OPNHOE87C5;@47AI>@G98F287OPOEW@G98J;@2 6 1c:v4F>@G>J 6HGCE>TH8$CE> T9;@2:<;@2 Nx; :<;HN O5;@2:M= 6982E8(NH698FTH8t4F>@G92PCPOP6HABO4F>@G :<;@2:I87CPO5;@2TH8 6HG ; :lda ;lN 87CE> Σ 2E8:B:<; XY; )%" %+3F " " > 3 %+(!X*,% )%'" !V33 TH8 Σ 6 K 8aOE87NHOE8F2E87GMC5; Om ; 4F>@X> 6 1"2v87R AITH87GCE8 M= 698 8F2v6HG*4F>@G>J 6HGCE>ABGX9Q GHABCE> 6 1cG*8Ra 8F47CE> 87G-87:#Nl8F>@Oi4J;@2E> 6HG-; :lΣa ;N 87CE>*TH8(6HGx;2E>@:<;*:I87ΣCPO5; Σ ; 92E8CE87G9T\Om;.M= 698 ∗ ∗ i i i i i i i 5 B i =M698ΣCPAI87G9=8 9{λ,>>NMa,RMA<; aa, X87aaa, GMCE8 x.6H.GV.}G HX87OE>ABGXQ9GHABCE>TH887:I87X87GCE>{2 6 g%!)%"\) !V3!2!" 1"2PC5;2E8F4F47AIW@G[TH8R9Q G98; :7{I 6HGx;@2>@N87O5;@47AI>@G98F225>>HN OE887:?4F>@G>J 6HGCE> TH8CE>MT9;@2:<;@2iNx;k :<;HN O5;@24F>@G92PCPOP6HAIT9;@2 4F>@G :<;@2:I87CPO5;@2iTH8(6HG ; :lda ;lN 87CE> Σ = {a , a Σ, a , . . . , a } 6 b% ) K 8J; GTH>{2YNx; :<;HN O5;@2 x, y ∈ Σ 6 K 8-2E6HNl>@G9T\Om; = 698 |x| = i 6 K 8$:B:<; XY;\)!X %'!) %D3 !" !V3!2!" x % y E 2E8$OE87NHOE8F2E87GC5;ABGXk |y| = j T\AI2ECPABGC5; X87GCE8 TH8:<;@2a >@OPXY;@2 > x.y ; >@CPO5;Nx; :<;HN O5; z = 698a2E8>>HN CPAI87G98; x;@T\AI87G9TH> :<;@2i:I87CPO5;@2 >*2 BXjlN >@:I>{2TH8 y ;*xy 4F>@GMCPABG6x;@47AIW@G[TH8:<;@2i:I87CPO5;@2i>2 BX@N >@:I>{2iTH8 x 6 =?>@O 8 J 87X*NH:I>92EA x = abbaa 8 y = bbbbb xy = abbaabbbbb 6 H ;G9>@C5;@47AIW@G x.y 2E86HCPAB:BA7h ; OmL; 476x; G9TH>LNH698FT9;NHOE>MT\6947ABOE2E8; :7{I 6HGx;L; XjHN A7I 98FT9;@T 2EAr2E8(6HCPAB:BA7h ;:<;*G9>@C5;@47AIW@G xy x>476x; G9TH>258$TH8F2E8F88F2ENl8F47AlxQ 4J; O:<;*>@Nl87O5;@47AIW@G[TH8$4F>@GXk 4J; CE87Gx;@47AIW@G-X8FT\A<; GCE86HG 2 BX@N >@:I> 6 W 9N 2ED7OPRS8F2E8\M= 698*:<;4F>@G94J; CE87Gx;@47AIW@G vTH8R9Q GHAIT9; ;@2 r8F2(6HGx;8 \CE87G92PAIW@G TH8*:<;4F>@GXk 4J; CE87Gx;@47AIW@GTH86HG*2 BXjlN >@:I>4F>@G*6HGx;Nx; :<;HN O5;@C5; :94F>@X>2E8TH8R9Q GHAIW(87G 5$6 PV675 NH698F2o6HG 2 BX@N >@:I>t>t:I87CPO5;t2E8 NH698FTH84F>@G92PAITH87O5; 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6HGV= 698t:<;8 N>{2PAI47AIW@G[2E8$NH698Rk T9;<x9 ;@4F87O 4F>@X>;>= 6 87G a >@OPXY;ABGCP6HABCPABR{; 6 =?>@O(87:B:I>L2E8tTH8 PJ ; G:<;@2TH87X>{2PCPO5;@47AI>@G98F2 a >@OPXY; :I8F2(Nx; O5;L:I>{28 J 87OE47AI47AI>{2 6 1"2PCE>LG9>V2PA7{I GHAlxQ 4J; M= 698*87:cCPO5;xN ;J > TH8ia >@OPXY; :BA7h ;@47AIW@G 258J;[X87O5; X87GCE8OE87CEW@OPAI4F>"8F287RMAITH87GMCE8 M= 6986HGx; Nl87OE2E>@Gx; 4J; Nx;h TH8V8Ra 8F47CP6x; OTH8Rk X>{2ECPO5;@47AI>@G98F2iOPA7{I 6HOE>{25;@2TH8NHOE>@NHAI8FT9;@TH8F2 2E87G947AB:B:<;@2 x4F>@X>*:<;@2FM= 698(2E8 ABG9T\AI4J; G;>M= 6 2587Om ; 4J; Nx;h TH8 ; Gx; :BA7h ; O87GNHOE>a 6HG9T\AIT9;@TL>@CPOE>{2aNHOE>>HN :I87XY;@2X<@; 2TH87:BAI4J;@TH>{2 = 6982E8:I8 NH:<; GMCE8F87G 6 H ;*4F>@G94J; CE87Gx;@47AIW@G[TH8Nx; :<;HN O5;@2iCPAI87G98(:<;@2i2EA7{I 6HAI87GCE8F2NHOE>@NHAI8FT9;@TH8F2 H ;4F>@G94J; CE87Gx;@47AIW@Gy8F26HGx;->@N87O5;@47AIW@G4F87OPO5;@T9;87G87:a4F>@G>J 6HGCE> TH8Nx; :<;HN O5;@2 • 6 Σ =cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; x(yz) = (xy)z 6 • =?>@O476HX*NH:BABO:<;@2aTH>{2"NHOE>@NHAI8FT9;@TH8F2i; GMCE87OPAI>@OE8F2 \:<;$4F>@G94J; CE87Gx;@47AIW@GVTH8:<;@2"Nx; :<;k HN O5;@2 TH86HG-; :lda ;N 87CE>*8F26HG " %+4 (T> & 6 ∗ ∗ 1 2 3 N ∗ ∗ i i i i i i i • • • • i 1 AI2ECE87G947A<;*TH887:I87X87GCE>*G9876HCPOE> 6VH ;tNx; :<;NHO5;tR@;@4 <; 8F287:/87:I87X87GCE>*G9876HCPOE> TH8":<;4F>@G94J; CE87Gx;@47AIW@GTH8Nx; :<;NHO5;@2 C5; GMCE> Nl>@O?:<; TH87OE8F4mλ9x;4F>@X> N>@Or:<; A7h =6HAI87O k T9; 6 1cGR8 a 8F47CE> S2E8J; x 6HGx; Nx; :<;NHO5;476x; : =6HAI87O5; 6 K 8476HX*NH:I8F=698 λx = xλ = x 6 H ; H:BCPABXY;7A I{6x; :IT9;@T[2E8tTH87X698F2ECPO5\; a ;@47AB:BX87GCE8tN>@OOE8F476HOE2EAIW@G[87G x l; 6HGV=698 8F2i>> NR A<;;GHABRS87:rABGCP6HABCPABRS> 6 =?>@Oa476HX*NH:BABO":<;@2cCPOE8F2"NHOE>@NHAI8FT9;@TH8F2a; GMCE87OPAI>@OE8F2 S:<;4F>@G94J; CE87Gx;@47AIW@GLTH8Nx; :<; NHO5;@2 8F26HG 4 ))% 6 H ;t4F>@G94J; CE87Gx;@47AIW@G TH8 Nx; :<; NHO5;@2G9>CPAI87G98 :<;NHOE>@NHAI8FT9;@T4F>@GHX6HC5; CPABR{; 6 132aCPOPlA k RMA<; : 9x; :B:<; OL4F>@GCPO5;@8 JP87X*NH:I>{2 ABG947:B692E>m4F>@GfNx; :<; NHO5;@2TH8V6HGx; 2E>@:<; :I87CPO5; ":<;@2 Nx; :<; NHO5;@2 a E b RS87OPlA Qx4J; G =M698 ab G9>8F2 7A I{6x; :c; ba 2PAI87X*NHOEi 8 =M698 a E b 2E8J; G 4J; O5;@47CE87OE8F2T\AI2ECPABGCE>{2 6 =cOE>@NHAI8FT9;@T TH8-4J; G94F87:<;@47AIW@G N>@O:<; 7A h =M6HAI87OET9; 6 =v; O5;mCE> TH> a, b, c ∈ Σ∗ 258 476HX*NH:I@ 8 =M698 ab = ac =⇒ b = c 6 `?; @X NHAID7G[2E8(476HX*NH:I8(:<;*:I,8 ETH8(4J; G94F87:<;@47AIW@GNl>@O:<;TH87OE8Fm4 9x; ac = bc =⇒ a = b 6 b% ) K 8J; 6 K 8T\AI4F8?=698 x 8F2 !2&%!TH8 z +E =698 y 8F2 3! HT 8 z 6 UTH87X<;@2 x 8F2S!2gz%=! xy !VTH8 2PA G9>Y8F2:<;*Nx; :<;NHO5;R@;@4 <; 6)M 8$A7I{6x; : XY; 9G 87O5; y 8F2F)3! !LTH8 z 2PA x G9>*8Fz2i:<;yNx; :<;NHO5;R{;@4 <; 6 =?>@O8 J 87X*NH:I>?2E8J; x = abc 6H ;@22EA7I{6HAI87GCE8F2Nx; :<;NHO5;@22E>@G 4J;Nl8,h ; TH8 x λ a 6XH ;@22P7A I{6HAI87GMCE8F2Nx; :<; NHO5;@22E>@G4F>@:<;tTH8 x λ c bc abc 6 `r> T9;@22E>@GNHOE>@NHA<;@2 ab abc 8 H4F87NHCE> abc 6 K 8V>> N92E87OPR{; mO ; =698L:<D ; a 6HG947AIW@G S\LHWhT3]sJqVFN NhfVXWECEHS\E&c% GQEVCPAI87G98 cOE8F2ENl8F47CE> ; :<; 4F>@G94J; CE87Gx;@47AIW@G ?6HGx;NHOE>@NHAI8FT9;@T 2E87X8 JE; GCE8;:<; a 6HG947AIW@G SmLZTeEHGp]qJq[RLOE8F2ENl8F47CE> ;:<; X6H:BCPABNH:BAI4J;@47AIW@GTH8$G HX87OE>{2iOE8J; :I8F2 |xy| = |x| + |y| 6 b% ) K 8:B:<; XY; &V %'! "Z 4!TH8 6HGx;Nx; :<; NHO5; a :<;t>@Nl87O5;@47AIW@G =M6984F>@G92PAI2PCE8L87G4F>@G94J; CE87Gx; OP:<;[4F>@G92P7A IS>X*AI2PXY; i RS8F4F8F2 6 a>@X>:<;-4F>@G94J; CE87Gx;@47AIW@G CPAI87G98t:<;NHOE>@NHAI8FT9;@T ;@2E> 47A<; CPABR@; lG9>L8F2NHOE8F47AI2E>L8F2PN8F47lA Qx4J; O(87:o>@OETH87G 87G =M698CPAI87G987G =M698(R8 a 8F47CP6x; OE2E8(:<;@2 i >@N87O5;@47AI>@G98F2 6 RS8F4F8F2 xi = xxx . . . x i 132PCPOPAI47C5; X87GCE< 8 9x; NH:<; G9TH> :<;Nl>@CE87G947A<;G9>L8F26HGx;>@Nl87O5;@47AIW@G GM6987R@; #2EABG9>6HGx; G9>@C5;@47AIW@G-Nx; O5;OE8FT\6947ABO; 7: I{6HG9>{24J;@2E>{2iTH8:<;*>@N87O5;@47AIW@G 4F>@G94J; CE87Gx;@47AIW@G 6 K 8$THR8 Q9G987GVC5; @ X NHAID7G x1 = x XE x0 = λ 6 132i87R AITH87GCE@ 8 =M698(2E8RS87OPlA Qx4J< ; =698 xi xj = xi+j (i, j ≥ 0) i i i i i i i 5,P 1YJ 87X*NH:I>2E8J; i 6 13GCE>@G94F8F2 x = abc 2 x = xx = abcabc x3 = xxx = abcabcabc `?; @X NHAID7G87G*8F2PCE84J;@2E> @:<;@2oNHOE>@NHAI8FT9;@TH8F2"TH8a:<; a 6HG947AIW@GS\LHWhT3]sJqVFN2E>@G*2E87X8 JE; GCE8F2 ; :<;@2"TH87::I>>I; OPABCPX> |xi| = i × |x| TH>@G9TH8 87:2PA7I{G9> × OE87NHOE8F2E87GMC5;$:<;$X6H:BCPABNH:BAI4J;@47AIW@G $ TH8$G HX87OE>{2 87GMCE87OE>{2 6 b% ) K 8J; x = σ σ . . . σ 6 K 8:B:<; XY; !V3!2! %%*,! > !V3 !2! %" ! TH8 x E 2E8*OE87NHOE8F2E87GMC5; x ; x = σ . . . σ σ /8F2TH8F47ABO v;:<;b=M698Y8F2ECm; a >@OPXY;@T9;N>@O:<;@2X*AI2PXY;@2 :I87CPO5;@2i87G >@OETH87GVABGMRS87OE2E> 6 H ;ia 6HG947AIW@GV:I>@G{I ABCP69T 8F2ABGMR@; OPA<; GMCE84F>@GVOE8F2ENl8F47CE>;:<;tOE8cx8 \AIW@G TH8(Nx; :<;HN O5;@2 1 2 R n R n 2 1 |xR | = |x| ! & -/'5%&+P-/4 b% ) K 8 :B:<; XY; 3%'()!+*,%*25>>NHOE8 87:#; :lad;Nl87CE> Σ ;476x; : M= 6HAI87Oi2P6lN 4F>@GXk J 6HGCE>YTH87:/:I87GI{6x;J 8$6HGHABRS87OE25; :rTH8 Σ 6 ENl>@OV87:B:I> 1cG Nx; OPCPAI476H:<; O 87:4F>@G>J 6HGMCE>R@;@4 I> 8F2V6HG 2P6N4F>@G>J 6HGMCE>yTH8 258*:B:<; XY;LC5; X@HN AID7G03%'()!+*,% ! 6 1"2P∅CE8:I87G{I 6x;JP8*G9> TH8,N84F>@GXa 6HΣG9T\ABOE2E84F>@Gm87: M= 6984F>@GCPAI87G98L4F>@X> HGHAI4F>-87:I87X87GMCE>[:<;VNx; :<;HN O5; R@;@4 <; {λ} =M698YC5; XjNHAID7G8F26HG 2E6lN 4F>@G>J 6HGCE>TH8 Σ JT\Ala 87OE87GCE8TH87: ; GCE87OPAI>@O 6 =o; O5;iT\AI2PCPABG{I 6HABOP:I>{2 ZxN ;@2PC5;iT9; OE2583476987GC5; TH8 M= 69887:lG HX87OE>*TH8 87:I87X87GCE>{2iTH8:I>{2TH>{24F>@G>J 6HGMCE>{28F2aT\AI2ECPABGCE> 5 87GL87:4J;@2E>TH8 E c 87G-87:/TH8 ∅ 6 {λ} W 9N 2ED7OPRS8F2E8@M= 698$C5; GMCE> 4F>@X> 2E>@GV:I87G{I 6x;PJ 8F2:= 698$NH698FTH87GTH8R9Q GHABOE2E8t2E>>HN OE8 476x; : M= 6HAI87O ; :lda ;lN 87CE> 6 =v>@Oi>@CP∅O5;tNx; OPCE8 {λ} H87:#NHOE>@NHAI>; :la ;N 87CE>NH698FTH8(4F>@G92PAITH87O5; OE2E8(4F>@X> 6HG9>-TH8*:I>{2$:I87G{I 6x;PJ 8F21SI 87G987O5;@TH>{2tN>@OD7: r;>M= 69D7:&M= 6984F>@GCPAI87G98CE>MT9;@2:<;@2$Nx; :<;HN O5;@2 N>{2PA7HN :I8F2iTH8$6HGx;2E>@:<;:I87CPO5; 6 K 8iABGCPOE> T\694F8;99 >@O5;(6HGx;(G9>@C5;@47AIW@G = 6982PABX*NH:BAlxQ 4J;$87G; :7{I 6HG9>{24J;@2E>{2:<;(TH8F2E47OPABNXk 47AIW@GTH8:I>{23:I87G{I 6x;PJ 8F2 6 1cG:I>$2E694F8F2PABRS> {2PA987:B:I>(G9>$T9; :B6I ; Oa;(8= 6 BRS> 4F> {2E8TH87G9>@C5; Om; 4F>@G 87:c2 BXjlN >@:I> x 87:v:I87G{I 6x;J 8 {x} a >@OPXY;@TH> HGHAI4J; X87GCE8*N>@O:<;YNx; :<;HN O5; x ∈ Σ 6 =v>@O8 J 87X*NH:I> l87:?:I87G{I 6x;PJ 8i= 6982EW@:I>Y4F>@GMCPAI87G98:<;Nx; :<;HN O5;R@;@4 <;Y2E8TH87G9>@C5; Om; 2PA?G9> T9;t:B6I ; O ;*4F>@GXa 692EAIW@G a4F>@G 87:/2 BX@N >@:I> λ X E 87:#:I87G{I 6x;PJ 81M= 698$4F>@GCPAI87G98 HGHAI4J; X87GMCE8 ;:<;Nx; :<;HN O5;<a >@OPXY;@T9;N>@O :<; HGHAI4J;:I87CPO5; σ 2E8tTH87G9>@C5; Om ; A7{I 6x; :BX87GCE8*N>@O σ 6W NXk 25D7OPRS8F2E8 = 6984F>@Gy8F2PCE>{24F>@GRS87GHAI>{2.x9 ; E 47AI87OPCE>D{I O5;@TH>[TH8; X@HN A7I 98FT9;@Ty87Gy; :7{I 6HGx;@2 8 \NHOE8F2PAI>@G98F2 9NH698F2 σ NH698FTH8TH87G9>@C5; O6HGx;:I87CPO5;H6HGx;Nx; :<;HN O5;Y>6HGV:I87G{I 6x;J 8 6 >@O k XY; :BX87GMCE8 87:94F>@GMCE8 CE>$87GK= 698i; Nx; OE8,Jh 4J;:<;8 \NHOE8F2PAIW@GTH87CE87OPX*ABGx; Om ; ;:= 69D2E8OE8RxQ 87OE8 87:/2 BX@N >@:I> V E 2EA#G9> a 698F258$;@2 /2E86HCPAB:BA7h ; Om*; :<;tG9>@C5;@47AIW@G-G9>; X@HN A7{I 6x; 6 ∗ ∗ ∗ i i i i i i i i [ g%!)%"\) 3%+(!+*%" b% ) ^ K 8J; GVTH>{2 :I87GI{6x;J 8F2 TH8RQ9GHAITH>{22E>>NHOE8(87:rX*AI2PX>; :la ;N87CE> 6 L ,L ⊂ Σ K 8(:B:<; XY;\) TH8:I>{2iTH>{2i:I87GI{6x;JP8F2 L + L x; :#:I87GI{6x;JP8$TH8RQ9GHAITH>;@2 8F2"THL8F47ABO +; L:x4F= >@G>J {x|x 6HGCE>:∈a >@LOPXY∨;@THx>(∈N>@LO3:<};@2cNx; :<;HN O5;@2&=M698N87OPCE87G98,hJ4J; GYABG9T\AI2PCPABGMC5; X87GCE8 ;t6HG9>6 >@CPOE>*TH8(:I>{2iTH>{2:I87G{I 6x;PJ 8F2 6 =?>@OoO5;Jh >@G98F2H9 AI2PCEW@OPAI4J;@2 2E8"2E6987:I836HCPAB:BA7h ; Oc87:M2 BXjlN >@:I>yNx; O5;OE8Ra 87OPABOE2E8a;i:<;i6HGHAIW@G TH8$:I87G{I 6x;J 8F2 H87G RS8, h TH87:/2 BX@N >@:I>TH8(6HGHAIW@G-TH8(4F>@G>J 6HGMCE>{2 6 M 8;@476987OETH>4F>@G8F2PC5;*TH8R9Q GHAI47AIW@G E :I>{2 4F>@GRS87GHAI>{2ABG9T\AI4J;@∪TH>{2 X<@; 2; OPOPA7xN ;x476x; :lk M= 6HAI87O :I87G{I 6x;J 8ja >@OPXY;@TH>N>@O 6HG[G HX87OE><9Q GHABCE>LTH8Nx; :<;HN O5;@2 {x , x , . . . , x } 2E8 NH698FTH88F2E47OPA7HN ABO 87GV:<;ja >@OPXY; 6 x + x + ... + x H ;6HGHAIW@G TH8(:I87G{I 6x;PJ 8F2CPAI87G98(:<;@22PA7{I 6HAI87GCE8F2 NHOE>@NHAI8FT9;@TH8F2 W Nl87O5;@47AIW@G 4F87OPO5;@T9; 9:<;t6HGHAIW@GVTH8(TH>{2:I87G{I 6x;J 8F2 2E>>HN OE8(87:#X*AI2PX>; :lda ;lN 87CE>*8F2 • C5; X@HN AID7G-6HGV:I87G{I 6x;J 82E>>HN OE8$T\AI4 99 >; :la ;N 87CE> 6 =cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; (L + L ) + L = L + (L + L ) 6 • =?>@O$476HX*NH:BABO(:<;@2TH>{2 NHOE>@NHAI8FT9;@TH8F2; GMCE87OPAI>@OE8F2 :<;6HGHAIW@G[TH8:I87G{I 6x;PJ 8F2(8F26HG " %'4 (T> g 6 1 AI2ECE87G947A<;TH8387:I87X87GCE> G9876HCPOE>J476x; : M= 6HAI87O5;FM= 698c2E8J;87:S:I87G{I 6x;PJ 8 L F87:{:I87G{I 6x;k • J 8R{;@4 I>476HX*NH:I81M= 698 ∅ + L = L + ∅ = L 6 =?>@O(476HX*NH:BABO :<;@2 CPOE8F2NHOE>@NHAI8FT9;@TH8F2(; GCE87OPAI>@OE8F2 l:<;6HGHAIW@GTH8:I87G{I 6x;J 8F28F2i6HG 4 % 6 =cOE>@NHAI8FT9;@T4F>@GHX6HC5; CPABR{; v476x; :I8F2m= 6HAI87O5; = 6982E8J; G L E L ?2E8RS87OPAlxQ 4J; M= 698 • 6 L +L =L +L =?>@O CE87G987O :<;@2 476x; CPOE>NHOE>@NHAI8FT9;@TH8F2; GMCE87OPAI>@OE8F2 :<;*6HGHAIW@G-TH8:I87G{I 6x;J 8F2 8F2i6HG 4 %e!2g%+3 !V 6 =cOE>@NHAI8FT9;@TLAITH87X*Nl>@CE87GMCE8 H476x; : = 6HAI87O5;iM= 698258J; L 2E8RS87OPAlxQ 4J;i= 698 L + L = L 6 • b% ) K 8J; G TH>{2 :I87G{I 6x;J 8F2-TH8R9Q GHAITH>{2-2E>>HN OE8 87:$X*AI2PX>f; :lda ;lN 87CE> TH8:I>{2 TH>{2:I87G{I 6x;PJ 8F2 L L > x; :BCE87OPGx;k 6 K 8(:B:<; XY;.!X %+!' L ,L ⊂ Σ CPABR{; X87GCE8 L .L 9; :r:I87G{I 6x;J 8$TH8R9Q GHAITH>;@2 8F2(THL8F47ABLO :<=;@2 {xy|x Nx; :<;HN ∈O5;@2$L TH,87y:c∈:I87GL{I 6x};PJ 84F>@G94J; CE87Gx;@47AIW@G TH8>@CPOE>{2(TH>{2(2E8ja >@OPXY; G[4F>@GXk 4J; CE87Gx; G9TH>Y6HGx;Nx; :<;HN O5;*TH87:/NHOPABX87O :I87G{I 6x;J 8$4F>@G->@CPO5;*TH87:/2E8,{I 6HG9TH> 6 1 ∗ 2 1 1 2 1 2 2 1 1 2 2 N N 1 2 3 1 2 3 1 1 2 1 2 i i 2 1 ∗ 1 1 2 2 1 2 2 1 2 i i i i i 5 i H ;*4F>@G94J; CE87Gx;@47AIW@G[TH8:I87GI{6x;J 8F2 CPAI87G98(:<;@2i2PA7I{6HAI87GMCE8F2iNHOE>@NHAI8FT9;@TH8F2 W Nl87O5;@47AIW@G 4F87OPO5;@T9; :<; 4F>@G94J; CE87Gx;@47AIW@G TH8-TH>{2:I87GI{6x;JP8F22E>>NHOE8-87: X*AI2PX> • ; :la ;N 87CE>8F2i>@CPOE>:I87G{I 6x;J 8$25>>HN OE8(87:/X*AI2PX>Y; :lad;Nl87CE> 6 =cOE>@NHAI8FT9;@T-;@2E> 47A<; CPABR{; (L L )L = L (L L ) 6 • =?>@Oo476HX*NH:BABOo:<;@2rTH>{2?NHOE>@NHAI8FT9;@TH8F23; GCE87OPAI>@OE8F2 :<;i>@N87O5;@47AIW@GTH8"4F>@G94J; CE87Gx;@47AIW@G TH8(:I87G I{6x; JP8F2 8F26He G " %+4 (T> g 6 1 AI2ECE87G947A<;TH8387:I87X87GCE> G9876HCPOE> J476x; : =M6HAI87O5;F=M698c2E8J;87:S:I87GI{6x;JP8 F87:{:I87GI{6x;k • J 8 a >@OPXY;@TH>*N>@Oi:<;tNx; :<; NHO5;R{;@4 <;*476HX*NH:I18 =M698 λL = Lλ = L 6 L =?>@Oo476HX*NH:BABOo:<;@2rTH>{2?NHOE>@NHAI8FT9;@TH8F23; GCE87OPAI>@OE8F2 :<;i>@N87O5;@47AIW@GTH8"4F>@G94J; CE87Gx;@47AIW@G TH8(:I87G I{6x; JP8F2 8F26HG 4 ))% 6 1c:"4F>@>G J 6HGCE> TH8:I>{2(:I87G I{6x; J 8F22E>> NHOE8*6HG ; l: ad; N87CE> XJ 6HGMCE>V4F>@G :<;@2>@Nl87O5;@47AI>@G98F2 6HGHAIW@G E 4F>@G94J; CE87Gx;@47AIW@G o8F2$6HG 2 % NH698F2 r4F>@X>2E. 8 9x;VRMAI2PCE> r4J;@T9;6HGx;-TH8 8F2 ;@2 >@N87O5;@47AI>@G98F2(8F26HGX>@G9>@AITH8 6 UTH87< X ;@2 l;Nx; OPCPABO(TH8:I>{2 :I87G I{6x; JP8FS2 a >@OPXY;@TH>{2 HGHAI4J; X87GMCE8Nl>@O:<;Nx; :<; NHO5;YR{;@4 <;Y>YNl>@O 6HGx; HGHAI4J;Nx; :<; NHO5;TH86HGx;2E>@:<;*:I87CPO5; )E X8FT\A<; GMCE8L:<;@2>@Nl87O5;@47AI>@G98F2TH86HGHAIW@G E 4F>@G94J; CE87Gx;@47AIW@G 38F2tN>{2P7A NH:I8 IS87G987O5; O476x; l: k =M6HAI87O:I87G I{6x; JP8(2E>> NHOE86HG ; l: a ; N87CE> Σ 8 H4F87NHCE> ∅ 6 =?>@Oi:I>C5; GMCE> 987:r; l: ad; Nl87CE>*8F26HG 4F>@>G J 6HGCE>*THS8 IS87G987O5;@TH>@OE8F2Nx; O5;t87:#4F>@>G J 6HGMCE> L TH8 CE> TH>{2:I>{2a:I87G I{6x; JP8F2 =698 NH698FTH87G THR8 Q9GHABOE258$2E>> NHOE8 Σ \N>@O:I. > =698 L 2E8(TH87G9>@X*ABGx\ ; 2 % 3 2%.IS87G987O5;@TH>YNl>@O Σ 6 b% ) T K 8:B:<; XY; gT %'!e "Z 4!*TH8i6HG*:I87G I{6x; J 8 ; :<;(>@Nl87O5;@47AIW@G =M6984F>@G92PAI2PCE8L87G4F>@G94J; CE87Gx; OP:I> 4F>@G92P7A IS>X*AI2PX> i RS8F4F8F2 6 a>@X>:<;-4F>@G94J; CE87Gx;@47AIW@G CPAI87G98t:<;NHOE>@NHAI8FT9;@T ;@2E> 47A<; CPABR@; lG9>L8F2NHOE8F47AI2E>L8F2PN8F47lA Qx4J; O(87:o>@OETH87G 87G =M698CPAI87G987G =M698(R8 a 8F47CP6x; OE2E8(:<;@2 i >@N87O5;@47AI>@G98F2 Li = LLL . . . L i RS8F4F8F2 6 132PCPOPAI47C5; X87GCES8 9x; NH:<; G9TH> S8F2PC5;(G9>(8F2c6HGx;>@Nl87O5;@47AIW@GG6987R{; S2PABG9>6HGx; G9>@C5;@47AIW@G =M698(2PABX*NH:BlA Qx4J;; 7: I{6HG9>{2 4J;@2E>{2iTH8:<;>@Nl87O5;@47AIW@G 4F>@G94J; CE87Gx;@47AIW@G 6 K 8(THR8 Q9G987GC5; @ X NHAID7G 1 E 0 6 13287R AITH87GCE?8 =M698(2E8 RS87OPlA Qx4J; =698 Nx; O5; CE> TH>:I87G I{6x; JP8 L ENx; O5;LCE>MTH=> Li, j ≥L 0 = λ 1 2 3 2 3 1 b% ) [ K 8:B:<; XY;i %'!' '%%>j3!V"Z!YTH86HGY:I87GI{6x;JP8 L 8F2rTHL8F47ABO =J87∪:S:I87GI{L6x;JP8g=698c258c>>NHCPAI87G98c6HGHAI87G9TH> 87:{:I87GI{6x;JP8 H 4F>@GCE>MT9;@2/2E692Nl>@CE87G947A<;@2 N>{2PA7HN :I8F2 \ABG947:B6925> L 6 W N R A<; X87GCE8 :<;47:<; 692P6HO5;TH8(476x; : M= 6HAI87Oi:I87G{I 6x;PJ 8$4F>@GCPAI87G98:<;Nx; :<;HN O5;R{;@4 <; 6 b% ) K 8$TH8R9Q G98(:<;.3!V"Z! g)"Z Z !TH8(6HG:I87G{I 6x;PJ 8 L ;@2 Li Lj = Li+j ∗ ∞ i=0 i 0 L+ = ∪ ∞ i=1 Li i i i i i i i 4 i H ;@2 2PA7I{6HAI87GMCE8F2iAITH87GMCPAIT9;@TH8F22E>@GV87R AITH87GCE8F2 L∗ = L + ∪ λ L+ = LL∗ = L∗ L H ;Y>@N87O5;@47AIW@G TH847:<; 692E6HO5;?J 692PCPAlQx4J;Y:<;*G9>@C5;@47AIW@G l6HCPAB:BA7h ;@T9;87G-:<;LK 8F4F47AIW@G 5$6 P TH8 Σ Nx; O5;*87:3%'()!+*,% )%" !V3vTH86HG-; :lad;Nl87CE> Σ 6 b% ) K 8J; L 6HG:I87GI{6x;JP8 476x; : =M6HAI87O5; 6 K 8 :B:<; XY;i3%+(!+*% %%*, E 2E8OE87NHOE8F2587GC5; L 9; >\ %" TH8 L X 8F2 TH{x 8F47ABO |xx; ∈:M= L} 698(4F>@GCPAI87G98$:<;@2iNx; :<;HN O5;@2iABGRS87OE2 ;@2;:<;@2iTH8 L 6 =c698F2PCE>?M= 698:I>{2o:I87G{I 6x;PJ 8F2c2E>@G*4F>@G>J 6HGCE>{2 @C5; XjHN AID7GNH698FTH87GTH8R9Q GHABOE2E8:<;@2o>@Nl87O5;k 47AI>@G98F2tTH8K %'" %') E)4 3%'4 %' !) 4F>@GmOE8F2PN8F47CE>-; :":I87G{I 6x;PJ 86HGHAlk RS87OE2 ; : 6 M ;@TH> M= 698DF2PC5;@2Y2E>@G>@Nl87O5;@47AI>@G98F2L47: @; 2PAI4J;@2LTH8 CE8F>@O <; TH8-4F>@G>J 6HGMCE>{2 aG9> 8F2NHOE8F47AI2E>TH87C5; :B:<; OP:<;@2$;>= 6 6)d 8F4769D7OETH8F2E8j= 69887:v4F>@G>J 6HGCE> %" %' TH8>@CPOE>{2 TH>{24F>@GMCPAI87G98L:I>{2t87:I87X87GMCE>{2\M= 6982E887G9476987GCPO5; G 87G; XjlN >{2 L ∪ L = {w|w ∈ E 87:4F>@X*NH:I87X87GMC5; OPAI>[TH86HG 4F>@G>J 6HGCE>4F>@G OE8F2ENl8F47CE>;V>@CPOE> M= 698:I> L ,w ∈ L } 4F>@GMCPAI87G988F2PCm; a >@OPXY;@TH> N>@O/:I>{2v87:I87X87GCE>{2cTH87: 2E8,{I 6HG9TH>SM= 6983G9>8F2PCm; G87Gt87:NHOPABX87OE> ∗ R R 1 1 2 2 Lc = Σ ∗ −L = {w|w ∈ Σ∗, w ∈ / L} D )f}=1}=Z E 6+M 8F47ABOa476T; :I8F22E>@GY:<;@2"Nx; :<;NHO5;@2]=M698 258 >>NHCPAI87G987GL4F>@X> OE8F2P6H:BC5;@TH>TH8(:<;@2 2PA7I{6HAI87GCE8F2>@Nl87O5;@47AI>@G98F2 xx xxx x x x 6 6T; :I8F2i25>@G 2P692:I>@GI{ABCP69TH8F2 M 8RQ9GHABO Σ 6 K 8J; Σ = {a, b, c} x = aa y = b z = cba 6M 8R9Q GHABOi:<;@2i2PA7{I 6HAI87GCE8F2 Nx; :<;NHO5;@2 2E>@G 2P692 :I>@G I{ABCP69 TH8F2x 4J;N 8,h ;@ 2 E 4F>@:<;@ 2 6 6T; :I8F2 xx xy xz yx yy yz zx zy zz xyz x x z (xz) (yxx) K 8J; Σ = {a, b, c} 6 132E47OPA7HN ABO2E87AI2TH8 :<;@24J;@TH87Gx;@2X<@; 24F>@OPC5;@2TH8 Σ E TH8 Σ 6 M 8R9Q GHABO(:<;4F>@G94J; CE87Gx;@47AIW@G TH8Nx; :<;HN O5;@2(TH8ja >@OPXY;OPA7{I 6HOE>{25;#TH8t;@476987OETH>4F>@G :I>{23; AI>@XY;@2cTH8:<; 4F>@G92PCPOP694F47AIW@GTH8aNx; :<;HN O5;@2 6 1 \NHOE8F25; OcTH8a:<; X*AI2EXY; XY; G987O5; :<;@2NHOE>@NHAI8FT9;@TH8F2 TH8$8F2PC5;>@N87O5;@47AIW@G 6 M 8R9Q GHABO :<;4F>@G94J; CE87Gx;@47AIW@G[TH8:I87G{I 6x;J 8F2TH8:a >@OPXY;*OPA7{I 6HOE>{25;9TH8;@476987OETH>4F>@G :I>{23; AI>@XY;@2cTH8:<; 4F>@G92PCPOP694F47AIW@GTH8aNx; :<;HN O5;@2 6 1 \NHOE8F25; OcTH8a:<; X*AI2EXY; XY; G987O5; :<;@2NHOE>@NHAI8FT9;@TH8F2 TH8$8F2PC5;>@N87O5;@47AIW@G 6 5$6 K 8J; Σ = {$} x=$ 3 BX6 5 0 ∗ 3 2 2 2 3 XG 6 PV6 LX6 i i + ∗ i i