20071104 verification konev_lecture13
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The document discusses formal logic concepts like temporal operators and model checking. It presents semantic conditions for formulas with temporal operators like EGψ (for all futures ψ will be true). Examples are given of model checking on finite state systems to evaluate whether formulas like Eψ1 Uψ2 (ψ1 will remain true until ψ2 becomes true) hold. The document also provides an example of a finite state transition system and how it can be represented mathematically.
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EGψ q0 , . . . , qi −1 , (qi , . . . , qj )∗
‡ Vpf xdhsr pPhP fUds
D¦¢iDiiBqei–WeSƨ t Äx
%Åf¦¾
i j : q i = qj S
Vsh f t t€ “P€ x
{gggDg€ ¥ gSW¦ˆR RtR Rt
ƒy˜ƒ’¾
(qi , qi +1 ) ∈ T . Ã (S, qi ) |= EGψ
q0 q1
‡ ‡h‡‡
¦y“uvuu‡ ¥ ¥ € U ÀPd r hsx€r pPhtd x fP psx‚ VshP fÂt€ x
…ƒSiE…iWgqgWi¦–q¦qWµsgi–gƒg£eh
(S, q0 ) |= EGψ
q0
„PP „U †
Su'…H6„ †€U ÀPd Âx€r pt“t€ Vpf U ps
ƒ…DSq•¦gguegÁ…¦¡quP
H(Z ) = [ψ1 ] ∩ EXZ
EGψ
UU TR€ Ÿ tR“ xh Vt€ ·Ud fx‚P€ V Àr px‰Ut€
…¦SƒD9¶y¦g¢ag¸gg4ugSWµat eBeSƒg¿¨ ‡ ‡
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EGψ
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20071104 verification konev_lecture13
- 1. Ž yy Œ‹‹‰ r ‰ Vx ‡†r ‰ Vh Šƒ¦a©ˆyƒ¦qiR … „ƒ‚ r€lvt sƒgyy{yxlvt sq wu r~ }|zz wu r –…pn¤lDD¤D¦W¤…¦S¦¦– oj m kj f — — i h ™ d g f e d ™ ˜ — dP€x ” “UdPr ’ sUx Sg¦©•gSiƒESƒ¦4‘ ‡gU¦eDg…i6B§DS'…ƒDƒDa¦ywvuDiiBqei¤ge¦aYW¦SSQI V TRths‰ ˆ ‡ †P „U‚h€ xR U tdhsr pPhP fUd c b `X VU TRP £5GG 8 7¨ $!C !3 8 7¨ 5$!$33 '4¦'©HFEE'D#BA@96%4'$ 2 1$0 )!£¡ £¨$¡ ! £ ¥ ¡ £¨£ ¥£¡ §¤©©¤('%# §©©§¦¤¢
- 2. Ž yu †P „ U‚h€ xR ¨ œ › ž Vp ƒS'ƒƒƒ…a¦ˆ¹¸²’w…f ¥ i…iqqgi–ggSDDyBB¤qWƒ•g¸u'Ys’ƒ¦¶ 4BµD´³ td hsr pPhP fUds r h V pd V ™ fP‚ x€ ·x „ p „ x Ÿ œ › ® V p ¥ Pps’„ƒ¦g²’B¢g…aSau…eSR ± ¬ϕ hxx‰xt€ SW¦gg£U ϕ x Ÿ† œ › ® t € h€ P ptd U d x t p „ x Ÿ† œ › ž eh ¥ ¤s’ƒ¦g 46°¨ Bq¦¯ U ps dxxhtP‚ a¦W¦gWgSW¦x EG EU EX EF ¬ † ƒ „ We§«WDDW e{ª x er p x h ¨ V U € t T U Sx ¦ ¥ € Pd Uh €d R ¥ ¥ ’ƒŠ€S¦'xP¤ 'S¦g¢ƒƒ¢ˆuerB§eˆŠBeeƒ©wgƒƒƒƒD’§¦U ¤¦ 4%® h U „ x ‚ U ‚ x R ex h ©D€ SWDigBˆ ƒ x Ÿ b œ › p P † €Pxhsx‚ † p „ dxxhtP‚ a¦W¦gWgSW¦x AG AU AX AF ¬ € € 'SPWe§« U ued{ª er p x h ¨ V U € t T U h x U ¦ ¥ P„ d € R ¥ ¥ 4†ƒƒS¦'¦W‚£Wƒ…hWyRg§gxx ˆh¡Beƒeƒ©w…WiDƒWƒƒDsS§¦¤¦ 46ž UŠ¤ x „ x ƒƒ¢x r pe P©† € SW h s x p ’ƒ x Ÿ b œ › U‚ €Px ‚ † „ b œ › tR †š V ™ fx‚ td f „Uhx „sst ¦ 4vyS¦%¤ug˜u4—ƒDS'Sug–• 7!£ !3! 5¨ 0 $¡ 5G £5£ ¥ ©Š6B#…”%6¦¤“ 4¦§’4¡ ‘
- 3. Ž yuÇ EGψ q0 , . . . , qi −1 , (qi , . . . , qj )∗ ‡ Vpf xdhsr pPhP fUds D¦¢iDiiBqei–WeSƨ t Äx %Åf¦¾ i j : q i = qj S Vsh f t t€ “P€ x {gggDg€ ¥ gSW¦ˆR RtR Rt ƒy˜ƒ’¾ (qi , qi +1 ) ∈ T . à (S, qi ) |= EGψ q0 q1 ‡ ‡h‡‡ ¦y“uvuu‡ ¥ ¥ € U ÀPd r hsx€r pPhtd x fP psx‚ VshP fÂt€ x …ƒSiE…iWgqgWi¦–q¦qWµsgi–gƒg£eh (S, q0 ) |= EGψ q0 „PP „U † Su'…H6„ †€U ÀPd Âx€r pt“t€ Vpf U ps ƒ…DSq•¦gguegÁ…¦¡quP H(Z ) = [ψ1 ] ∩ EXZ EGψ UU TR€ Ÿ tR“ xh Vt€ ·Ud fx‚P€ V Àr px‰Ut€ …¦SƒD9¶y¦g¢ag¸gg4ugSWµat eBeSƒg¿¨ ‡ ‡ yR¾ EGψ ½¥ $¡5G £5£ ¥¡G 0G ¼%¦¤“ 4¦§’4©»'©#¸º
- 4. yŽ Ì i f 1 ËÊ –¤É (S, qi ) |= ψ1 à (S, qf ) |= ψ2 q0 , q 1 , . . . , q f Eψ1 Uψ2 xh  xRt ge“ ¥ ¦ˆƒSh ¥ rh ‚ rhUxhsx …Wu¢De¦WDig‚ Vp D´³ (S, qf ) |= ψ xh  xRt ge“ ¥ ¦ˆDeh ¥ q 0 , q1 , . . . , q f rh ‚ rhUxhsx Dgu¢De¦WDiW‚ EXψ EFψ¥ Vp D´³ EXψ EFψ Eψ1 Uψ2 È È ½¥ $¡5G £5£ ¥¡G 0G ¼%¦¤“ 4¦§’4©»'©#¸º
- 5. yŽ Ñ ¬ϕ ϕ † p „ x Ÿ œ › s’ƒ¦¶ 4%® Vpf xdhsr pPhP fUd Ð D¦¶iDiiBqei¤geSs Ð † p „ x Ÿ œ › ž V p f P „ U‚ h€ x Bs’D¦¡²’˜D¦ŠS'ƒƒƒ…a¦©” $¡5G“ £5£ ¥¡G 7!£ !3! 5¨ %¤¤Q¦’¦§44'ÏÎ'¸6B#ŠÍ
- 6. Ž yuâ áØÙ áÕÝßÕß Ü ÞÚ ÕÝÕÔÜÙ ÖÜÛÚ ØÙ Ø × Ö ÕÔ ue…iqsesàBeƒqW{ai¢aˆqeƒ%Dv4Ó VUTRths‰ gSDgDi6ˆ P „U‚h€x S'ƒƒDDa¦ˆR UhÂt€ † ‰xh tRU pPd „ xR ÀU ps d x fx¬PP‚ t „PhsU Deƒg¦ge“ ¥ y…qq¦ˆ…q¸¦–uayWSW˜©SgiSc I AGϕ ½¥ !£ !3! 5¨ Ò'¸6B#ŠÍ
- 7. yŽ æ S’ h h h h h S 8 7¨¨$ ¥ ½ 10$! 5Gä 9à姩46¦˜ã
- 8. 'Švº !£ ! …, -2, 0, 2, 4, … Even x:int …, -3, -1, 1, 3, … Odd …, -3, -2, -1 Neg y:int 0 Zero 1, 2, 3, … Pos yuç Ž
- 9. Ž yuö õd x fx¬PP‚ PU€P À x€hx rhU pP fP‚ x Rt u¦–uayWSWEeƒSQ¦gDSDeBq–qWƒ¦ŠDn” 1 1 0 0 0 1 0 0 0 0 1 −→ 1 2 0 0 0 x x 0 0 0 0 x4 x3 x2 x1 ¥ S'ƒƒDn‡ P„U‚t U „ †€€P „PP 6§ƒƒS'SWSW‚ U6„†6„U’fged¢6ƒ……aS–W¢£WƒueiuôSuuƒƒDasc U U „ †€ “ U h€P fU s VU€ Vxhsxs „P U‚‚ Ä óïí '–iuaÜ ð ò Ü Þ ÖØìØñ ï í ÜÞÖØì nÅî²'–iØ ë tdhsP ·x€ „ td f t€ P †€€P „PP‚ „P ™‰xšt uDiuêuW6˜u’¸g˜©ƒƒS'SgSWS’SuW–• ¥8¥3 é 7¨ ½è3 BnB(v! ƒ%#¸º
- 10. Ž yaÅú S’ I h S I ‡4auygSW¢DW¦q–{vW¸u'„ ¥ SeDBueiuùƒ…iƒg…iWt fx¬PP‚ r ht pP fs x€ ·x „PU € Vxhsxs „ † € hR t hs‰ „ †€€t f ¬ †€€P pdths fP ÂU€ Vxhsxs ¬ †€hPR€ x ƒDƒ%§’ƒƒS¦ˆDei–qWD‚ ¥ …ƒueiu§’DDiWSƒ¦ˆR ¬Psd V p U ps V U € Vx hs xs xÄ x € hR t hs ‰ t š U U h  P P yuqµD¦f BquP ¥ WƒueiuE¦W…qDg…iW¸S£DeSWS©ø ½ 10$!5Gä$ ½$¨“ $G!£¡ ¨ #''¤nB(–'4Q–©¤µ÷
- 11. Ž yq S’ I h S I Rx‰U Àx RsUx ¦e¦ŠSS¦©ø S |= ϕ =⇒ SA |= ϕ ϕ S t p „ x Ÿ œ › ® ¦s’D¦¶²’B¿¨ SA † „PhsUs VUTRths‰t Vt€r ptsPdU€ 6SgiSS£g¦eDg…iWÁageBaaWSqƒD©¨ 7 ƒ9à¼H¡F´ýû#''¤nB(–'4Q–©¤µ÷ è ! ) ÿ þ ü ½ 10$!5Gä$ ½$¨“ $G!£¡ ¨
- 12. Ž yaÅ S’ I h S I „PU€ Vxhsxs „ †€hRths‰t „ †€€t SSƒugiuÒƒ…qDg…iW»§ƒƒD–f xÄ x € € P pd t hs fP P U € Vx hs xs xÄ x € hP R € x xÄ x € fx † ‰ Vh x š ¦WƒS¦ˆDei–qWD‚ ¥ eDBu¥eiu˜¦gDiWSƒ¦ˆR ¦W4uÒ¦£qSa¬ eU U h  P P ‚ x € ·x „ U ps V U € Vx hs xs xÄ x € hR t hs ‰ t š U U h  P P …gSWSg•W¸a'£BquP WƒueiuE¦W…qDg…iW¸S£DeSWS©ø ½ 10$!5Gä ½$¨ $ 1¨£5G£ #''¤nB$ –44“ –n#'’¤'©¤0 ¢
- 13. Ž yaÅÇ d xP „ U‚h€ xR ¬ †€ ·x t „P p‰x ¦WS'ƒƒƒ…a¦ˆ•’ƒ¡up ©S¦eWø S’ I h S I VU TtRU ŸUP W¦Dy¢eƒS4§ SA |= ϕ =⇒ S |= ϕ ϕ S t p „ x Ÿ œ › ž ¤s’ƒ¦¢ 46ƨ SA † „PhsUs VUTRths‰ Vt€r ptUT€PhsUš 6SeieS¢W¦SDgDigt ugeBaDSgiSSiR ± ¨ è ! ) ÿ þ ¥ ½ 10$!5Gä ½$¨ $ 1¨£5G£ 7 ƒ9à¼H¡Fv¦ #''¤nB$ –44“ –n#'’¤'©¤0 ¢
- 14. yŽ Ì 10$!5Gä$ £¨£¨ ¨ 5 #''¤nBÒ©©§©’˜÷
- 15. yŽ Ñ tuqsQFpnV‚ F— QP‰ˆFQPQ9IE’ƒ ” gFQXh rqo ‚ m l i™ yki “ˆ‚˜j•ˆ †ji˜ π – Eˆ ” † d† y y f•˜ † ˜ ™† d‚™‘™ˆ˜ S H(π) = πA π eQ’‰ˆgg”‚ ©I’P‰ˆQ™ — I• — $Qe999P’V• — ¬ϕ • SA πA P’E5‰ˆEƒP‰ˆy ‡„ƒ©8€Ix “ˆƒ‘ † † … y ‚ y ϕ ñ iØ wÛ ìÜ f vÝÛ í ÚÜ Ö ØÚtÚÝÙ ÕÔ ØÝig fÜ p ï Š(êsƒ{sñ ñ ViqutFa%egsgqDv¡Was9Ù aV¦PEÙegØr ß dq SA |= ϕ ÖDÕsWˆWsaàyihy(VWÕ XaqsWaFq9WyqWgDvÔ f ñÝßiÕñß ÝÜg ìge× g ÖÜÝßØß ágÙÝÜÚÔÙ Õ ϕ SA xdhsÂxds œ › ž rhUPdx q…ie¦iS¢ 46ŠDeƒSi¦Wø c „PhsUs b €hRths‰t rhUxhsx ’SgiSSH8aDiƒeDig…g¦gDi©ø a Y ž W ® ) T R 6H !DB @ 7 6 3 2 0 ) '% # ! 4`©X› ¦ 5VUFSA QPIG'FECA987 (54 1($ © 10$!5Gä$ £¨£¨ ¨ 5è £0G£¨ 5$ 5¡ #''¦Bô©©§©’BÁn…'–©%¶#’Áã
- 16. Ž yaÅâ 1 1 0 0 0 0 0 1 1 0 0 1 −→ 0 0 0 0 0 0 0 x1 x2 x3 0 0 0 x4 x3 x2 x1 Âx „UfU ¦'…4ggd ¬ † € € P „ P P ‚ ¬ † „ U f U d P € š U x h† b R t „ P t pP f € P ·x p U ps ’ƒDS'SWSWw46…4ggSWEe¢ep68qDyR Sg¦q–{s ¥ Sêu¶q¯ 1 1 0 0 0 1 0 0 0 0 1 −→ 1 2 0 0 0 x x 0 0 0 0 x4 x3 x2 x1 P „U‚h€ xR U p €P ·x „UPd x S'…ƒDƒDa¦yw¶S´ap ¥ ƒSi¦Wø £ 7¨¨£ £!£3 £ 7 ¥¡£¨ £ 7 ¥ ¶'Š'’'’¡E§ ’4'’”¶E 4¢
- 17. 'Švº !£ ! AG(go → AXstop) AGAFgo go, stop, stop, stop, stop, stop . . . yŽ æ
- 18. à%¤ ¦’3 §©46¦˜ã ¡ 5$0 ¥£! ½ 10$! 5Gä Wegƒ6¦nø õPhU€ „x do { Rel KeAcquireSpinLock(); Acq Unlocked Locked nPacketsOld = nPackets; Rel if(request){ Acq request = request-Next; Error KeReleaseSpinLock(); nPackets++; } } while (nPackets != nPacketsOld); KeReleaseSpinLock(); yaÅç Ž
- 19. Ž yaÅö $ $!C !3 $¡£ ¡E'D§å%©Dè v
- 20. Ž ya ú ‡ h b V€P „ †xhtP‚x Rt „U fPpsx UuS'HD¦gWgSW¦ŠDyR ¥ 4qqWƒ‚ Œ € f { z™ –m k y ˜ | | { z™ –m k y ˜‰ ~D¤W¤hh‹~¤W¦¦h{Š| ‡ ‡ b €€P „PP‚ dP p ‰ „P fPd ˆ8aƒS'SWSW£ˆS¦se¼S–qq§ † †{ z™ –m k y s PhsP „d xRr pxh VshU € px‚s 8s~¤W¦¦h˜ Šeiu'g•ˆgeg¢{egDBeWSSU hP ·x –m—…˜g i„™{k™f™ƒ x €f{ z™–mky˜ || {z™–mky i´a'„ W¤hg%8D…D¦(™ ‚g€ ¥ (…(~¦g¦¦h8}(¦g¦e˜ t pR U T š U t fx¬ † d P U d x ps x h xÄ x h tš†š U € P ·x p P „ U ‚ h€ x ¦qEew%ua’e˜Se¦qu¶ge“ ¥ ¦g¡aqSwSêu¦¶S'…ƒDƒDa¦n” !£ !3!5¨ 0 é 7¨ x 'ŠB#…”vX w
- 21. Ž yq $ $!C ! $¡£ è ½$¨¨£¨ ¨ 5 ¶E'D§3 %©D4ä –'©§©’˜÷