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20071104 verification konev_lecture11

  • 1. ‘gg Ž Œ‹‹‰ ˆ… … Vh ‡†ˆ… … Vf ```Šv¦r©ˆgv¦„‰R ƒ ‚€ p~}jtr qvggywvjtr qo us p| {zxx us p ”tnl¤jDD¤D`¦W¤t¦S¦`¦” mh k ih d • • g f — ˜ e d ™ ˜ — – • dPsh ’ ‘UdPˆ… ‡ †U…h S¦©“S‰vESv¦4„ ƒ €h€hx w usfq p hfd b aXX VU TRP '‚rtrWyvtr©¦igec``YW¦SSQI £5GG 8 7¨ $!C !3 8 7¨ 5$!$33 '4¦'©HFEE'D#BA@96%4'$ 2 1$0 )!£¡ £¨$¡ ! £ ¥ ¡ £¨£ ¥£¡ §¤©©¤('%# §©©§¦¤¢ 
  • 2. ‘g‚ Ž ¹¸ ·±°¶ ´ ² ± ¯ ® ­ ¬ © « ª © ¨ §¦ ¥ ¤£ º tWgn„‚~µe³rn°Sv6riri‚rvˆ@¢ ¡ t F(t ∧ Gt) hd f†w hd† œ žœ ‰D‰S¦‰S ŸqU t ¬t d h €hP…Px ˆ šP €h p pU…fh p††q ¦c‚rgWSW›„¤r'™vDS'S‚c˜ !£ !3 – • 1$0 )!£¡ ’ “ ©Š—F#¦H''¤”Yv’
  • 3. ‘g‚ Ž ƒ f† d U €P q…x p VUd h š†Ê w U Ï Ò V… hdfP šd h € Ut‰`h„B𦄉dvDS†ÓBi¦„‚ЊQˆ¦‰D‰¦ˆ¦cqÊ ÎiDWÊ‚Ñwv‘SW¦hˆS‚`…›BšÐ‰‰gD‰‚ŸD„B6S¦ˆS‰¦Wà ˆf x us Ps R†P U fPÊdf†P Ïʆ Î pP V…Pd h… UsP €PdÍUh…x pUh… PSvSc„‰Si¦WDÌt¦Wtf È DSNF (¬ϕ) Vš€ h€  … x Ê pPf†U† pUh… D¦d¦c‚hrgPWSPW˼S‰SSÉt¦Wtf È Ç Æ Å Ä R pU €hdU… À¼¢Ht4r„và ¬ϕ = ¬F(t ∧ Gt) ≡ G(¬t ∨ F¬t) qdf†whd† PUsqTU…fh pPqdU…fq p††q ‚D‰S¦‰SÂSvvvvDSÁSW‚vDW©S‚Wc˜ £$ £ ¥ 7 5 'À§¿¾½¼Š»
  • 4. ‘gŽ Þ  GF¬w G(w → X(¬t ∨ w ))∧ G(t → w )∧ R pP €PdU… ™Sc„„và G(t → F¬t) pP ÝUxP…P SQvSWS©Ã ƒƒƒU € Ò š u p hf‘ Î pUs phx† ‚‚4lBÉ6yil6tD6¦WS4Ü ¬ϕ = G(¬t ∨ F¬t) Û Ú Ù × 0 ¥¡ ! “EØÖ ¦ÕÔ
  • 5. 'Dá§''c''§¤¤#ÂÔ )$!C à 0G£ß¨£ ¥£¡ U = {G(t → w )} E = {GF¬w } S = {G(w → X(¬t ∨ w ))} I = {} t, w ¬t, w t, ¬w ¬t, w ¬t, ¬w ‘gŽ â 
  • 6. 'Dá§''c''§¤¤#ÂÔ )$!C à 0G£ß¨£ ¥£¡ U = {G(t → w )} E = {GF¬w } S = {G(w → X(¬t ∨ w ))} I = {} t, w ¬t, w ¬t, ¬w ¬t, w t, ¬w ‘gŽ â 
  • 7. 'Dá§''c''§¤¤#ÂÔ )$!C à 0G£ß¨£ ¥£¡ U = {G(t → w )} E = {GF¬w } S = {G(w → X(¬t ∨ w ))} I = {} t, w ¬t, w ¬t, ¬w t, ¬w ‘gŽ â 
  • 8. '%c§§#ÀD0 ã £å! ¨ ä t, w ¬t, w t, w ¬t, w t, ¬w ¬t t t, ¬w ¬t t ¬t, ¬w t, ¬w ‘g‚æ Ž
  • 9.  ‘gŽ ï ¬w EG U f† hd U š €Pd q… x† pPUd h š†Ê í ìë D‰`‰t¦„„DvSîSii¦„‚9† tt‚ê Ê šÊ p… h é œ ž è pP V…Pd h… Dq¼v¦¢Ÿ¼SgS„¦W§Ã t, ¬w ¬t, ¬w t ¬t t, w ¬t, w £¨£ ¥£¡ç ! ''§¦¤l4#ÕÔ
  • 10. Ž ‘g‚¡ (qi , wi , qi +1 ) ∈ ∆ q0 ∈ Q 0 q0 q1 q2 w U A a ¦w‘‚t‚‚ ΃ Î Î ƒ ƒfƒƒ wUvsBrh‰†`‚†Âˆt‰†`hWiˆB„PW‚¦¤„¦„`Wx û s iW©¦„Dq ivô yñ ‚r‰W§Ã V f h f s š fqdh €Pš†h Σ N q qfq phfd øÿ þ €hh… w ƒ ø d SýVSvt9n“‰¦„Â`tSW¦gS‚`… û t‰‰'Ã Í U U T R s Ê éñ hd h š† P h s ‘ P s hR †P ˆf†Ê F w s Vhf h U Ï Òq pUsU… hdf†P ühs tUv‚i‰†`‚† WŠQD©tDtvDx ‰D‰‚—‚W6p û Q0 wvBrf‰`‚†Õ¼viBr`iÑ‰D‰‚—‚6p û Us Vh †h  usˆ šq‘qs hdf†P ühs ∆⊆Q ×Σ×Q q € P…P P s Ý hsf %rh‚gWSx›iUDSPQ¦WDSh û Q w Vhf†h† hdf†P ühs tUvs‚i‰`‚¾‰D‰‚—‚W6p û Σ fUdq éšq w us‘Psh ivЦ‚¾§vtS¦ˆR û P c€ ú A = (Σ, Q, ∆, Q0 , F ), ùø`SW§qyt©ÕW©¦„eb ÷ ö õ óô ò ñ U  Ò „ f q p h f d ð ω 7 5$ 5¡ 6%Ð#¼$
  • 11. Ž ‘g‚ L(A1 × A2 ) = L(A1 ) ∩ L(A2 ) A2 A1 U 6DiW©¦„v¾¼vvD%‚rB¦ˆ§SÂivSSS‚‚WSWx U pqfq p hfdq  ussq €qÍ Îd hR uÍ V PUsP‘P†P…P wU Ï Ò Vš†U‘ ud Î pfU… šq w usˆ šqU p hsU šhx fPÊdf†P Ï ŠQD„St§l6Div¦h ú r˧v‚`6¦WBiWÕ‰‰gD‰‚ŸDÊ È L(A) = L(A) = Σω L(A) L(A) A qR uÍV PUsPs šhx h € wU Òq pUsU…x g§SSvSWiW¦c tŠÏ v©vvv4Î A f hfdq ˆfU h…f†hx hs üh rq©p¦igDÂDi¦WD‰`WÂr'p t©ËrW©¦„D¢¼¦WvD%¢©Ã U Ò „ Êfq phfdq Ê phssq € h a ufRq §D‰D¢ w} ¨©¨¦ §§§§¥L(A) = {w |A R uÍ S¤ w A A ƒ is r‚‚WvyŠQD©tvvvx q €hh…x wU Ï Òq pUsU… w f Ê f† Ï Ê† U š†P p hfq p hfd V†fPq p Us U… ‰P‚ˆdt‰‚P D„›B„‚¤Î ¦W©¦„vq q‰©tvvvx „¦¦š È r hdh qf ∈ F ƒ hf†q‘ hs‘Ps hR†P ¤‰SÑWtS¦ˆS‚`… d q‰SSWt‰„d r V†fPq‘P… f† h f† w hR qR U š† €h h … x w U Ò q p U s U … ˆ f† € hh … inW¦gvgy„‚P ‚r‰vytŠÏ v©vvvx D‰‚P r‚‚Wà 7 BBc©Ó£ÐÕÔ ¥8!3 £ ¢ 2$ ¨ !
  • 12. Ž ‘gr$# ... b n1 ab n2 ab n3 b ω b n1 ab n2 b ω pU…fh p††q tvtS'S‚Wc˜ b n1 ab ω pU…fh p††q tvtS'S‚Wc˜ pU…fh p††q tvtS'S‚Wc˜ bω L pU…fh p††q tvtS'S‚Wc˜ a SEh ú P QD©vtDvx R uÍV Ï ÒqpUsU… Î U Ò „ qfq p hfd hssqd h… UsU p…PfP fPÊdf† Ïʆ P t©ËiW©¦„vq h ú WDv‚¦Wvt6DS‰c€ ‰‚ˆt‰‚P D„Šl‡ a w ∈ L(A) ⇐⇒ w wUsP € ühd h š†U ‘ Phs ‘Ps hR fU ü…P €h tvScQr‚ˆ¦„iE`WSW¦ˆyiÂgS¤r‚† aR uÍ `S¦V b a, b fq phfd W©¦„eb $5$ 5¡ 7¨ ߣ¨ 0 G 0$0 ! 5G!3 0$5 £G¡ £ ¾'%ÐÀ$ EË'¼#tHÁ¤Q¦©Éc¤Ð
  • 13. ‘g„‘ Ž a ∧ G(a → Xb) ∧ G(b → Xa)? C©U£§ª ¥§ Vr0‚‰Ÿiy¥ ¹C©¬S¤¬¤Q © ˆPD‰FD„‰R‰ªT a ∧ G(a → X¬a) ∧ G(¬a → Xa) IG E¤CA PHF„DB¤ @  us fP‘ V†ˆ fq‘P… f†d Ps fP üh p P qs h 8 VUs Vhf†h ¼vt‰i d q„tr`SWt‰„›Õ‰—‚'›s i¦%9vv‚‰`‚† a  us fP‘Ps d V†ˆ fq‘P… f†d fP üh hf‘ Î p U fP p ¼vt‰iSW yU q„tr`SWt‰„y‰—‚'p i'6tt‰'vq 7 σ2i = {a} σ |= ϕ ϕ ƒ hf icÎ U š† „‚P hf‘ Î w hRq u šÊ p … h é œ žœ fPÊd f† Ï Ê† P a fR q i4t¦gvSf Bq¼D¦ Ÿqy‰‰gD‰‚P t„ŠWs t‰D¢ a ƒ  VU s Vhf†h†  u s f5‘ d V†fPq‘P… f† p h… hfhR d ÎR uÍ V fPq p U s U … vv‚‰`‚›¼vt‰6¢Yy‰`SWD‰‰d ¦W¦Sˆ¢4SS¦Ð‰©vtDvx 3 ) 4210( a ' fq phfd W©¦„eb $ G •$¨% £5ç$! 7 À©ác'44¼©B'Ë¢ 
  • 14. ‘gr  Ž FA = Q ∪ {ι} Q0,A {ι} α = L(s ) ¨ ∆A : ((s, α, s ) ∈ ∆) ⇐⇒ (s, s ) ∈ T QA = S ∪ ι ΣA = 2Prop P c€ ú A = (ΣA , QA , ∆A , Q0,A , FA ), U Ò „ fq p hfdq RqR qsPšdqf† €P…x ˆf u … fP üh t©ËW©¦„DDgyiSgvS‰¤„vÐt'¦›‰er'p S = (Q, T , q0 , L) d h €hP…Px q pPf† ¦cr‚gWSE©Si‰iU È → 8 2 W 75$ 5¡ §Š¢X%ÐÀ$ ¡ ¥8£!£3 7 £5G #BD¼'¼ÖE'4¦' ã
  • 15. ‘gr  Ž r qi qi Pi ∈ F ƒ hf†q‘ hs ‘Ps hR†P d V†fPq‘P… f† ¤‰`ÑWSW¦ˆS‚`… Yy‰`SWD‰‰d a qÝvS‰dýqf‰cWwDs s U …P V† P € q p h € üqR d U š† €hh…x wU Ï Òq pUsU… ¦cQrg¢yB„‚P r‚‚WvŠQD©tDtv§Ã F ⊆ 2Q wvBVr‰`h‚† dt‰‚PerhW6pyWŠQÒD©vtDvx ‰D‰‚—‚W6p û Us hf† f† ü s U Ï q pUsU… hdf†P ühs Q0 wU Vhf† †  usˆ šq‘qs hdf†P ühs vsBr‰`h‚Õ¼viBr`iÑ‰D‰‚—‚6p û ∆⊆Q ×Σ×Q q €hP Px P s Ý hsf %r‚gW…S›iUDSPQ¦WDSh û Q w Vhf†h† hdf†P ühs tUvs‚i‰`‚¾‰D‰‚—‚W6p û Σ fUdq éšq w us‘Psh ivЦ‚¾§vtS¦ˆR û P c€ ú A = (Σ, Q, ∆, Q0 , F ), 8 2 W 75$ 5¡$ £ 7¨¨ ¢ ä §Š¢a%ÐÀîÐ'£ `l©ä Y
  • 16. ‘gŽ Þ  ∼ ƒUÒ ¦t©„ fq phfdq w uss Ï…h… W©¦„D›§vvSP ¦```h éq wUR†P‘sP €Pdhx w ussqdh…UTÊ€P … ú SS‚ivSc„‰¦WѧvvD‰¦W¦D¦„¤˜ eeec £ß$ fd©'4ç b
  • 17. 'D@''c©©§¦¤#3 §'ŠÕÔ )$!C à 0G£ß¨£ ¥£¡ – !£ ! D‰‚‚¦Qg pUf†Êxh U = {G(t → w )} E = {GF¬w }, GFw S = {G(w → X(¬t ∨ w ))} I = {} t, w ¬t, w ¬t, ¬w t, ¬w ‘gŽ â 
  • 18. §Š¢i%Ð#¼ØË'ýh©l©§'ŠÕÔ 8 2 W 5$ 5¡$ à 7¨¨£ ¢ ää – !£ ! a1 a2 a2 s0 s1 a1 a1 = {t, w } a1 a2 a3 a2 = {¬t, w } a1 a3 a3 = {¬t, ¬w } a4 = {t, ¬w } a4 a2 s2 ι a3 a3 P1 = {s0 , s1 }, P2 = {s2 } ‘gr æ Ž
  • 19.  ‘gŽ ï  x =y ¹ I U§ B©rr‰ª q ¹ x =n y =0 ¤ „C «¦y£­ ¹ y =i ¤C „I « q ∈ Pi x = i − 1 «­ ¦y£ pp (q, a, q ) ∈ ∆ ((q, x), a, (q , y ) ∈ ∆ U U š† B„‚P P c€ ú B = (Σ, Q × {0, . . . , n}, ∆ , Q0 × {0}, Q × n) pU h…f†h à ƒ U Ò „ fq p hfdq w uss Ï …h… ¦WD‰`l'¦t'Ër©¦igD¾vDSP ¦```h A = (Σ, Q, ∆, Q0 , {P1 , . . . , Pn }) ˆf†Ê t‰‰'à → 8 2 W 7 5$ 5¡ Š›a6%ÐÀ$ 8 2 W 75$ 5¡$ £ 7¨¨ ¢ ä §Š¢a%ÐÀîÐ'£ `l©ä Y
  • 20. Ž ‘gr$¡ Gª«tG  v ¥£«¤Q S ˆªª© ‡¤ ‘`v6­ RR‚WD„‰Bir`­ § ®u¦§ „ EC«¦  ­§ uC§v ¤€u ­¤Q ‚ªyBi¤…r0iv¤ ƒ„§rˆwqVƒr‰‚£ q† « © ¤ h V ϕ ©riªr‰ª¦­r‰QF¤D¬Ÿ¤ 0#B§yu® SVÂiUvS„SxP‚‚WwSgP¤…EP WDyÀr¢D„Di„idiv«S‰¦Wà © d¦uˆRC € Í vy« P s ¤ P Q S † §B C x 6§ u š 0« ¤‰¦‚C‚£ ©¤ Î Q DfWU0… P d h … ‘C v U Chf†Êx ˆ¤t ¬ϕ ¤ C§ ¤ v uC« ¤¦C£¤Q ¤t „QxBEwgCD6§Àr0D„Di„dW0« ss A(ϕ) fq phfdq ˆfUh…f†h W©¦„Dt¦D‰`©Ã A(S) ϕ S fq phfdq ˆfUh…f†h W©¦„Dt¦D‰`©Ã q šÊ p… h é žœ ¦ˆ¼v¦¢œ qyU d h €hP…Px q pPf†U† us ¦¤r‚gSWÕ©S‰SSÓvDq g C 5 r§¼Šb
  • 21. Ž ‘gr$ O((|S| × |ψ|) × 2|ϕ| )luC£¤ª ‡¤ rBDi„ŸS­ k ƒqdf†P ühs p t‰t‰‚erW6“h ú P QD©tvvvx Ï ÒqpUsU… P €Ud d ˆfUÍq… ud hs üh p Uf†hdU š€Pdq…x† VUdhš†Ê P uf†h… cWi¢¢DiSW§ÑW‚'ÑD‰`‰t„‰DvS“Wi¦„‚F©t‰`Wà O(|S| × 2|ϕ|+|ψ|) luC£¤ª ‡¤ rBDi„ŸS­ k ϕ ⇐⇒ S |= (Gψ) → ϕ j™ih e™˜— f‚‘–gfd–•” S |=ψ Êdf†w hd† R ˆfUdq…h ˆt‰i¦‰S tiDv`¤€ hs üh p Uf†hdU š €Pdq…x† VUd h š†Ê žœ d hf‘ Î pPqsÍ u WÂr'D‰`‰t¦„„DvS“¦¦„‰Ðœ ˆ¢yi©6SWSÓ¾“ 5G¡ ¥£¡$!3G •¡ G 6¦l¦§Q¤¤4'½4#E’
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