Implementing Fault-Tolerance in Real-Time Systems by Program Transformations
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Implementing Fault-Tolerance in Real-Time Systems by Program Transformations Tolga Ayav — Pascal Fradet — Alain Girault. N° 5919 May 2006. ISSN 0249-6399. apport de recherche. ISRN INRIA/RR--5919--FR+ENG. Thème COM.
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(127) . . . . . <. . ~yi ~{ ªGd!egiPr¡i8|¥{+r¡suz+wsu|§xY{+r wxyzg¤}a©Riyi{givu~y¤§| iP$sxh8zg¤§s|¥g¤¥iuiv~Ygr?~y{+a°
(128) u|¨suivr?xY sux k xYRiui~s|¥xy{ r
(129) o +~yu~yh0isiv| iP$°
(130) |§se~«RxY¡sPª d!eg|}r ¤¥~y{gyz+~yyi|¥r °%i¤¥¤§^q{gx<°
(131) { ²r¡|¥h0g¤§i¦~{ ai Iguivrur¡|¥³yi iv{gxyzgYe ªUd!egi7iP~yIiv!h0~´p«ui¢£iv!s+ x |¥i¤}rxy{~{+ |¥i¤}r¡xY{ ²PyY ¢£xY ~0wxyh0g¤¥isui7IiPrw|¥Isu|§xY{ ª d!egi7¢£xy¤¥¤¥x´°
(132) |¥{g2i g~h0g¤¥i7+xYy*~<h iP~ygr?~{azg{+r|§Y{giv|¥{Ysuiyiv ³~u|¥~y©g¤¥i7~{ $+¤¥~Ywivr |§s |§{ ª [sG©Rxyzg{+gr suegi8³<~y|}~y©g¤§i ©Qp ~{+Fac ZwxYh¦+zIsiPr
(133) suegi¢c~YwsxY|}~¤µx¢ ²R°
(134) eg|}wueZ|§s?Ã+{+~y¤§¤¥p$°
(135) u|¨suivr?~Yr i i |§sur8xyzIsugzIsPª iuiy² regxyiz+¤¥ 10 ©Ririiv{½~yr8~ayi{+iu|¥w2wxyh0gzIs*~<su|§xY{E r|§h0g¤¥ii{gxyz+ye9suxZguivri{Qs Fac wxy{+w|}ri¤¥pxYzg suivw*eg{g|}¾Yz+ivrvª0k?¢®wxYzguriy² ~yrG¤§xY{g$~Yr?suegip~yii IgiPrrivZ|¥{segi0gui³q|¥xyz+rGrpq{Ys*~ ² h8z+w*eh0xyuiwxYh0g¤§i ~y{+iP~¤¥|¥r¡s|}wGwxyh0gzIs*~<su|§xY{+r
(136) w~{$©Risuiv~siv$~Yr®°®i¤¥¤Bª. . . \. . . . *. Fac = read(i) ; if i > 10 then i := 10; o := 1; else o := 1; for l = 1 to 10 do if l <= i then o := o ∗ l; else skip; write(o);. 1 :9. . d!egi2r|§h0g¤¥ivr¡sr¡su~sivh¦iv{Ysx¢%suegi0¤}~{+yz+~yyi|}r ·£segi ¿|¥{+r¡suz+wsu|§xY{ ¸²µ°
(137) e+|¥w*e9i q|}r¡surxY{ skip ~y¤§¤
(138) guxIwivrur¡xyurvª i$s*~yi$se+i `x¢ segi wxyh0h2~{+¿sx ©Risegiazg{g|§s«xy¢ s|¥h¦iZ~y{+¼°%i ~Yrrzgh0i2sue+~<ssuegii IivwzIs|¥xy{Esu|§h0ivrxy¢?~¤¥¤%xsuegskip i¦r¡su~sivh¦iv{Ys*r~yi«h8z+¤¨su|§+¤§i$x¢ 6· ¸ª h0xyui7¢£zg{+g~h0iv{Ysu~y¤:~Yrrzgh0Is|¥xY{|}r se+~s?segii Iivwzgs|¥xy{su|§h0ivr¦·£©Ri8|§s "² "²Rxy skip¸
(139) xy¢ ~y{Qpr¡su~sivh¦iv{Ys8·cxy
(140) i q+ivrur|§xY{ ¸ wv~{$©Ri7i³~¤¥z+~<suivªMd!egi ~y{+~¤¥pIr|¥r®|}r!segisuxyg|}wx¢:h8z+w*e °%xYZ·cri i Mz+ruwueg{+i?~{+@ ?®zgu{+rvS²vYyq² | is?~¤^ª¥²+YY ¢£xy!|§{ rts*~{+wiP¸ "I°%i7r¡e+~y¤§¤{gxys
(141) I°%iv¤§¤zgRxY{ sueg|¥r
(142) |}rur¡zgi7~y{Qp«¢ z+¡suegiPª ¹gxy!se+iuih2~|¥{g|¥{g¦x¢se+i7~su|¥w¤§iY²q°%iGà «suegii qiPwzIsu|§xY{su|§h0iPr!xy¢:r¡su~sih0iv{Ysur!sx0©Ri2·c|§{$s|¥h0i zg{g|§s*ru¸%suegxYrix¢"d:~y©g¤§i0Yª. . aic3ade. . #. . 2. 1. 2. 1. 2. 1. 2. 1. . 2. 1. 2. 1. 2. 1. 2. fhglade. . adc3aie (skip) = jkgAaie (skip) = fhglade (skip) adc3aie (read) = jkgAaie (read) = fhglade (read) adc3aie (write) = jkglade (write) = fhgAade (write) adc3aie (x := e) = jlglade (x := e) = fhgAaie (x := e) adc3aie (S ;S ) jkgAaie fhglade (S(S ;S;S )) fhglade (if B then S else S ) jkgAaie (if B then S else S ) adc3aie (for i = n to n do S) jkgAaie i = n to n do S) fhglade (for (for i = n to n do S) 1. adc3aie adc3ade lfhgAade jkglade. . = = = = = = = = = = = =. 1 3 3 3. da c3aie (S ) + adc ade (S ) jkgAaie jkglade fhglade (S(S ))++ fhgAaie (S(S) ) fhglade (S ), fhglade (S )) 1 + max( k j gAaie (S ), jkgAaie (S )) 1 + min( adc3ade (S)) (n − n + 1) × (3 + jlglade (S)) (n − n + 1) × (3 + h f gAaie (S)) (n − n + 1) × (3 + 1. 2. 1. 2. 1. 2. 1. 1. 2. 2. 2. 1. 2. 1. 2. 1. d:~©g¤¥i$D! E g~ywsP² °%xYur¡sw~Yr¡iY² ~{+©RiPrts[wv~yrii IivwzIs|¥xy{su|§h0iPrGxy¢%xyzgguxyYu~h0h0|§{+¤}~{gyz ~yiYÁ r r¡su~sih0iv{Ysurvª k?¢MwxYzguriy²+°
(143) egiv{suegi adc3ad9 e x¢¬~«rts*~<sivh0i{Ys |}r?Q{+x´°
(144) { ² |§s |¥r ~¤}rx2iv¾Qz+~y¤ sx«|§s*r fhglade ~y{+|§sur jkglade:ª9d!egi~y©Rx´³yi2à yzguivr~ui«³<~¤¥|}9¢£xY¦~{Qp #¡r|§h0g¤¥i%iIguivrur¡|¥xy{+r e xY b ª r¡|¥{gZsuih0Rxy*~up. GBGIH1JKL6M:L.
(145) "' 7) ' "& + /2)7' > + ³<~y|}~y©g¤§iPr²+|¨sG|¥r ~y¤§°®~´pIr
(146) RxQrr|§©+¤§i7suxrg¤¥|¨sGwxyh0g¤¥ia~u|¨suegh0isu|¥w8~{+a©Rxqxy¤¥iv~y{iI guivrur¡|¥xy{ r
(147) rx2se ~<s suegip»uih2~|¥{`r|¥h0g¤§i9i{gxYzgye±·c~Yr|¥{ suegivi¯~yggivrurwxIIi´¸ª d!egi fhglade ·ciPr¡ ª jlglade ¸x¢suegi . C. . . sadc3ade fhglade. rts*~<suih0i{Qs
(148) |}r wxyh0gzIsuiv$|§{$se+i7ru~h0i°®~´py²I©qp«ui+¤¥~Yw|¥{g ©Qp |¥{se+i7|¥yeQs¡[e+~{+ +~y¡s·ciPr¡ ª ¸"+ru~h0iGse+|§{g0¢£xy
(149) segi Á¥ª ; |¨suesegiPr¡i8à yzguivrv²R°%iYis ª ¯{Zsegi¦uivr¡s xy¢Usuegi0~s|}w¤¥iy²°%i¦wxY{+r|¥Iiv?segi (Fac) = 84 s*~yr ²Isue+~<s |}r®sux2r~´p °
(150) |¨suea~0Iiv~YI¤§|¥{gi ´Riu|§xg«xy¢ su|§h0i7zg{g|§survª 200 d!(Fac, egi7iP~200) ¤§Bsu|§h0iguxyRiv¡stp2¢£xy?Fac ~0rpIrtsuih xy¢ s*~yrIr |¥r!sue+~<s?iv~yw*es*~yr {(S1 , T1 ), . . . , (Sn , Tn )} h8z+r¡s h0iis |§sur IiP~yI¤¥|¥{giyª¬oq|¥{+wi7iv~Ywue$guxIwiPrrxyn®uzg{+r
(151) ~0 r|§{+y¤¥iGsu~yr²I|¨s?~yh0xyzg{Qsur®sux !. Ajlglade. for. fhglade. ∀i ∈ {1, 2, ..., n},. fhglade (S ) ≤ T i. ·¡P¸. i. d!egirih2~y{Ys|}wrx¢~½r¡su~sivh¦iv{Ys |}r«Y|¥³yi{»©Qp¼suegiZ¢£zg{+ws|¥xy{ ª, [[S]] : State → State r¡su~si h2~y+r guxyYu~h³<~uS |}~©g¤¥ivr sxsegiv|§³~¤¥zgivrvª7d!e+i0 r¡ivh0~y{Ysu|¥w8¢£zg{+wsu|§xY{s*~yivrG~ r¡su~sih0siv∈{Ys State ²Q~{0|¥{g|§s|}~¤ rts*~<sui ~y{+¦pq|§iv¤¥grUVsegi uivr¡z+¤¨su|§{+7r¡su~si xY©Isu~y|§{+iv0~<¢ sivUse+i
(152) i Iivwzgs|¥xy{ S s0 xy¢Gsuegirts*~<sivh0i{Y6s ! ª,oqi³yivu~y¤ iv¾Qzg|¥³<~y¤§iv{Ys«¢£xyuh0~y¤?sgfiÃ+{g|§s|¥xy{+rx¢ ·£xYRi*~<su|§xY{+~¤^² [[.]] 0 = sf Iiv{gxs*~<su|§xY{+~¤^²I~ q|¥xYh0[[S]]s ~s|}wv¸® w~{$©Ri¢£xyzg{+$|¥{ |¥i¤}rxy{~{+ |¥i¤}r¡xY{ ²PyY [ª d!egi k,rih2~{Ysu|}wr!x¢"~s*~yr |}r!Y|§³yiv{©qp~¦+~|¥
(153) x¢"r¡suiv~yh0r. . ;. . . (S, T ). (i1 , . . . , in , . . .), (o1 , . . . , on , . . .). °
(154) egivi |}r®suegi7|¥{ggzIs?gux´³q|}Iiv©Qp«suegi7i{q³q|§uxy{gh0iv{Ys Iz+|¥{g¦se+i seRiv|¥xI$~{+ |}r®suegi7¤}~yr¡s Yx zIsgzIs®ik°
(155) |§s¡sui{«gzg|¥{g7se+i sue«Riu|§xI ª"oqx ²Y|§¢ ri³yivu~y¤ ~uki
(156) Ri¢£xyuh¦iP0Izguo|¥{gk 8~7Riu|§xI ² suegir¡ivh2~{Ysu|¥wvr8~y{+9suegii{q³qk|§uxy{gh0iv{Ys8°
(157) |¥¤§¤!wxy{ r¡|}Ii8xywrite(o) {+¤§psegi¤¥~Yrtsxy{giYª i$~¤}r¡x~yrur¡zgh0i2se ~<s suegi7iv{Q³q|§uxy{+h¦iv{Ys gxYRxYrivr%suegiru~h0i|¥{ggzgs?gzg|¥{g«~¦Riu|¥xI4 !Mri³yi*~¤ Izgu|§{g0se+iru~h0i read(i) Riu|¥xI°
(158) |¥¤¥¤uivrzg¤§s
(159) |¥{suegir~yh¦iuiv~YI|§{+Yrvª ¹gxy
(160) i I~yh0g¤§iY²I|§¢suegi7i{q³Q|¥uxy{gh0i{Qs
(161) guxyRxYrivr ~yr!|¥{ggzIs segiv{$segi7guxyYu~h. . 2. read(i); o := i; write(o); read(i); o := o ∗ i; write(o). g uxIIz+wivr ~yr7xyzIsugzIsIz+|¥{gse+~sr~yh¦i2Riv|¥xI²:~{+9{gxys ª$ rrzgh0|¥{gsegiriv¾Qzgiv{+wi2xy¢ 4 (2, 4) |¥{Ysuiyivur
(162) ~yr! |¥{ggzIs*r²gsegi k,rih2~{Qs|}wr!x¢ |}r ! Fac. (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, . . .), (0, 1!, 2!, 3!, 4!, 5!, 6!, 7!, 8!, 9!, 10!, 10!, 10!, . . .). . . « #. $: :8. :8 1µ;< " : $1. . ¹ ~|¥¤§z+i¦Iisuivws|¥xy{~y{+a¢c~|¥¤¥zgih2~Yr¡q|¥{gui¤¥pxy{|¥{+ris|¥{gegiv~y¡su© iP~<su|§{g$~{+w*egivw*qRxy|¥{Ysu|§{g|¥{I r¡suz+wsu|§xY{+r|¥{EguxyYu~h2rvª$d!e+ivri«|§{ rtsuz+ws|¥xy{+r8h8z+rts¦© i|¥{+risivErz+w*e sue+~<s8se+ip ~yi«i IivwzIsiP & <ª ise+iui¢£xyui«suu~y{+rt¢£xYh ~as*~yr gxYy*~h rz+w*e½se+~s0~egiP~s©Riv~s¦~{ ¿~w*egivw*Q RxY|§{Qs
(163) ~ui i IivwzIsuiviv³yiup ~{ Riv|¥xI«xy¢µs|¥h0iiPr¡RiPws|¥³yi¤¥pYªMj%xy{+I|§su|§xY{+~¤r¡su~<suih0i{Qsur THB CP wxyh0g¤¥|¥wv~<si«se+|¥r|¥{+r¡iv¡su|§xY{ ª d!e+ip ¤§iPT~y sxh2~{qp9+~se+r°
(164) |¨sue¿g| iui{Ys¦i IivwzIsu|§xY{Es|¥h¦iPrª [s¦|¥r suegiui¢£xyui!|¥h0RxYrur¡|¥©g¤¥i®sux7|¥{+r¡iv¡sM|¥{+r¡suz+wsu|§xY{+rM~<s¬wxy{+r¡su~y{YsUs|¥h0i |§{Qsiu³<~y¤¥rU°
(165) |¨suegxyzgs¬Iz+g¤§|}w~s|¥{gsuegi wxIIiyª8dx$~´³yxy|}sueg|}rGguxy©g¤¥ih²°%iÃ+*rtsGsuu~y{+rt¢£xYh¶suegi¦guxyYu~hÂ|§{xYugi sxà asuegi¦i Iivwzgs|¥xy{ su|§h0i7x¢U~¤¥¤ wxY{+I|§s|¥xy{+~y¤ ~{+¤¥xqxY+r%sux¦se+i|¥
(166) °®xy*rts w~yrii Iivwzgs|¥xy{s|¥h0iyª ¯{Ysuzg|¨su|§³Yi¤¥py²g|§s?~h0xYzg{Ys*r. 9#& ),/ : ;/ 6(';>2> . . . # . "! G#"$.
(167) 9 >& & 1+0/2143 ' > + :*>?&)7'*1 6&@- )7:73")7' )7';1 :*) '*+0/ :*1. . . .
(168) . . . . . . sux~y+I|§{+Izgh0h8pZwxIIisuxwxy{+I|§s|¥xy{ ~¤:~y{+a¤¥xqxyrts*~<suih0i{Qsurvªoqz+w*eZs*~{+r¡¢£xyuh2~<su|§xY{+r rzggRxQr¡i sux©Ri~©g¤¥i«suxi³<~y¤§z ~<si«suegi »x¢?guxyYu~h2rª¥ª ¢ sivsueg|}r8su|§h0i$iv¾Qz+~¤¥| v~<su|§xY{ ²Uw*egivw*qRxy|¥{Ys*r ~y{+$egiv~su© iP~<s*r
(169) w~y{$©Ri7|¥{YsuxgIz+wiP$r|¥h¦+¤§pz+r|§{g0suegiru~h0is*~{+r¡¢£xyuh2~<su|§xY{ ªMdµx2i{ r¡zguiGsu|¥h¦|¥{g wxyuivws{giPrr%|¥{segiPr¡iGs*~{ rt¢£xYh2~<su|§xY{+r²Q°%i7~y¤¥rx¦~Yrrzgh0i segi7z+r¡ix¢:gxgwivrur¡xYurMsue+~<s Ix0{gxs
(170) e+~´³yi wxyh0g¤¥i ©Rive+~P³q|¥xy*rrz+w*e¿~yr8g|¥Ri¤¥|¥{g|§{+~y{+½w~Ywueg|¥{g ªd!e+|¥r¦~yrurzgh¦gs|¥xy{½|¥riP~¤¥|¥r¡s|}w«r|¥{+wi«suegi uiv~y¤¨^s|¥h0iGih8©RiPgIiv~ygg¤¥|¥wv~<s|¥xy{ r®Yi{givu~y¤§¤¥p«Ix0{gxys
(171) ui¤¥p«xy{suegivriG¢£iP~<sz+iPrª `s*~{+r¡¢£xyuh2~<s|¥xy{«h0~´p0|¥{+wiP~yri segi axy¢µguxyy*~h2rª"oqx ²I~<¢ siv®iP~ywue«suu~y{+r¡¢£xyuh0~s|¥xy{ ² suegiiP~¤§Bsu|§h0iwxy{ rtsuu~y|§{Qs h8z+r¡sG© i¦w*egivw*yiv4 "se+~y{gIr sxxyzgG~yrurzgh0Is|¥xy{+r?Txy{ (T (S)) ≤ T "²Iseg|}r w~y{©RiIxY{gi7~zgsxyh2~s|}w~¤¥¤¥pyª. fhgAade. . fhglade. fhglade . fhglade.
(172) !". . . E¬¾Qz+~y¤§| |¥{g segi%i. IivwzIs|¥xy{7s|¥h0i%x¢+~
(173) guxyYu~h wxy{ r¡|}rts*r |¥{+~YgI|¥{g?gzgh0h8p7wxIIi¬|¥{¤§iPrrµi IRiv{+r¡|¥³yi ©g*~{+w*egiPrª d!egi¦Iz+h¦h8pZwxgIi~ygIiP¢£xY +~ygg|§{g|¥rGriv¾Qzgi{+wivr?xy¢ r¡su~<suih0i{Qsurvª i°
(174) |§si skip s x v i g P i ¡ r v i Y { G s ~ ¡ r P i Y ¾ + z i + { w i y x ¢ ¡ r u s < ~ u s i 0 h i Q { u s r. ! ªGd!eg|}r suivw*eg{g|}¾Yz+i8|}r n skipn r|§h0|¥¤¥~y®sux¦suegi7xy{giz+riv$|§{ #r¡|¥{gy¤¥iG n~<sskip egxYy*~h0h¦|¥{g1 % Mz+ruwueg(skip {+iP²+yy)Q= ^ª n d!egiy¤¥xy© ~¤¬iv¾Qz+~y¤§| v~s|¥xy{Egxgwivrur|¥rIiÃ+{giPE|¥{+Iz+ws|¥³yi¤¥p ©Qpsuegi¢£xy¤¥¤§x<°
(175) |¥{gas*~{+r¡¢£xyuh2~<su|§xY{ uzg¤¥ivrv²I{gxsuiv ªMd!egiuzg¤¥ivr!©Ri¤¥x<°`h8z+r¡s ©Ri7zg{+Iivur¡sxqxg¤§|¥yi7~0wv~yriGi Iguivrur¡|¥xY{|¥{suegi7gxYy*~h¦ h0|¥{g¤}~{+yz+~yF yi0l lZ|§¤¥{gi is?~¤^ª¥²Pyy6 /!8w~yrivr7~ui0i³<~y¤§z+~siPZ¢£uxyhnsxYsux©Rxys¡suxyh²~y{+suegi suu~y{+rt¢£xYh2~s|¥xy{9z+¤§iwxYuivrRxy{+I|¥{g$sxasuegi«Ã+ur¡s8+~s¡siv{ se ~<sh0~suw*egivr7suegi|§{+gzIsguxyYu~h |¥r Ri¢£xyuh0ivª. adc3ade. . . *. # < % $ Ê ½g Ê . * B Æ . . 1. F[if B then S1 else S2 ] 2. F[for i = n1 to n2 do S] 3. F[S1 ;S2 ] 4. F[S]. /2+<$. :*). if B then F[S1 ]; skipmax (0,δ2 −δ1 ) ; else F[S2 ]; skipmax (0,δ1 −δ2 ) ; δi = (F[Si ]) i = 1, 2 = for i = n1 to n2 do F[S] = F[S1 ]; F[S2 ] = S =. fhglade. :;+2$&) / &. . j%xY{+I|§s|¥xy{+~y¤¥r ~uisegixY{g¤¥pr¡su~sih0iv{YsurGrzg©ItivwsGsx$wxgIi8h0xII|§Ã w~s|¥xy{¿·c zg¤§i$P¸ªGd!egi8s*~{ rt¢£xY¡ h ~<su|§xY{~Yggr"~yrh2~y{Qp 2 ~yr{+iivIiPsxGh2~<s*wue8suegi®i qiPwzIsu|§xY{su|§h0i!x¢gsuegi!xse+i:©g*~{ wue !µegi{ wi suegi |¥{$skip suegi ©gu~y{+wueµª¬d!egi #th0xYr¡s i IRi{+r|¥³yi%©g*~{+w*e$uih2~|¥{+r
(176) zg{+wue ~{gYiv² °
(177) eg|¥¤¥max(0, isegi #¡¤§δiP2rr
(178) −i Iδ1R)i{+r|¥³yi©g*then ~{+w*e(%i{++r
(179) zg$s*~q|¥{g¦suegiru~h0is|¥h0i~Yr®suegi7h0xQrts i IRiv{+r¡|¥³yi7xy{giYª d!egiasuu~y{+r¡¢ xYh2~s|¥xy{ |¥r~ygg¤¥|§iP |¥{+Iz+ws|¥³yi¤¥p¼sxEsegirts*~<sivh0i{Ysx¢7iv~yw*eº©g*~{+w*eºgu|§xY0suxEseg|}r iP¾Yz ~¤¥| P~<s|¥xy{µª i {gx´°»gx<³yi
(180) se+~sv²Y¢£xY%~y{Qp¦guxyy*~h ²Yse+i?©Rivr¡sv²q~y{+2°%xy*rts%wv~yri
(181) i Iivwzgs|¥xy{0su|§h0iPr¬x¢ S F[S] ~yiG|}Iiv{Ys|}w~y¤ !. . & Ê Æ ´ Ç ∀S, jkglade (F[S]) = fhglade (F[S]) = adc3ade (F[S]) . ( ' d!e+iguxqxy¢|¥r
(182) ©Qp|¥{+Iz+ws|¥xy{xy{se+irtsuz+wszguix¢suegi7guxyy*~h. )7::. GBGIH1JKL6M:L. . S. ª. .
(183) "' )7' "& + /2)7' > +. P. . . . ª d!egi|§{ Iz+wsu|§xY{¿eQpqRxsuegivr|}r8|}rsue+~<s<jlglade µis = if B then S else S ]) = fhgladSe (F[S d a 3 c d a e ~{+ adc3ade jkglade (F[S ]) = fhglade (F[S ]) = (F[S ]) = δ (F[S ]) = (F[S ]) = adc ade (F[S ]) = δ ª !wvwxYuI|¥{gasx
(184) zg¤¥i~y{+Ed:~©g¤¥iZY²:°®i2sueQz+re+~´³yi<fhglade (F[S]) = 1. 2. 1. 2. 1. 1. 1. 2. 2. 2. 1 + max δ1 + max(0, δ2 − δ1 ), δ2 + max(0, δ1 − δ2 ). 2. ª. |§segxYzIs¬¤¥xYrurUx¢Yi{givu~y¤§|§stpy²y~Yrrzgh0i!se+~s · se+i?r¡pqh0h0isu|}w~y¤+w~yri!pq|¥i¤}grMr|¥h0|§¤}~ δ1 ≥ δ 2 wxyh0gzIs*~<s|¥xy{ ru¸ª d!egiv{ ²~y{+ δ2 + max(0, δ1 − δ2 ) = ª ?i{+δw1i + max(0, δ2 − δ1 ) = δ1 + 0 = δ1 ª (F[S]) = 1 + max(δ1 , δ1 ) = 1 + δ1 δ2 + δ 1 − δ 2 = δ 1 j%xy{q³yi*r¡iv¤§pY²P°%i®~y¤¥rx?e+~´³yi δ +max(0, δ2 −δ1 ), δ2 +max(0, δ1 − ª:d!egi{µ²yrtsu|§¤¥¤q©Qp8~yrur¡zgh0|¥{g?se+~s (F[S]) =²<1+min °®i®~y¤¥rx à {+1 δ1 ≥ δ 2 (F[S]) = 1+min(δ1 , δ1 ) = δ2 ) ª 1 + δ1 ¯{7wxy{+w¤§z+r|¥xy{ ² ~{+ suegiui¢£xYiU|¨sµ|}r ~y¤¥rx!iv¾Qz+~y¤<sx ª. R\. fhglade jkglade. jlglade. "jkgAaie (F[S]) = fhglade (F[S]). adc ade (F[S]). µis = for i =adc3nade to n do S ªªd!
(185) egwi2wxy|¥{+*IIz+|§{+ws|¥xysuxa{
(186) eqpQzgR¤¥ixys$egiP~yr¡{+|}r9|}d:r~su©+e+¤§~<is Yjk²µgl°%adi0e (F[S fhgladSe (F[S seqz+r7e+])~´³y=i ]) = (F[S ]) = δ adc ade ªMoq|¥{+wi ~y{+ ~ui?wxY{+rts*~{Qs%~y{+2©Qp0|¥{+Iz+ws|¥xy{ (n − n + 1) × (3 + δ ) eqpQRxys(F[S]) egiPr¡|}rv²qs=eg|}r
(187) |}r
(188) ~¤}rx¦iv¾Qz+~y¤sx<jkglade ~n{+ fhgladne ª (F[S]) (F[S]) µadcis ade S = S1;S2ª d!~yeg{+i |§jl{+glgadz+e wsu|§xY{¼eqpQRxysfhegiPglr¡ad|}r¦e |}rsue+~<sjkgladadec3(F[S fhglade (F[S ]) = ]) = i a e ª?
(189) wwxy*I|§{+ sux2
(190) z+(F[S ¤§i 2~]) {+=dδ~y©g¤¥i0y²g°%iseq(F[S z+r
(191) e ~P³y])i =aic3ade (F[S ]) = ª ?%(F[S p|¥{+I])z+w=s|¥xyδ{eqpqRxse+ivr¡|}rv² sueg|¥r
(192) |}r ~¤}r¡x¦iP¾Yz ~¤sux<jkgAade ~y{+ fhglade (F[S]) = ª δ +δ (F[S]) (F[S]) d!eqz+r²g°%iwxY{+w¤¥z+IiGsue+~<s
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