A CLP(FD)-based model checker for CTL
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(3) Linköpings universitet Department of Computer and Information Science. Final Thesis. A CLP(FD) based model checker for CTL by. Marcus Eriksson LITH-IDA-EX— 05/056— SE 2005-06-08. Supervisor: Vladislavs Jahundovics Department of Computer and Information Science at Linköpings universitet Examiner: Ulf Nilsson Department of Computer and Information Science at Linköpings universitet.
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(20) ¾. G 1¡¢YY£XYD¤1Y¡¥ F ¡FÕ=D&Y/Q0Gë´KHpȤ1Y¤>\J WNó;æåçæ è*éIî,êëäFï ïfô!úðçúMäWè4ïì/ä1é3ô ìí^úîJû,úðYôçð<îðYô&æ ì úî÷@òSðYò/äð ä1ìõ V ä1è4ïO÷Zä1ê &ïõnø4ä1è*ôëä1ì>ðçúWô åWðYò>ïè4ï ä.JkD 0*ô iìIôç<9ðp</ïX40md4!æ ",é^ä1"(ô ìN,<EAD$%p,-1$8",[6g L\NiA_Tj OPl!k hgigm 0@R;i?0*j 9!PlEk ",<m#40=R.Jb,,0*$> J0*4I"1(07J<4!@0hg02*';!",d0æ FìJ*08ú-1ðYIèZ",ä1<'*ô4!ì>",0 ðnRi= æíFôY÷cj;è4æ'íF",èZä1(é37,$M1é3-(Ã6ô ì'DíW(<æ(( ø1)3",ï è -1", LCEb5%<40^!O,$G56<((L L!O0* (9/J'*)3 0MLJ60*3,$GJ" "ö-(d4(Ã6J@0*(' ",", FI0 4h9",'*'**B< ",( J%E"-1(",%'*!",4(RJ O@0*)+'",", F909A F 0*J"1F0M0"1'*0*J(K'E0**42'* 4R/(JB@0* '", 0*F0 !M0*0*'*L<!(MLA J<%0*WR J",'*?O4!< (K!",('",) E C W 3 ) 9 3 ) 9JÃ6"((1 ",(",BXÃ6-1",LJ a:Lab? ;RPYL<0IT",þ ÿ%5%$&( 'I0*c",n<(4)3hB9!",< J)3"190* (On)3,$dÃ6 ./ J"1",0*'* ?X(Ja3<L4!<99/(Q('<0^a!",",(?X'",<)+40 V21-(' ,$:4(J@0*'", F0W<L99/('<0* b K0*BRJ"(<!^(9/'"10* (J b b b ",J b*"10*!@$&? U0*G$&( 15% o F1 F2 F1 θ = F 2. θ. VAR {true, f alse}. VAL. sol(C). C. VAR → VAL. σ :. C. CON ⊕ : CON × CON → CON ⊗ : CON × CON → CON Π : VAR∗ × CON → CON : CON × CON → CON ¬ : CON → CON : VAR × VAL → CON • sol(C1 ⊕ C2 ) = sol(C1 ) ∪ sol(C2 ). eK0*'*d4h!@0. • sol(C1 ⊗ C2 ) = sol(C1 ) ∩ sol(C2 ) • θ ∈ sol(Πx C). θ0 ∈ sol(C). <LJI0*J"10. ∀xi ∈ x : θ(xi ) = θ0 (xi ). • sol(¬C) = sol(true) p \ sol(C). e <?)3CEÃ6)3J4 "110*0H5'*!G!",b'*2E e>56<(T'*)3'*J64F4dLh",",'*J)3TO9R !q A@JLDJin Z0*3-1 ",(, NLJ$=bU"1-10*",4 ((L\ @60LJ*+Z0*-1 (",N'*!b2",R9 '*TA@Z0* (Nbd4()39 )3!D5F%0"1'*0*0< 0*(N b "(Z0%"(%"3<A#(ic'<0E-1$&(",'* L) %$&,($ ' ,0*=0*J"106",?2-1", L%A !#9/'*)3 0<0*b< • sol(C1 C2 ) = sol((C1 ⊗ ¬C2 ) ⊕ (¬C1 ⊗ C2 )). • θ ∈ sol(x v). (n1 , n2 , n3 ). θ(x) = v. {V1 → n1 , V2 → n2 , V3 → n3 } ∗ VAL {(1, 2, ∗)} {(1, 2, X) | X ∈ V AL}. VAR = {V1 , V2 , V3 } VAL = {1, 2, 3} sol((V1 1) ⊗ ¬(V2 2)) = sol(V1 1) ∩ sol(¬(V2 2)) = {(1, ∗, ∗)} ∩ ({(∗, ∗, ∗)} \ {(∗, 2, ∗)}) = {(1, ∗, ∗)} ∩ {(∗, 1, ∗), (∗, 3, ∗)} = {(1, 1, ∗), (1, 3, ∗)} sol(Π{V2 ,V3 } ((V1 1) ⊗ ¬(V2 2))). L$&J(<'dBM<LJö.J'*2"B'd"",4'*(,3J'*3õ(@0*ô 4è4'*hïLJ÷0*Zðp0dïJõ^!<:0-(éI*Jæ1 ?"1ú0dð#LJí56<ï3ì>Hï5%èZ0* ä1 ;ê;'*(î,Lì(ô O>Kï(è0*Lú0:bJ40*(('dF+õ(0*'*)3éU(í/=î(5%ú:7$&!(':S*<-14'*",( '*9'<!00*",A%R( CEN:A32CE)34+L,'d0*L -1 H"1 .J0*0* (' 0*'*)+E56d4'*"10*(A r. = {(∗, 1, ∗), (∗, 3, ∗)}. true. sutSvwsutSxytSz{yx"{y|wt3}S~zwxu{ysy z{1{y|\{" x1 S x1xZ{y xu\tS.
(21) 0GëöËW/£T³=8¤Y/n°71¡¡M7/À/YH*ËW°J ´ F ¡FÕ=D&Y/0 G!] ïó#èZòä1ôYê;÷@òKî,ìä1ô O>ê #ïø4è ä1è*ôëäÊ û1ê&ï4H@Dñ@ðp&æ,1ñ@¡ø4,ä1æ è*å6¡ôëäðpQó;û1ê&æI£Xï6é^ðp/ï¥è*äWé+ùù>À/ú ô ¡ìDíä1¡FúMì1ä1õ¤è4¢EïMDåëô!è4ú:Dæ äU&é Õ=éI¡Fø4aæ1ä1úè*H@ð=ôëDäí£Xû1ïê&ì>ïZÀ/ú6ïcèZô Jä1ì Jdê#î,L ìðpæ^ô õ(O>ôø4ïè4ä1èdïè*÷ôëæ ðpäå ïû1õ%ê&ïZéIä1ú6ìæ1ô ìõúðíV ôïì ì/ñ <(LÂQ!O<1-(5'M0*0*!%'*F20*L4h 0*!@0-(:" ?(,AéUí/îbJ0*'*U",!<+4h!@0:"Bõ(éUí/îACEW4h",)39 2R/ 15 dmgu(s, t). s. t. s. t s. t. L .J'. s = f (X, Y ). {X → S, Y → g(Z)} {S → X, Y → g(Z)}. . t = f (S, g(Z)). ff. mgu(s, t) . . G ª f. dmgu(s, t) . . 78 9 ")"! #@ *:< -!2?A7C:B. ,ª0* [6?T'EC%"H0*\r)3R!'*9/^" 4($S'",<L0*'* )3(-((9/!?S('b,",$d%<[6 3(C%(þ !¨\,ÿ]AA !3CE<L!9+)93 X",J30'**(JAö"10^`( 0O'W$&L",c'<0*4h'U9'* J*<39(F0'*(W9/(Q'<0*[6C%+"\,R/",(JL 0 9(X'*<0*FB0M9/",FJ*^< $&RL 0*BL9J'*d"10*@0J"1W00**öA#(]40M0U",0*!<H30*<L9<9/(@0'<0"1*0*FALJn",Fi|0* $&?@0 "10*3O4h!@0*[6C%F0*\!",! +"?+H('%-1",L LJ "1-(0* '(*", ,? $ 0*(^)+@0", "1B0*U",J-1^",'*!",RO 3PU&5%)3(!'*S0*)3J",( F0dR/ @0"10*VZA; 5%$&? 3S(56U:((9/'E)3'"1(0*'*3d @0H"10*0*:+-1Ã6",'*!(", R", )31-(`(6L'7J4%h9F'*0*+*<",d,0*0*2'M[6@0C%"1\K0*((A 9/'"10*('b(56^"2R J",'*?U0*'",J< 0* (+'*!"10* ( 0*J"10 *4'* R/I5%P&!('b9J",",0* ''*IJ,"10*$U -(@0"1 ?+0*4I0h9*J'*"1*0B<4hb 9'**I9/V6F* <eHR 0M K!%90*'/",FJ*<< 0*R ( :JI0*3 (U($&L'*'O() )3@0"10*ç A 0*<'4",0UJ0*,< $0* (n.J QP& 9/0*^F"T*0*<<'R",J(?K< K0* F @0(0**XB9N9J0A"1<0*]J 0I$VZ!A^^"CE(*<LU0*)3'",b#JX(< 0*-(0*J ("1H0I",'*T-(!"1'*0*? 0*(!@0K",%"10*0*@0B"1ö9/0* )3'*)39 AH0!I"4p 0*n1 0^?HCD ",4R"(.J@0 B+(d"A b )3",J%0*J"100* 9<LJ'*(K9/",'<K0@? LJ('*!? EIO@00*"120*!G@00*"1'0*", J< 0*A ('*!"10* (B!:"(*<L)3A ee b J ",J , " ee (' P&('%R/,0*/V eKeK0*0*'*'*dd!!%"%"UU9J9J"1"10*0* <L<LJJOO0*0*JJ"1"100 $&('%-('*? eK0*'*d!$&%"('%U9J-("10*'* ? ",J",JB ", <LJI0*J"10 eK0*'*d!%"U9J"10* <LJO0*J"10 B1 , . . . , Bn. 2n. R. hσ1 , σ2 i ∈ R σ1 σ2. σ1 Rσ2. σ0 σ1 . . .. F. σ0 σ0 σ0 σ0 σ0 σ0 σ0. |= x = v |= F1 ∧ F2 |= F1 ∨ F2 |= ¬F |= ex(F ) |= eg(F ) |= eu(F1 , F2 ). σ0 |= ef (F ). σ0 σ0 |= F. σ0. x ∈ VAR v ∈ VAL σ0 (x) = v σ0 |= F1 σ0 |= F2 σ0 |= F1 σ0 |= F2 σ0 6|= F σ 0 σ1 . . . σ1 |= F σ 0 σ1 . . . σi |= F σ 0 σ1 . . . i≥0 σj |= F1 0≤j<i σi |= F2 σ 0 σ1 . . . ∃i ≥ 0 : σi |= F. i≥0. "n ,A3 ¡c¢£¤¦¥§&¨©S£§Sª. ,"p+J "( 0*AG (CE30*G ':0*)3M(",9/'O"10*('e>=':!@0*$&'*(U) ^CD",b R b A b,",0*J'*M",'*(M", ?B!<G K(9/0*J'"1"100*D@( '$;0*'*b W! b ax ag. au. af. ex eg eu. ef.
(22) Ê G1¡¢YY£XYD¤1Y¡¥ ^" 9J"10* )3",<LJJhD0*J"10S(EW!!G2öJ",-(('*?S @0F0*"1«0*3Dp$&0*(J'G"1",0U D",9Jö"10*R/J^'*"(KEZ$&Ad'*(`) (':4h", T)39(( b @0*9 C8EZ:A '*)3)WR/'80*E)3", ,$0*[6C%\I(9/'"10*('b(E4(L!W"(*<4!"10*#0*E 40<0*' O0*G$&( 15% U)+",'A "h 0*$& O('M0*'*iMGd #h !¬M@00 !!%*"Ä 1 0*(!RJ@.J", O?IM*"1F0*0*! @. JSP&15 ",JO O0*G$&L0*L'* V Lß $¼!%*"10*!@.JO<()3:0* )3d O0*2`L0*L'*. σ0 σ1 . . . σ0 |= ax(F ). ®y®&¯. σ 0 σ1 . . .. F. σ0. °<±²´³Aµ¶h³Aµ1·¸µ1¹Xº¦»"ºA¼S½X¹(½3¾d°<¿±. ÂQ(GI460.J*^ W40*.Jd2[6"W\N$&_5 '*99'*'*<!F"100*"10*A (3,$0*2[6C%\K(9/'"10*(';4.J+ OCD",R :A b )3",0",( 3;"+,$ P&('*(LJJV;[6C%\O!3D$&('*)WL(!"d!",EJ+O"d0*@02"1@00*M"10* *4EZ'*A 90* (+"(;",'*(L)3F0A=CE 0*'0",",J(<0*G0 (@I56H$&'*(@0)"10*0*IG.J*'4@0'* 9@00* "1(0*J:2"0*(UW"0*,2'*(<L)34(JF02"A ,Jö'*40*L'*Jd0*'*L^ en0*'*I!W" "5E,!'*20" (?(L*b)3<OJFú4ä0:GðY",ô! &ú 0JO>MïZ<ú4o]h)+99'*",'*F0**!!<"1%""0*(,IAB'*G4](0WJ e>0@0",*'('*",FW"F0%0BA$&'*$&CE(()'*)W!90*L!d"'*,,0*$!4"1('%0*JW9@0'*!*M0'",*! '*F"102"10*0*-1",'*A !",HR "I2"<9/(24!0", f ()3CE+,O!$6340*(F)3^L9 (-1 9/4",0* I'F"1 0U0*!@0(U0*' b,8$G",[6'*+C%4\h9('*9/bE*<'<"1J0*H4(' Sd)Dp0*+0*" '*'*?H)+O20*4h,$E!@0,3"R/0*I,n4'2h( 99/'*.J'*"1< 0*0*B(T'9JA"("1^20*n`n(5%'2CD4",hR'*",BO)3`l9A (! Ab ",!5E**"" 1?0*I!@$&*"?I10*` !@.EZJA ¸ EH!IFL -1", F0O0* D@ 0B!O,0I0*'*LH0*J"10", G9J"10*JB<('I('O!"10*' (RO?B0*0@CE56WO#'*@00* ('"1",F0*J0M3<J-10*",",(J'*:!",'*R<! !"1UP(0*A% Y("IiM )+"14hh", )W)3L!9)x 2"(,,<$;$#$&"3@0(8L0*''d",@0J*4<"1'*0*0* R/ (VB%:0*'*RJ?!"1"1J0d0*" 9/(-B '*)31%-("%0:0'M< *@0'(","1 J0* <M0*4 (hoç$&J9(M$'*'*)W&*'*<(L)!" 0* 0* ",JS$&'*() 0* ",JS0*S", 7,0*'U@0"10*G0*K0*)+< -(À+!. holds/2. holds(F, S). F. S. step/2 sat/1. ¬af (¬F ). eg(F ). step/2. h1, 1i. h1, 0i. h1, 0i. sat. h1, 1i. step([S1 , S2 ], [S10 , S20 ]) ← sat(or(and(S1 = 1, and(S2 = 1, and(S10 = 1, S20 = 0))),. ('b B .hO,0"10* (. ÁÂ. or(and(S1 = 1, and(S2 = 0, and(S10 = 1, S20 = 1))), or(and(S1 = 0, and(S2 = 0, and(S10 = 0, S20 = 0))) and(S1 = 0, and(S2 = 1, and(S10 = 0, S20 = 1))))))).. xwtÃ\zwtS zÄtS}Ã{y ~z ÅÆ ÅÇ\ ¸xu{Z{ytSx"ÈÊÉwxu{|\SËtX3{ysyzwxy {y ~z{y~xy~È*t xu{Z{ytÆ.
(23) 0GëöËW/£T³=8¤Y/n°71¡¡M7/À/YH*ËW°J. Ö. holds(or(F, G), S) ← holds(F, S). holds(or(F, G), S) ← holds(G, S). holds(and(F, G), S) ← holds(F, S), holds(G, S). holds(not(F ), S) ←∼ holds(F, S). holds(xor(F 1, F 2), S) ← holds(F 1, S), ∼ holds(F 2, S). holds(xor(F 1, F 2), S) ← holds(F 2, S), ∼ holds(F 1, S). holds(ex(F ), S) ← step(S1, S2), holds(F, S2). holds(eg(F ), S) ← holds(not(af (not(F ))), S). holds(eu(F, G), S) ← holds(G, S). holds(eu(F, G), S) ← holds(F, S), holds(ex(eu(F, G)), S). holds(ef (F ), S) ← holds(eu(true, F ), S). holds(ax(F ), S) ← holds(not(ex(not(F ))), S). holds(ag(F ), S) ← holds(not(ef (not(F ))), S). holds(au(F 1, F 2), S) ← holds(and(not(eu(not(F 2), and(not(F 1), not(F 2)))), not(eg(not(F 2)))), S). holds(af (F ), S) ← holds(F, S). holds(af (F ), S) ← holds(ax(af (F )), S). holds(true, S). A. "n ,A AX ¢YÌaÍw¥§SªÃÎ £Ï«£Ð1 ¡c¢£¤¦¥§&¨©S£§Sª. step([S1 , S2 ], [S10 , S20 ]) ← sat((S1 ∧ S2 ∧ S10 ∧ ¬S20 ) ∨ (S1 ∧ ¬S2 ∧ S10 ∧ S20 ) ∨ (¬S1 ∧ ¬S2 ∧ ¬S10 ∧ ¬S20 ) ∨ (¬S1 ∧ S2 ∧ ¬S10 ∧ S20 )). ('b B('",9O,0"10* ( 00. 01. 10. 11. *0WJAT0*CECEG;@0B4"1h0*49 ('*J@0**<'(",J FN"0,WbR/R(1LT-(0M#0*",!N5B?3%@0'*F"10<L0*0*O -1W",-1"(",N"'*F!",0%RÃ6!q@d0(L*JZ", 0*!^UL (4d(JJ,$J@0*@0''3*",5% 9JF$06!&4'*hn(9)4'*(0**J<6( @00*^ ("1560*J (U"LH!^@0","1!<0* 0*'",J< 0* (O!%", 156A (S10 , S20 ). (S1 , S2 ).
(24) G1¡¢YY£XYD¤1Y¡¥. Ñ. Ò ¡À/¤Y/c(/D(¡F1D¥ -1JCE"," '*-(I!",LJR0*^*"G,(<L+",RJ,@0$M*5% 0* 0*LNF0*A"1(0*J (%TU Qì>[6æð7C%*"\Q10*'*!@$&F?3L0* '*!2LE$&J(20'*)W*SL!"(" < <EZA=FCEL@0!E'* (",J!2<,6"$%30*4(^J$&4('*'*)Ò^0*D@JÂQ"10%!56 ÂQJ"10M $=56GJ" -(:0*d LJ@0*'"10* -(:4h",)39 d9'*(('",)g `D (L'*2A . p(X) ←∼ q(X). q(1). r(2). r(3).. Ó3Ô ÕAÖ× ØAcÙyÚ Ú Û¸ªZ©Ã§&¨©SÎ Íw¥*¥Ü¸¨Ý ¤¸Ú ¥¤`§S£Þ§&¨Ý. 0*0*0*$M4'*((F)3b(L956@0E*)356(( FL0*!UU (0*AH620*CEd",(JI4<50#6!"10*0<'0*M70'U",*J)3<56",J' W5%", 0*!4L O!("1'0*0* !%L0*,-($M'<('* RA $'56",%Jö<LW ",-(+'",<JI<5",6J ' "(,)3F",(CEçLA F:0E0* J++0",,$N +F"10*0*U(^4(56)3M9 )3^F!60,",$= 0*BU÷4æ !ìJq@úLðYèJîZ÷0*ðY(ôçø1ï6,ì>$7ï*",íäDðY",ô&æ Jì<56þ 'ÿN60",*JI+"+RJ9/"(F<!<0*", -(? F ¡FÕ=D&Y/Q0 GëK æè4 ïå;êëäåpî,ðYô&ìæ ÷ìJðpúIæ èûú ï P!ðpúpó;üïÖ éOïìcûH@¤æ æ¢&ê!³=ú2û e4²Dõï¤ï÷ì>Ï7ðçSú æ¡FRIðY` ô ìæ ^ åíHÔ DæDûV ,e4ïL J÷ðç` úSåpJ R2î,ìä1WD÷ìò>ðpõ3æ ïBèBù>ä1ð]è4ê \äù>ïßõ(ò/ïZôYä÷Zú ûäà1ï ðpïdïðUè4úpæHæ üådéOæ äèWûù>éIæ ê!è4úØæ æè4íFP!ï^èZúpüä1ä1ééOè<í/ûî,æ éIê!ú2ô!ïúì>õïðçðYú2ì>ò>æïBåëè4ðYôæ úZìéï í ð ÷4ðYò>æ ìJï^úæð]ä1û ì>e4ïð÷V ðGõæ é^ä1ô ìcðpæH÷4æ ìJúðYèî÷ðä1ìæû e4ï÷ð V L åpî,ì÷ðpæ è^ó#ôçðYòä1è*ôçð <ü à1ïè4æ^ô!ú+÷Zä1ê &ïõHä F ¡FÕ=D&Y/Q0 GlÑ]HSá¡F ÏD1¤DDD&È/¡F1¥A¡ J WDò> ûæ å ï ð V ô!úðYò>ïMúZï ð;æ åDä1ê ù=æ1ú*ú<ô]û1ê&ï6íFè4æî,ìõIðpïè*é+ú=ù=ï æ1âGú*ú<ïô]è û1û1ê&èZïGä1ðpìæWõHïî,f(ìù>ôçè4ø1ïZïú*èúúZïî(ú<ô ìíOæ ðYò>å7ï%ä1ìöåpî,ä1ìê ÷ù>ðpò/æ ä(èúñ X 6= 1. X ∈ {2, 3}. p(X). q(X). P. P. UA. A A. P1 P2 P3 P4. : : : :. holds(and(F1 , F2 ), S) ← holds(F1 , S), holds(F2 , S). holds(a, [S1 , S2 ]) ← sat(S1 = 1). holds(b, [S1 , S2 ]) ← sat(S2 = 1). query([B1 , B2 ]) ← holds(and(a, b), [B1 , B2 ]).. &$9L'*JZ0*!("1'o0*d<?)WR/(!o. bb b b. query/1 holds/2 sat/1 and/2 a/0 b/0. Ó3Ô ÕAÖ× ,A A3ã0ܸ¨Ý ¤¸Ú ¥Ê¤`§S£Þ§&¨Ý. _#",'<0%,$D0*$M'*R'",JIL -('<:,$D0*G9'*(('",) `D (L'*2A 256(L!OR/ {a, b, and(a, b), and(and(a, and(b, a))), and(and(a, a), and(b, b)), . . . }.
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(26) ¦ç. G 1¡¢YY£XYD¤1Y¡¥ ô!údäW÷Zä1ù=ê Zæ1V ú<ôçðYôçø1ïGê ôçðpïèZä1êDô ìSúZïé3ôëñëíFè4æî,ìõ÷Zä1ì>æ ìôY÷Zä1êDåçæ è*é,Uè4ï@ù>è4ïZúZïì>ðYô ìíHðYò>ïdò>ï4äõæ å6ä ô!úäWì>æ ì/ñ]ïé2ù=ð üBúZïNîïì÷4ïGæ åDê ôçðpïèZä1ê!úUðYò>ïUûæõ1üBê ôçðpïèZä1ê!úæ å=äI÷Zä1ê ZV ô!úGðYò>ï2÷4æ ìJúðYèZä1ô ì>ð=úZæ2åYä1èä÷÷î,éIî,êëäðpïõ3åçæ èdðYòô!úG÷Zä1ê #ð<îù>ê&ï\V ô!ú2ðYðYò>ô&æ ïdìMúZï êAð V ëRdó#ôçðYòöðYòô!úG÷Zä1æ ê åN#ù>ð<îè4ù>æíFê&ï\èZV ä1éx÷*êëäî(úZïZúdðYò/äðM÷Zä1ìXûïUî(úZïõIåçæ è ìXí g8P!úZï ï éJï÷Zñ ", >0*L< 9 BM"0*,J+O9'*",(J(<5'6",)x'60* LT9`D E0(L*J'*^"10M)+A O" "(?Id",('*L!<'dG ",'*9`Lo 029'*(('",)Bb8<()33,$60*+9/F*< R . H B C P. {P1 , . . . , Pn }. [6", 0*à L9 $. query([B1 , B2 ]) holds(and(a, b), [B1 , B2 ]) holds(and(a, b), [B1 , B2 ]). $. [. (holds(and(a, b), [B1 , B2 ])) (holds(a, [B1 , B2 ]), holds(b, [B1 , B2 ])) (holds(b, [B1 , B2 ])). iMJ<56'60*L9 . holds(a, [B1 , B2 ]) holds(b, [B1 , B2 ]) holds(and(a, b), [B1 , B2 ]). _. true true. {P1 } {P2 }. true ⊗ (B1 = 1). {P3 }. i. B1 B2 true ⊗ (B1 = 1) ⊗ (B2 = 1). 9f '*Z*0*CE< (BX!W0 *A A [6 <V7\N!3_5%4PY 1`=8-(aG'JVU2A )3(4 ?ö0* 0*SI0*0*B)3'*9/F(L'", '*E)3((F!B0U[6,C%$G\;"Sb=R4(L'*0+'*",Z0U?ö 0*9)3L0^9/(9''*",(6(' (",()¹!^PY4<h o &%. îðï"ï!ñò ó"ô,2Ê 9 ")"! #õ> +,*. *0CE3)+!iM6",<=(:Z40*h0* 0*(IdJ<45%h( "( K>Z(0M( 9/-(:4'.J","1^0* 0*( J($&B%('*(!)+"19/0*",'/'%"140*(hWO9(!",H"(J<","1S0*'6 (ä0*++û,ú0*LðYWèZJä0*÷ðMG'@0ð]4ä(",û1)3Jê&ï 9A:A LCE0"10*3 ("(IZ)30*LJ",8 ç)3b9 I ()3''o 0LJ"1<0* K(O$&'*)+ F" L?+FLJ0*< d?(bD"3",<J4S0%('%"(",S,0*F0*'*'%?K!d"10("3LJ@0",*''*",LJFZ0*0*L'*(A#0*B\NR/(+LL9Ol ",FJLI ö^ J<<O'<0*0%+(9/ )3'9"10* ()3JE"F,0'*"1o 0*$&( ('MKR/,0**4'*J R/<'<O0* (B0*",J!%0B*(<!MLL9/JVZ<AEMCE<d-(0*'",'*N)ý<4LJ02P<&I( $&-( 'M^0* !("F,",RJ'*@00**'"()3Z!0M0*"1 )30"^W@04*('*)3LJZ90*L4'*hd 0@!? < )39CE?KBð]äû1Fê&0*ï '*A W0*J"10+9/(9L!"10*I0*O0",R B",'*B", 60*L9 +",J",J<56'W0*L9 AcCE (",J<5?^6 O'>0*0*L9d 5ED" 4?3()30*J;"10%G0*0@56E?Oj/"," '*-(:(LJL<'Ã÷A ", 0*'*J"10* -(>",J+P&L99/'ZV>R/(LJÃ÷Q5%!W e>' ø ù z{ysu tSxX1|w~xyt*}S ÉwxytSxsut*Ësu z{yxsutÈ*tSsuútS¸ÇA1|w tûtStSvw zwú zü ~suÈ{y ~z«ýT~É{ÊsutSz\È* zwú xyÉwýwxu{y {yÉ{y ~zwx{y~*xytSv\sy{ytc{y|wtSÈÆ.
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