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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(19) 0* ('*'* (L)3!I : 9J+'*"((LE" ( J',", I)½'2"(*5%< (4!!"1UT0*4^(( J4(@0",* J '@0",?T* ' F",(0 d" ( F!0 b *" ? D<CE!E'*L d!%",99 !",R G $D0*!M4( J@0*'", F0E!M*"10*!@.JA E F ¡FÕ=D&Y/0 GYI ú<ô!úðçú æ å#äUúZï ð6æ å7 î,H@ì(ôç/ø1Dïè¥úä11ê ¤Mü Y>N+îFä1¢Y/ì>À/ðYô YO>IïõW³D1ù>/è4ÀJæíF1¤èZä1K£ égJMèLî,ê&ïZQú÷4æ PYìJ÷*úêëäðYèZî(ä1úZô ïZì>Sú ðWR ê&æíFôY÷+ù>è4æíFèZä1é ÷4æ ì/ñ æó#ó#å8ò>ò>ðYïïò>ì>è4ï;ï ï èø1î,ïê&è ïô!R7úDä1ä1ä1ììUìõ äõ ðpæ é3ô!ú;ôYä2÷8åç÷4æ æ è*ìJéIúðYî,èZêëä1äQä1ô è4ì>PçïMðYðò>VXúï=äWDðYò>ò>ô!ú&ï4ï;O>ä éIïõTõYRï4UV ä1ìô ìíUæ åJúî÷@ò+ä1è4ä ï6èûî,æê&õ1ï#ü2ô!úEê ôçðYðpò/ïäèZä1ð ê!ú*P&æ ô!è=ú=úúîJäû ðYô!íú&æ,O>ä1ïê!õ ú F ¡FÕ=D&Y/0 GëK ðì>pïï*è*íé+äú#ðpïù=õ^æ1äú<ôçðpðYæ ôçé3ø1ïMôY÷% ê ôçåçðpH@æ ï¢Yè*èZ&éIä1êN¡Fî,1ä1êë¤ìäF¢Z\ïõUJMV ì>L ï*íäê ôçðYðpôçïø1èZï%ä1ê:ê ôçðpô!úïèZä1äHê=ù=ó#æ1ô êú*#ú<ô]ûû1ï2ê üî(úZì>ïï*õBíäðpædðpïè4õnïåçäïèGðpæ ðpé3æ2ôYì>÷Oæ åçì/æ ñ&è*ì>éIï*íî,äêëäAðpïV[õ3WDä1ìò>õ ï F ¡FÕ=D&Y/0 G!] ÷ä1Zä1ìê &ä1ïìJõBúäBó;ï÷Zè ä1êæ ;è æ äIèG¦ î,ä+ìH@(ì>ôç¤ð%æ ¢Yì/÷*S¢ êëñ@^=äî,¤î(ìDúZôçï\ðM¥ V _W÷*êë¡Fä`î(JaúZ\ï LV L ù>è4ù>æè4íFæèZíFä1èZéä1é èî,èê&ïGî,ê&ó#ï3ôçó#ðYòBôçðYæ òKì>ì>ïMææ è7ûéIæõ1æ üHè4ï:ê ôçûðpïæèZõ1ä1ü^ê!úMê ôçô!ðpú2ïèZ÷Zä1ä1ê!ê ú;&ïô!õ ú F ¡FÕ=D&Y/0 G b ðYôçø1ï é^äùù>ô ìí ¾ Hp1¡ D¤£XYDÀSæ ìS¥8ðYÏ=ò>¥ï:úZ&ï 8ð Y/c Jdæ L å6ø4è4ä1ïè*ì/ôëä1äé3û1ê&ôïZìúíUV úîJû,úðYôçð<îðYô&æ ì ô!úEä+û1ô e4ï÷Zñ A0 ← C | B1 , . . . , Bn . (n ≥ 0). A0. B1 , . . . , Bn. C. C B 1 , . . . , Bn. A0. θ. VAR → VAR. VAR.
(20) ¾. G 1¡ ¢YY£XYD¤1Y¡ ¥ F ¡FÕ=D&Y/Q0Gë´KHpȤ1Y¤>\J WNó;æåçæ è*éIî,êëäFï ïfô!úðçúMäWè4ïì/ä1é3ô ìí^úîJû,úðYôçð<îðYô&æ ì úî÷@òSðYò/äð ä1ìõ V ä1è4ïO÷Zä1ê &ïõnø4ä1è*ôëä1ì>ðçúWô åWðYò>ïè4ï ä.J kD 0*ô iìIôç<9ðp</ïX40md4!æ ",é^ä1"(ô ìN,<EAD$%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(07 J<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,$GJ" "ö-(d4(Ã6 J @0*(' ",", F I0 4h9",'*'**B< ",( J%E"-1(",%'*!",4(R J O@0*)+'",", F909A F 0*J"1 F0M0"1'*0* J( K'E0**42'* 4R/( JB@0* ' ", 0* F0 !M0*0*'*L<!(MLA J<%0*WR J",'*?O4!< ( K!", ('",) E C W 3 ) 9 3 ) 9JÃ6"( (1 ", (",B XÃ6-1",LJ a:La b? ;RPYL<0IT",þ ÿ%5%$&( 'I0*c", n<(4)3hB9!",< J)3"190* (O n)3,$dÃ6 ./ J"1",0*'* ?X( Ja 3<L4!<99/( Q('<0^a !",", (?X'",<)+40 V21-(' ,$: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@ JLD Jin Z0* 3-1 ",(, NLJ$=bU"1-10*",4 ((L \ @60L J*+Z0*-1 (", N'*!b2",R9 '*TA@Z0* ( Nbd4()39 )3!D5 F%0"1'*0*0< 0*( N b "(Z0%"(%"3<A#(ic'<0E-1$&(",'* 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$&J(<'dBM<LJ ö.J'*2"B'd"",4'*(,3 J'*3õ(@0*ô 4è4'*h ïLJ÷0*Zðp0dï Jõ^!<:0-(éI*Jæ1 ?"1ú0dð#LJí56<ï3ì>Hï5%èZ0* ä1 ;ê;'*( î,Lì (ô O>Kï(è0*Lú0:bJ40*(('d F+õ(0*'*)3éU(í/=î(5%ú:7$&!(':S*<-14'*",( '*9'<!00*",A%R( CE N: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 ¡ ìDí ä1¡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 ìõúð íV ôïì ì/ñ <(LÂQ!O< 1-(5'M0*0*!%'* F20*L4h 0*!@0-(:" ?(,A éUí/îbJ0*'*U",!<+4h!@0:"Bõ(éUí/îA CEW4h",)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. ,ª 0* [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/",F J*^< $&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", "1 B0*U", J-1^",'* !",RO 3P U&5%)3(!'*S 0*)3J", ( F0dR/ @0"10*VZA; 5%$&? 3 S(56 U:((9/'E)3'"1(0*'*3d @0 H"10*0*:+-1Ã6",'* !(", R", )31-(`(6L'7J4%h9 F'*0*+*<", d,0*0*2'M[6@0C%"1\K0*((A 9/'"10*('b(56 ^"2R J",'*?U0*'", J< 0* ( +'*!"10* ( 0*J"10 *4'* R/I5%P&!('b9J",",0* ''*I J,"10*$U -(@0"1 ?+0*4I0h9*J'*"1*0B<4hb 9'**I9/V6F* <eHR 0M K!%90*'/",F J*<< 0*R ( : JI0*3 (U($&L'*'O() )3@0"10*ç A 0*<'4",0U J0*,< $ 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 ("1 H0I",'* T-(!"1'* 0*? 0*(!@0 K",%"10*0*@0B"1 ö9/0* )3'*)39 AH0!I"4p 0* n1 0^?HCD ",4R"(.J@0 B+ (d" A b )3", J%0*J"10 0* 9<LJ'*(K9/",'< K0@? L J('*!? E I O@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 (CE 30*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",<L JJhD0*J"10S(EW!!G2 öJ", -( ('*?S @0 F0*"1«0*3Dp$&0*(J'G"1",0U D",9J ö"10*R/J^'*"(KEZ$&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*$& O('M0*'*iMGd # h !¬M@00 !!%*"Ä 1 0*(!RJ@.J", O?IM*"1 F0*0*! @. JSP& 15 ", JO O0*G$&L0*L'* V L ß $¼!%*"10*!@.JO<()3:0* )3d O0*2`L0*L'*. σ0 σ1 . . . σ0 |= ax(F ). ®y®&¯. σ 0 σ1 . . .. F. σ0. °<±²´³Aµ¶h³Aµ1·¸µ1¹Xº¦»"ºA¼S½X¹(½3¾d°<¿±. ÂQ( G I460.J *^ W40*.J d2[6"W\N$&_5 '*99'*'*<! F"100*"10*A ( 3,$0*2[6C%\K(9/'"10*(';4.J + OCD",R :A b )3",0", ( 3;"+,$ P& ('*(L JJV;[6C%\O!3D$&('*)WL(!"d!",E J+ O"d0*@02"1@00*M"10* *4EZ'*A 90* ( +"(;",'* (L)3 F0A=CE 0*'0",", J(<0*G0 (@ I56H$&'*(@0)"10*0*IG.J*'4@0'* 9@00* "1(0* J:2"0*(UW"0*,2'* (<L)34( J F02"A , Jö'*40*L'* Jd0*'*L^ en0*'*I!W" "5E,!'*20" (?(L*b)3<O J Fú4ä0:GðY",ô! &ú 0 JO>MïZ<ú4o]h)+99'*",'* F0**!!<"1%" "0*(,IAB'*G4](0W J 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*<'<"1 J0*H4( ' Sd)Dp0*+0*" '*'*?H)+O20*4h,$E! @0,3"R/0*I, n4'2h( 9 9/'*.J '*"1< 0*0*B(T'9JA"("1^20*n` n(5%'2CD4",hR'*",BO)3`l9A (! Ab ",!5E**"" 1?0*I!@$&*"?I10*` !@.EZJA ¸ EH!IFL -1", F0O0* D@ 0B!O ,0I0*'*LH0*J"10", G9J"10*JB< ( 'I('O!"10*' (R O?B0*0@CE56WO#'*@00* ('"1",F0* J0M3<J-10*",",( J'* :!",'*R<! !"1UP(0*A% Y("IiM )+"14hh", )W)3L!9)x 2"(,,<$;$#$&"3@0(8L0*''d",@0 J*4<"1'*0*0* R/ (V B%:0*'*R J? !"1"1J0d0*" 9/(- B '* )31 %-("%0:0'M< * @0' (","1 J0* <M0*4 (hoç$& 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,Wb R/R(1L T-(0M#0*",!N5 B?3%@0'*F"10<L0*0*O -1 W",-1"(",N"'* F! ",0%RÃ6!q @d0(L* JZ", 0*! ^UL (4 d( J J,$J@0*@0''3*",5% 9J F$06!&4'*hn(9)4'*(0** J<6( @00* ^ ("1560* J (U"LH!^@0","1!<0* 0*'", J< 0* ( O!%", 156A (S10 , S20 ). (S1 , S2 ).
(24) G1¡ ¢YY£XYD¤1Y¡ ¥. Ñ. Ò ¡ À/¤Y/c(/D(¡F1D¥ -1JCE"," '*-(I! ",LJR0*^*"G, (<L+",RJ,@0$M*5% 0* 0*LN F0*A"1(0* J (% TU Qì>[6æ ð7C%*"\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* $M4'*((F)3b(L956@0E*)356(( FL0*!U U (0*AH620*CEd", ( JI4<50#6!"10*0<'0*M70'U",* J)3<56", J' W5%", 0*!4L O!("1'0*0* ! %L0* ,-($M'<('* RA $'56",% Jö<LW ",-( +'",< JI<5",6 J ' " (,)3F",(CEçLA F:0E0 * J++0",, $N + F"10*0*U( ^4(56)3M9 )3^ F!60 ,",$= 0*BU÷4æ !ìJq@úLðYè JîZ÷0*ðY(ôçø1 ï6,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>æ ïBåëè4ðYôæ úZìéï í ð ÷4ðYò>æ ìJï^úæð]ä1û ì>e4ïð÷V ðGõæ é^ä1ô ìcðpæH÷4æ ìJúðYèî÷ð ä1ìæû 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!ú ðYò>ïMúZï ð;æ åDä1ê ù=æ1ú*ú<ô]û1ê&ï6íFè4æ î,ìõIðpïè*é+ú=ù=ï æ1âGú*ú<ïô]è û1û1ê&èZïGä1ðpìæWõHïî,f(ìù>ôçè4ø1ïZïú*èú úZïî(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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