Implementing fault-tolerance in real-time systems by program transformations

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appor t de r ech er ch e

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Implementing Fault-Tolerance in Real-Time Systemsby Program Transformations

Tolga Ayav — Pascal Fradet — Alain Girault

N° 5919

May 2006

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Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe, 38334 Montbonnot Saint Ismier (France)

Téléphone : +33 4 76 61 52 00 — Télécopie +33 4 76 61 52 52

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d�x�r¡|¥h0�g¤¥|¨¢£p�s�egi��g�uivr�i�{Qsu~�s�|¥xy{ ²"°%i�su~y­yi«s�egi$�giv~y�I¤¥|¥{gi�iP¾Yz�~�¤¬suxZs�e+i��Riv��|¥xI�E©gzIs¦xYzg�¦~��g�g�uxY~Ywue�IxIivr®{gxys��Iiv�Ri�{+��xY{�s�eg|}r�~Yr�r�zgh0�Isu|§xY{ ª \i�{�w�iy²Is�egi��uiv~y¤¨�^s�|¥h0iGw�xy{+r¡s��*~�|¥{Ys�~yrur�xIw�|}~<suiv�2s�x¦s�egiGs*~yr�­(S, T )

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S ::= x:=A assignment

| skip no operation

| read(i) input read

| write(o) output write

| S1;S2 sequencing

| if B then S1 else S2 conditional

| for i = n1 to n2 do S iteration

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xY�Ri��u~�s�|¥xy{�r��+~y�u~yh0i�s�iv��| ��iP�$°�|§s�e�~«�RxY�¡sPª d!eg|}r ¤¥~y{g�yz+~y�yi�|¥r °%i�¤¥¤§�^­q{gx<°�{ ²�r¡|¥h0�g¤§i¦~�{��ai��I�g�uivrur¡|¥³yiiv{gxyzg�Ye ªUd!egi7��iP~y�Iiv�!h0~´p«�ui�¢£iv�!s�x+� � |¥i�¤}r�xy{�~�{+� � |¥i�¤}r¡xY{ ²��P�y�Y�� ¢£xY�~0w�xyh0�g¤¥i�sui7�IiPr�w���|¥�Isu|§xY{ ªd!egi7¢£xy¤¥¤¥x´°�|¥{g�2i��g~�h0�g¤¥i7�+��xY�y�*~<h

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|§sur8xyzIsu�gzIsPª \i��uiy²Fac

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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);

d!egi2r�|§h0�g¤¥ivr¡s�r¡su~�s�ivh¦iv{Ys�x�¢%suegi0¤}~�{+�yz+~y�yi�|}rskip

·£s�egi 1 :�9¿|¥{+r¡s��uz+w�su|§xY{�¸�²µ°�e+|¥w*e9i��q|}r¡sur�xY{~y¤§¤��g�uxIw�ivrur¡xy�urvª � i$s*~�­yi$s�e+i aic3ade`x�¢ s�egi

skipw�xyh0h2~�{+�¿s�x�©Ri�s�egiazg{g|§s«xy¢ s�|¥h¦iZ~y{+�¼°%i

~Yr�r�zgh0i2sue+~<s�suegi�i��Iivw�zIs�|¥xy{Esu|§h0ivr�xy¢?~�¤¥¤%x�suegi��¦r¡su~�s�ivh¦iv{Ys*r�~y��i«h8z+¤¨su|§�+¤§i$x�¢ adc3aie6·skip

¸�ª��h0xy�ui7¢£zg{+�g~�h0iv{Ysu~y¤:~Yr�r�zgh0�Is�|¥xY{�|}rs�e+~�s?s�egi�i��Iivw�zgs�|¥xy{�su|§h0ivr¦·£©Ri8|§s adc3ade"²lfhgAade"²Rxy� jkglade�¸�xy¢~y{Qp�r¡su~�s�ivh¦iv{Ys8·cxy��i��q�+��ivrur�|§xY{�¸

Swv~�{$©Ri7i�³�~�¤¥z+~<suiv��ªMd!egi fhglade�~y{+~�¤¥pIr�|¥r®|}r!s�egi�suxy�g|}w�x�¢:h8z+w*e

°%xY��­Z·cr�i�i�� �Mz+ruwueg{+i��?~�{+�@?®zg�u{+rv²��v�Y�y�q²#� | i�s?~�¤^ª¥²+���Y�Y���¢£xy�!|§{�rts*~�{+w�iP¸�"I°%i7r¡e+~y¤§¤�{gxys��I°%iv¤§¤�zg�RxY{sueg|¥r�|}rur¡zgi7~y{Qp«¢ z+�¡suegi��Pª

¹gxy�!s�e+i��ui�h2~�|¥{g|¥{g�¦x�¢�s�e+i7~���su|¥w�¤§iY²q°%iGÃ��«suegi�i��qiPw�zIsu|§xY{�su|§h0iPr!xy¢:r¡su~�s�i�h0iv{Ysur!s�x0©Ri2·c|§{$s�|¥h0izg{g|§s*ru¸%suegxYr�i�x�¢"d:~y©g¤§i0�Yª

adc3aie(skip) =

jkgAaie(skip) =

fhglade(skip) = 1adc3aie

(read) =jkgAaie

(read) =fhglade

(read) = 3adc3aie(write) =

jkglade(write) =

fhgAade(write) = 3adc3aie

(x := e) =jlglade

(x := e) =fhgAaie

(x := e) = 3adc3aie(S1;S2) =

adc3aie(S1) +

adc ade(S2)jkgAaie

(S1;S2) =jkgAaie

(S1) +jkglade

(S2)fhglade(S1;S2) =

fhglade(S1) +

fhgAaie(S2)fhglade

(if B then S1 else S2) = 1 + max(fhglade

(S1),fhglade

(S2))jkgAaie(if B then S1 else S2) = 1 + min(

jkgAaie(S1),

jkgAaie(S2))adc3aie

(for i = n1 to n2 do S) = (n2 − n1 + 1) × (3 +adc3ade

(S))jkgAaie(for i = n1 to n2 do S) = (n2 − n1 + 1) × (3 +

jlglade(S))fhglade

(for i = n1 to n2 do S) = (n2 − n1 + 1) × (3 +fhgAaie

(S))

d:~�©g¤¥i$�D! E �g~yw�sP² °%xY�ur¡s�w�~Yr¡iY² ~�{+��©RiPrts��[wv~yr�i�i��Iivw�zIs�|¥xy{�su|§h0iPrGxy¢%xyzg���g�uxy�Y�u~�h0h0|§{+��¤}~�{g�yz�~��yiYÁ rr¡su~�s�i�h0iv{Ysurvª

k?¢Mw�xYzg�ur�iy²+°�egiv{�suegi adc3ade9x�¢¬~«rts*~<s�ivh0i�{Ys |}r?­Q{+x´°�{ ²�|§s |¥r ~�¤}r�x2iv¾Qz+~y¤ s�x«|§s*r fhglade�~y{+��|§surjkglade:ª9d!egi�~y©Rx´³yi2Ã��yzg�uivr�~��ui«³<~�¤¥|}�9¢£xY�¦~�{Qp #¡r�|§h0�g¤¥i�%�i��I�g�uivrur¡|¥xy{+r

exY�

bª�� r¡|¥{g�Zsui�h0�Rxy�*~��up

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³<~y��|}~y©g¤§iPr�²+|¨sG|¥r ~y¤§°®~´pIr��RxQr�r�|§©+¤§i7sux�r��g¤¥|¨sGw�xyh0�g¤¥i��a~��u|¨suegh0i�su|¥w8~�{+�a©Rxqxy¤¥iv~y{�i��I�g�uivrur¡|¥xy{�r�r�x2s�e�~<ssuegi�p»�ui�h2~�|¥{`r�|¥h0�g¤§i9i�{gxYzg�ye±·c~Yr�|¥{ sueg��ivi��¯~y�g�g��ivrur�w�xI�Ii´¸�ª d!egi fhglade ·c��iPr¡� ª jlglade ¸�x�¢�suegifor

rts*~<sui�h0i�{Qs�|}rw�xyh0�gzIsuiv�$|§{$s�e+i7ru~�h0i�°®~´py²I©qp«�ui��+¤¥~Yw�|¥{g�sadc3ade�©Qp�fhglade�|¥{�s�e+i7��|¥�yeQs¡�[e+~�{+��+~y�¡s�·c��iPr¡� ªAjlglade ¸�"+ru~�h0iGs�e+|§{g�0¢£xy��s�egi��

;紻

� |¨sue�s�egiPr¡i8Ã��yzg�uivrv²R°%i��Yi�s fhglade(Fac) = 84

ª �¯{Zs�egi¦�uivr¡s xy¢Usuegi0~���s�|}w�¤¥iy²�°%i¦w�xY{+r�|¥�Iiv�?s�egis*~yr�­

(Fac, 200)²Isue+~<s|}r®sux2r�~´p

Fac°�|¨suea~0�Iiv~Y�I¤§|¥{gi��´�Ri��u|§xg�«xy¢

200su|§h0i7zg{g|§survª

d!egi7��iP~�¤§�Bsu|§h0i��g�uxy�Riv�¡stp2¢£xy�?~0r�pIrtsui�h xy¢ns*~yr�­Ir {(S1, T1), . . . , (Sn, Tn)} |¥r!sue+~<s?iv~yw*e�s*~yr�­h8z+r¡sh0i�i�s|§sur�IiP~y�I¤¥|¥{giyª¬oq|¥{+w�i7iv~Ywue$�g�uxIw�iPr�r�xy�®�uzg{+r�~0r�|§{+�y¤¥iGsu~yr�­�²I|¨s?~yh0xyzg{Qsur®sux�!

∀i ∈ {1, 2, ..., n}, fhglade (Si) ≤ Ti·¡�P¸

d!egi�r�i�h2~y{Ys�|}w�r�x�¢�~½r¡su~�s�ivh¦iv{YsS|}r«�Y|¥³yi�{»©Qp¼suegiZ¢£zg{+w�s�|¥xy{

[[S]] : State → Stateª,�

r¡su~�s�is ∈ State

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S²Q~�{0|¥{g|§s�|}~�¤�rts*~<sui

s0~y{+�¦pq|§iv¤¥�grUs�egi�uivr¡z+¤¨su|§{+�7r¡su~�s�i

sfxY©Isu~y|§{+iv�0~<¢ s�iv�Us�e+i�i��Iivw�zgs�|¥xy{

xy¢Gsuegi�rts*~<s�ivh0i�{Ys6![[S]]s0 = sf

ª,oqi�³yiv�u~y¤iv¾Qzg|¥³<~y¤§iv{Ys«¢£xy�uh0~y¤?�gi�Ã+{g|§s�|¥xy{+r�x�¢[[.]]

·£xY�Ri��*~<su|§xY{+~�¤^²�Iiv{gx�s*~<su|§xY{+~�¤^²I~;�q|¥xYh0~�s�|}wv¸®w�~�{$©Ri�¢£xyzg{+�$|¥{ � � |¥i�¤}r�xy{�~�{+� � |¥i�¤}r¡xY{ ²��P�y�Y��[ªd!egi ��k,r�i�h2~�{Ysu|}w�r!x�¢"~�s*~yr�­

(S, T )|}r!�Y|§³yiv{�©qp�~¦�+~�|¥��x�¢"r¡s��uiv~yh0r

(i1, . . . , in, . . .), (o1, . . . , on, . . .)

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ks�e��Riv��|¥xI�$~�{+�

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suegi�r¡ivh2~�{Ysu|¥wvr8~y{+�9suegi�i�{q³q|§�uxy{gh0iv{Ys8°�|¥¤§¤!w�xy{�r¡|}�Ii��8xy{+¤§p�s�egi�¤¥~Yrts�xy{giYª � i$~�¤}r¡x�~yrur¡zgh0i2s�e�~<ssuegi7iv{Q³q|§�uxy{+h¦iv{Ys�g��xY�RxYr�ivr%suegi�ru~�h0i�|¥{g�gzgs?�gzg��|¥{g�«~¦�Ri��u|¥xI�4!Mr�i�³yi��*~�¤

read(i)�Izg�u|§{g�0s�e+i�ru~�h0i

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2~yr!|¥{g�gzIss�egiv{$s�egi7�g�uxy�Y�u~�h

read(i); o := i; write(o); read(i); o := o ∗ i; write(o)

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|¥{Ysui��yiv�ur�~yr!|¥{g�gzIs*r�²gs�egi ��k,r�i�h2~�{Qs�|}w�r!x�¢Fac

|}r!

(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!, . . .)

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(skipn) = nªGd!eg|}rsuivw*eg{g|}¾Yz+i8|}r

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# �<� � �%$ Ê �*�½�g��� Ê � �* B Æ � �

1. F [if B then S1 else S2] = if B then F [S1]; skipmax (0,δ2−δ1);

else F [S2]; skipmax (0,δ1−δ2);� /2+<$

δi =fhglade

(F [Si])��:*)

i = 1, 22. F [for i = n1 to n2 do S] = for i = n1 to n2 do F [S]3. F [S1;S2] = F [S1]; F [S2]4. F [S] = S

:;+2$�&�) � / ��&

j%xY{+�I|§s�|¥xy{+~y¤¥r ~��ui�s�egi�xY{g¤¥p�r¡su~�s�i�h0iv{YsurGr�zg©I�tivw�sGs�x$w�xg�Ii8h0xI�I|§Ã�w�~�s�|¥xy{¿·c�zg¤§i$�P¸�ªGd!egi8s��*~�{�rt¢£xY�¡�h2~<su|§xY{�~Y�g�gr"~yr�h2~y{Qp

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suegimax(0, δ2 − δ1)

|¥{$suegithen

©g�u~y{+wueµª¬d!egi #th0xYr¡si��I�Ri�{+r�|¥³yi�%�©g�*~�{+w*e$�ui�h2~�|¥{+r�zg{+wue�~�{g�Yiv��²°�eg|¥¤¥i�s�egi #¡¤§iPr�r�i��I�Ri�{+r�|¥³yi�©g�*~�{+w*e(%�i�{+�+r�zg�$s*~�­q|¥{g�¦suegi�ru~�h0i�s�|¥h0i�~Yr®suegi7h0xQrtsi��I�Riv{+r¡|¥³yi7xy{giYªd!egiasu�u~y{+r¡¢ xY��h2~�s�|¥xy{ |¥r�~y�g�g¤¥|§iP� |¥{+�Iz+w�s�|¥³yi�¤¥p¼s�xEs�egi�rts*~<s�ivh0i�{Ys�x�¢7iv~yw*eº©g�*~�{+w*eº�g�u|§xY�0suxEs�eg|}riP¾Yz�~�¤¥| �P~<s�|¥xy{µª� i {gx´°»�g��x<³yi�s�e+~�sv²Y¢£xY�%~y{Qp¦�g�uxy�y�*~�h

S²Ys�e+i?©Rivr¡sv²q~y{+�2°%xy�*rts%wv~yr�i�i��Iivw�zgs�|¥xy{0su|§h0iPr¬x�¢ F [S]~y��iG|}�Iiv{Ys�|}w�~y¤ !

& � Ê � Æ �´� Ç � ∀S,jkglade

(F [S]) =fhglade

(F [S]) =adc3ade

(F [S])�

)7:�: �(' d!e+i��g�uxqxy¢�|¥r�©Qp�|¥{+�Iz+w�s�|¥xy{�xy{�s�e+i�rtsu��z+w�s�zg�ui�x�¢�suegi7�g�uxy�y�*~�hSª

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��µi�s

S = if B then S1 else S2ª�d!egi�|§{��Iz+w�su|§xY{¿eQpq�Rx�suegivr�|}r8|}r�sue+~<s<jlglade

(F [S1]) =fhglade(F [S1]) =

adc3ade(F [S1]) = δ1

~�{+� adc3ade(F [S2]) =

jkglade(F [S2]) =

fhglade(F [S2]) =adc ade

(F [S2]) = δ2ª��!wvw�xY�u�I|¥{g�as�x���zg¤¥i���~y{+�Ed:~�©g¤¥iZ�Y²:°®i2sueQz+r�e+~´³yi<fhglade

(F [S]) =1 + max

(

δ1 + max(0, δ2 − δ1), δ2 + max(0, δ1 − δ2)) ª

� |§s�egxYzIs¬¤¥xYrurUx�¢��Yi�{giv�u~y¤§|§stpy²y~Yr�r�zgh0i!s�e+~�sδ1 ≥ δ2

· s�e+i?r¡pqh0h0i�su��|}w�~y¤+w�~yr�i!pq|¥i�¤}�grMr�|¥h0|§¤}~��w�xyh0�gzIs*~<s�|¥xy{�ru¸�ªd!egiv{

δ1 + max(0, δ2 − δ1) = δ1 + 0 = δ1²�~y{+�

δ2 + max(0, δ1 − δ2) =δ2 + δ1 − δ2 = δ1

ªR\?i�{+w�i fhglade(F [S]) = 1 + max(δ1, δ1) = 1 + δ1

ªj%xy{q³yi��*r¡iv¤§pY²P°%i®~y¤¥r�x?e+~´³yijkglade

(F [S]) = 1+min(

δ1+max(0, δ2−δ1), δ2+max(0, δ1−δ2)

) ª:d!egi�{µ²yrtsu|§¤¥¤q©Qp8~yrur¡zgh0|¥{g�?s�e+~�sδ1 ≥ δ2

²<°®i®~y¤¥r�xÃ�{+� jlglade(F [S]) = 1+min(δ1, δ1) =

1 + δ1ª

�¯{7w�xy{+w�¤§z+r�|¥xy{ ²"jkgAaie(F [S]) =

fhglade(F [S])

~�{+� suegi��ui�¢£xY��iU|¨sµ|}r ~y¤¥r�x!iv¾Qz+~y¤<s�x adc ade(F [S])

ª

��µi�s

S = for i = n1 to n2 do S1ª�d!egi2|¥{+�Iz+w�s�|¥xy{�eqpQ�Rxys�egiPr¡|}r�|}r�sue+~<s jkglade

(F [S1]) =fhglade(F [S1]) =

adc3ade(F [S1]) = δ1

ª���w�w�xy�*�I|§{+��suxa��zg¤¥i��$~y{+�9d:~�©+¤§i��Y²µ°%i0s�eqz+r7e+~´³yiadc ade

(F [S]) = (n2 −n1 +1)× (3+ δ1)ªMoq|¥{+w�i

n1~y{+�

n2~��ui?w�xY{+rts*~�{Qs%~y{+�2©Qp0|¥{+�Iz+w�s�|¥xy{

eqpQ�Rxys�egiPr¡|}rv²qs�eg|}r�|}r�~�¤}r�x¦iv¾Qz+~y¤�s�x<jkglade(F [S])

~�{+� fhglade(F [S])

ª

��µi�s

S = S1;S2ª d!egi�|§{+�gz+w�su|§xY{¼eqpQ�Rxys�egiPr¡|}r¦|}r�sue+~<s�jkglade

(F [S1]) =fhglade

(F [S1]) =adc ade(F [S1]) = δ1

~y{+� jlglade(F [S2]) =

fhglade(F [S2]) =

adc3aie(F [S2]) = δ2

ª?��w�w�xy�*�I|§{+�sux2��z+¤§i �2~�{+��d�~y©g¤¥i0�y²g°%i�s�eqz+r�e�~P³yi aic3ade

(F [S]) = δ1 + δ2ª ?%p�|¥{+�Iz+w�s�|¥xy{�eqpq�Rx�s�e+ivr¡|}rv²

sueg|¥r�|}r~�¤}r¡x¦iP¾Yz�~�¤�sux<jkgAade(F [S])

~y{+� fhglade(F [S])

ªd!eqz+r�²g°%i�w�xY{+w�¤¥z+�IiGsue+~<s�¢£xy�~y{Qp

S²Ajkglade

(F [S]) =fhglade

(F [S]) =aic3ade

(F [S])ª

¹gzg��s�egiv��h0xY��iY²M°%ia~�¤}r¡x9�g��x<³yi�sue+~<s0suegi$su�u~y{+rt¢£xY��h2~�s�|¥xy{ F �IxIivr¦{gxys2w*e+~�{+�yi�s�egi fhglade xy¢�g�uxy�Y�u~�h2r�!

& � Ê � Æ �´� Ç � ∀S,fhglade

(S) =fhglade

(F [S])�

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��µi�s

S = if B then S1 else S2ª d!egi�|¥{+�Iz+w�s�|¥xy{`eQpq�Rxys�egiPr¡|}r�|}r�s�e+~�s fhglade

(S1) =fhglade(F [S1]) = δ1

~y{+�)fhglade(S2) =

fhglade(F [S2]) = δ2

ª½��w�w�xY�u�g|§{g��s�x9�zg¤§i9�$~y{+�d:~�©g¤¥i0�y²g°%iGsueQz+re+~P³yi !

fhgAaie(F [S]) = 1 + max

(

δ1 + max(0, δ2 − δ1), δ2 + max(0, δ1 − δ2))

= 1 + max(

max(δ1, δ1 + δ2 − δ1),max(δ2, δ2 + δ1 − δ2))

= 1 + max(

max(δ1, δ2),max(δ2, δ1))

= 1 + max(

δ1, δ2

)

laxY��ivx´³yiv�v²v~yw�w�xy�*�I|¥{g�%suxd�~y©g¤¥i��y²v°®iM~�¤}r�x!e+~P³Yi]fhglade(S) = 1+max(

fhglade(S1),

fhglade(S2))

ª?®p�|§{+�Iz�w�s�|¥xy{�eqpQ�Rxys�egiPr¡|}rv²qs�eg|}r�|}r!iv¾Qz+~y¤�s�x

1 + max(δ1, δ2)²Isue+~<s|}r fhglade

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S = for i = n1 to n2 do S1ª»d!egi�|¥{+�Iz�w�s�|¥xy{ºeQpq�Rx�suegivr�|¥r0|}r¦s�e+~�ssfhglade

(S1) =fhglade(F [S1])

ª��!wvw�xY�u�I|¥{g��sux$�zg¤§i«��~�{+��d:~�©g¤¥i$�y²�°%i�sueQz+r�e+~´³yi fhglade(F [S]) = (n2 −

n1 +1)×(3+f glade

(F [S1]))ªUlaxy�ui�x<³yi��v²�~Yw�w�xy�*�I|§{+�?sux�d�~y©g¤§i7�y²y°%i~y¤¥r�xGe+~´³yi fhglade

(S) =(n2−n1 +1)× (3+

fhglade(S1))

ª;?®p�|§{��Iz+w�su|§xY{2eqpQ�Rxys�egiPr¡|}rv²�sueg|¥r¬|}rMiv¾Qz+~y¤gs�x fhglade(F [S])

ª

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S = S1;S2ª d!e+ia|¥{+�Iz+w�s�|¥xy{»eQpq�Rx�suegivr�|}r2|¥r2s�e+~�s�fhgAaie

(S1) =fhglade

(F [S1])~�{��

fhglade(S2) =

fhgAaie(F [S2])

ª:��w�w�xY�u�I|¥{g�®sux��zg¤¥i ��~�{+��d:~�©+¤§i��Y²v°%iUs�eQz+rµe+~P³yi fhglade(F [S]) =fhglade

(F [S1])+f glade

(F [S2]).laxy�ui�x<³yi��v²´~ywvw�xy�*�I|¥{g��s�xGd�~y©g¤¥i?�y²<°%i%~y¤}r¡x?e+~P³yi fhglade

(S) =fhglade(S1) +

fhgAaie(S2)

ª ?%p�|¥{+�Iz+w�su|¥xy{�eQpq�Rx�suegivr�|¥rv²qsueg|¥r�|}r!iP¾Qz+~�¤�suxhfhglade(F [S])

ªd!eqz+r�²g°%i�w�xY{+w�¤¥z+�IiGsue+~<s�¢£xy�~y{Qp

S²ifhglade

(S) =fhglade

(F [S])ª

d!egiGsu�u~y{+r¡¢ xY��h2~�s�|¥xy{ F ~��g�+¤§|¥iv�$xy{$xyzg��i��I~yh0�g¤§iFac

�g�uxI�Iz+w�ivr®s�egi7{giv° �g�uxy�Y�u~�hFac1

!

Fac1 = F [Fac] =read(i);if i > 10 then i := 10; o := 1; else o := 1; skip3;for l = 1 to 10 do

if l <= i then o := o ∗ l; else skip3;write(o);

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# �<� � �%$ Ê �*�½�g��� Ê � �* B Æ � �

1. ITc (S, t) = S

/ � adc3ade(S) < t

2. ITc (S, t) = c;IT

c (S, T − adc ade(c) + t)

/ �t ≤ 0

3. ITc (a, t) = a;c

/ �0 < t ≤ adc3ade

(a)';1 �a/ � '*+.:�� / 6

4. ITc (S1;S2, t) = IT

c (S1, t);ITc (S2, t1)� /2+<$

t1 = t − adc3ade(S1)

/ � adc3ade(S1) < t� /2+<$

t1 = T − adc3ade(c) − r

/ � adc3aie(S1) = t + q(T − adc3ade

(c)) + r� /2+2$q ≥ 0 0 ≤ r < T − adc3aie

(c))

5. ITc (if b then S1 else S2, t) = if b then IT

c (S1, t − 1) else ITc (S2, t − 1)

6. ITc (for l = n1 to n2 do S, t) = Fold(IT

c (Unfold(for l = n1 to n2 do S), t))

��zg¤¥i��8r¡su~�s�ivrs�e+~�sv²R°�e+i�{as�e+i�rts*~<sui�h0i�{QsSÃ+{+|¥r�egivr?©Ri�¢£xy�ui7s�egi�{gi��qs |¥{+r�i���s�|¥xy{�s�|¥h0i

t·�-/.F0�.&2

adc ade(S) < t

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S|¥r¬�Y��iP~<s�iv�Us�e+~y{

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��zg¤¥i��¿~��g�+¤§|¥ivr�°�egiv{ s�e+i�su|§h0i�w�xYzg{Ysui��t|¥r�{+i��Y~�s�|¥³yiyª d!eg|}r$wv~yr�i�h2~Pp»~y��|}r�iZ°�e+i�{ suegi

|}�Iiv~y¤M�RxY|§{Qs�¢£xy�8|§{�r¡i���su|§{g��suegi�w�xyh0h2~�{+�c|}r #t|¥{9suegi«h0|}�g�I¤¥i�%«x�¢®suegi«©RxIxy¤¥iv~�{�i��I�g�uivr�r�|¥xy{9x�¢~

w�xy{+�I|§s�|¥xy{�~�¤�r¡su~�s�i�h0iv{Ysifª � egi�{

t|}r?{giv�Y~<su|§³yiY²+s�egi¦w�xyh0h2~�{���h8z+r¡s ©Ri8|¥{+r�i���s�iv�a��|¥�yeQs ~´°®~´pyª

d!egi¦su�u~y{+rt¢£xY��h2~�s�|¥xy{Z�+��xIw�i�iP�gr °�|§s�e9s�egi0�uivr�zg¤§s�|¥{g�$�g�uxy�Y�u~�h ~y{+�Zsuegi¦su|§h0i¦s*~��u�yi�sG¢£xy��s�egi2{gi��qs|¥{+r�i���s�|¥xy{�|}r"�uivr�i�s"s�x

T − aic3ade(c)+t

²�s�e+~�sU|}rv²�|§sM|¥rMw�xyh0�gzgs�iv� � . ��.�� .µs�egi�|}�Iiv~y¤I�g�ui�³q|§xYz+r:|¥{+r�i��¡su|¥xy{�RxY|§{Qs!sux2~´³yxy|}��~�{qp�w�¤¥xIw*­��I��|§¢ svª��zg¤¥i��«r¡su~�s�iPr�sue+~<sP²R°�egi�{asuegi��g��xY�y�*~�h |}r ~�{�~<s�xYh0|¥w�w�xyh0h2~�{+�

a·c°�egxYr�i adc ade9|}r?�y�uiv~<sui��

sue+~�{ZxY�?iv¾Qz+~y¤ suxt¸�²Rs�egi¦w�xyh0h2~�{��

c|}r?|§{�r¡iv�¡suiv���u|¥�yeYsG~�¢}sui��

a²�s�e�~<s |}r

(adc3ade

(a) − t)zg{g|§sur x�¢

su|§h0i7¤}~<s�iv�!s�e+~y{�suegi7|¥�IiP~�¤ �Rxy|¥{Ysvª��zg¤¥i�8rts*~<suivrUs�e+~�sMsuegi|¥{+r�i���s�|¥xy{0|¥{«~�r¡iP¾Qzgi�{+w�i

S1;S2|}rUÃ��ur¡s%�Ixy{gi|¥{

S1ª"d!egi?��iPr¡|}�Iz+~y¤Is�|¥h0i

t1z+r¡iP�Z¢£xy�Gs�egi0|¥{+r¡iv�¡su|§xY{�|¥{

S2|¥rGiv|¨suegi��

(t − adc3aie(S1))

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|¥{+r�|¥�IiS1xy�

(T − adc3aie(c) − r)

|¨¢r|¥r�s�egi7s�|¥h0i���iPr¡|}�Iz+~y¤��ui�h2~�|¥{g|¥{g�2~<¢ sui���suegi

q + 1|¥{+r�i���s�|¥xy{+r

|¥{+r�|¥�IiS1·�-/.F0�.&2+|¨¢Radc3aie

(S1) = t + q(T − adc3aie(c)) + r

¸�ª��zg¤¥i¬�!r¡su~�s�iPr�s�e+~�sv²v¢£xY�µw�xY{+�I|§s�|¥xy{+~y¤�r¡s*~<s�ivh0i�{Ys*r�²vsuegiM|§{�r¡iv�¡su|§xY{G|}r��Riv�¡¢£xY��h0iv�G|§{7©Rx�sue�©g�*~�{+w*egiPr�ª

d!egi su|§h0iGxy¢ s�egiGs�iPrts�~y{+�«©g�*~�{+w*eg|¥{g��|¥r%su~y­yi�{�|¥{Ysux0~yw�w�xyzg{Qs®©Qp«�givw��ui�h0i�{Qs�|¥{g�8suegi su|§h0iG�uivr�|¥�gz+~�¤·t − 1

¸�ª��zg¤¥i��0~y�g�g¤¥|§iPr®sux2¤§xIxy��rts*~<sui�h0i�{Qsurvª �[szg{+��xY¤§¤}r®suegi7¤¥xqxy��w�xyh0�g¤¥i�sui�¤¥p�· s�e�~�{g­Ir!sux¦suegi

UnfoldxY�Ri��u~�s�xY�*¸�²%�Ri���¢£xy�uh2r¦s�egiZ|§{+r�i���s�|¥xy{ |¥{ suegiazg{+��xY¤§¤¥iv�¼�uivr�zg¤§s�|¥{g���g�uxy�Y�u~�h ·£egiv{+w�ias�egi 〈S′|t′〉 =IT

c (S′′, t)¸�²Y~y{+��suegi�{�¢c~Yw�suxy�u| �vivr"w�xI�Ii!©Qp7¢£xY¤¥�g|§{g�7w�xg�Ii®|¥{

for¤¥xqxy�+r"~Yr:h8z+w*e0~yr:�RxQr�r�|¥©g¤§i�·£s�e�~�{g­Ir

sux�suegiFold

xY�Ri��*~<s�xY�*¸�ª7d!egiUnfold

~y{+�Fold

xY�Ri��*~<s�xY�ur ~��ui¦�Ii�Ã+{+iv��©qp�suegi¦¢£xy¤¥¤§x<°�|§{+��s��*~�{+r¡�¢£xY��h2~<su|§xY{��uzg¤§iPr�!

# �<� � �%$ Ê �*�½�g��� Ê � �* B Æ � m

1. Unfold(for l = n1 to n2 do S) = l := n1; S; l := n1 + 1; . . . l := n2; S

2. Fold((for l = n1 to n2 do S);l := n2 + 1;S;) = for l = n1 to n2 + 1 do S

�!w�s�z+~y¤§¤¥py²"|§s�°®xyzg¤}��©Ri��RxYrur¡|¥©g¤¥i2s�x�i��I�g�uivrur�suegi«s��*~�{�rt¢£xY��h2~<su|§xY{ I r�z+w*e�sue+~<s�|§s�h0|¥{g|§h2~y¤§¤¥pzg{g�uxy¤¥¤}r!¤¥xqxy��r�~�{+���IxqiPr!{+x�s{giviv��¢£xY¤¥�I|¥{g��ª \?x´°®i�³yi��P²qsuegi7s��*~�{+r¡¢£xy�uh2~<s�|¥xy{��uzg¤¥ivr�°%xyz+¤¥�$©Ri7h8z+w*eh0xy�ui�w�xyh0�g¤¥i���²g~y{+��°%i�wuegxQr¡i�|§{�rtsuiv~y�$~0r�|¥h¦�+¤§iv���g�uivr�i�{Ys*~<su|§xY{�|¥{Q³yxy¤¥³q|§{+��s�e+i

Foldxy�Ri��*~<suxy�Pª

dµ�*~�{+r¡¢£xy�uh2~<s�|¥xy{a�uzg¤¥ivr��«~Yr�r�zgh0i�sue+~<sGs�egi¦�Ri��u|§xg�T|¥r �Y��iP~<sui��sue+~�{Zsuegi¦i��Iivw�zIs�|¥xy{Zs�|¥h0i�xy¢

suegi�w�xYh0h0~y{+�c²�-/.F0�.&2

T >adc3ade

(c)ª®k?suegi��u°�|¥r�iy²qsuegi7|¥{+r¡iv�¡su|¥xy{$h0~´p�¤§xIxy�$©Qp�|¥{+r�i���s�|¥{g�

c°�|¨sueg|§{

c~y{+�$r�x¦xY{ ª

& � Ê � Æ �´� Ç�m ��1 '�+0)7'*1�����:*)��@& � 9�)7:73")7'�� ITc (F(S), T )

� +<$#&='"6�+ ��';>�+ / � & /21 +%&�) �*'*>∆-(&�+�� &(&�1�+<$#&

-(&03"/21�1 /2143 : � + � :�����6�6(&�� �,/"� &@6�:���� '*1����c/ � ����6�$ +<$#';+ '

T − ε ≤ ∆ < T + ε

� /2+<$ε-�&�/2143 +2$�& adc ade ::� +<$#&�� :��,+ &:_,9�&�1��,/ �*&@';+%:1� / 6 /21��,+0)���6�+0/ :*1 � '�� �,/ 3;1 �@&�1�+ :;)�+.&��,+��@/21 +2$�&

9 )!:�3;)7'1� � >�&�'���&'*>3��: 1�:*+.&@+2$�'*+ ��:;) +<$#&�� ) �(+c/21���&�),+%& � /21�+<$#&59 )7:73;)7'1� �

∆/ � �"&�� 1 & � '������(�(+

+<$#& -�&%3;/21 1�/2143+0/ � & : �c�

� i�¢£xy�uh0~y¤§| ��i7~�{+���+��x<³yiG�g�uxy�Riv�¡stp �¦|¥{$s�egi�~y�g�Ri�{+�I|���ªj%egivw*­q��xY|§{Qs�|¥{g�8~�{+�0egiP~���s�©Riv~�s�|¥{g��~��ui��Ri���¢£xy�uh¦iP�¦z+r¡|¥{g�7s�e+i�su�u~y{+rt¢£xY��h2~�s�|¥xy{ I ª¬j%egivw*­q�Rxy|¥{Ys*r~y��i8|¥{+r¡iv�¡suiv�aÃ+�*rts�~�{��aegiP~���s�©RiP~<sur?¤}~yr¡svª7d!egi��Ri��u|¥xI�a©Ri�st°%i�iv{Zst°%x$wue+ivw*­Q�RxY|§{Qsur h8z+r¡s s*~�­yi�|§{Qs�x

~Yw�w�xYzg{Ys?s�egi�x<³yi��uegiv~Y��s�e+~�s °�|¥¤¥¤�©Ri�~Y�g�IiP��©qp$e+iv~���s�©RiP~<s*r?~<¢ s�iv��°!~��*��ªd!egi�x´³yiv��e+iv~y��~Y�g�IiP��©qpegiP~���s�©Riv~�s�|¥{g�a�Izg�u|§{g�

Xzg{g|§sur�x�¢®s�|¥h0i2|}r Xh

THB−h

°�|§s�eh =

fhglade(hbeat)

ª$oqx+²�|§¢TCP

|¥r�suegi�IiPr¡|¥�uiv���Riv��|¥xI��x�¢"w*egivw*­q�Rxy|¥{Ysurv²I°%i7h8z+r¡sz+r¡i�s�egi7�Riv��|¥xI�

T ′CP

�gi�Ã+{giP��©qp«s�egi7iP¾Yz�~<s�|¥xY{ !

T ′CP = TCP − T ′

CP h

THB − h⇐⇒ T ′

CP

(

1 +h

THB − h

)

= TCP ⇐⇒ T ′CP =

TCP

1 + hTHB−h

·^�y¸

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�v� � �"'���� � )7'��"&�+�� /2)7'���> +

� |§s�e�s�egiPr¡i2{gxysu~�s�|¥xy{+rv² s�egi«|§{+r�i���s�|¥xy{9xy¢!w*egivw*­q�Rxy|¥{Ys*r7~y{+��egiP~���s�©Riv~�sur�|}r��Iivruw��u|§©RiP��©Qp�s�egi0¢£xY¤¨�¤¥x´°�|¥{g�2l ��w�xI�Ii !

¤¥i�s(S′,−) = IT ′

CP

checkpt(S, T ′CP )

|¥{¤¥i�s

((S′′;hbeat),−) = ITHB

hbeat(hbeat; S′, THB)|¥{

S′′;hbeat(k)

d!egiGÃ+�*rts�egiv~y�¡su©Riv~<s!|}r!~Y�g�Iiv���u|¥�yeYs~<s®suegi�©Ri��Y|§{+{g|§{g�0x�¢S′²qs�e+i�x�s�e+i��*r®~y��iG|§{+r�i���s�iP��©Qp I ²qs�egiv{suegi2¤}~yr¡s7egiv~y�¡su©Riv~<s7|}r7��iv�g¤¥~Yw�iP��©Qp

hbeat(k)ª � i«wv~�{�~y¤§°®~´pIrGiv{+r¡zg�ui¦s�e�~<s

S′Ã+{g|}r¡e+ivr7°�|¨sueE~

egiP~���s�©Riv~�sµ©qpG�+~Y�g�I|¥{g� �Izgh0h8p7w�xI�Ii¬~�sµs�e+i¬iv{+��ª:d!egi%w�xyh0h2~�{+�hbeat(k)

|}r�~?r��Rivw�|}~�¤YegiP~���s�©Riv~�ssue+~<sr�i�sur®s�egi�³�~��u|¥~y©g¤§i?sux

k|¥{+rtsuiv~Y��x�¢

1²9-/. 01.32 �����

[i] := kªM¹+xy¤¥¤§x<°�|§{+�8s�e+|¥r!¤}~yr¡s!egiv~y�¡su©Riv~<sP²Qs�e+i

h0xy{+|¨suxy�M°�|§¤¥¤gs�e+i���i�¢£xy�ui��IiPw��uiv~Yr¡i®suegi?r¡e+~y��iP��³<~y��|}~�©+¤§i�~�{+�0°�|§¤¥¤+��iPr¡zgh0i�iv���uxy�M�Ii�suivw�su|§xY{0°�egiv{¦suegi³<~y��|}~y©g¤§i0©Rivw�xYh¦iPr

0~y�Y~�|¥{ ª�d!eg|¥r�h¦iPwue�~�{g|}r¡h ~ywvw�xyzg{Qsur�¢£xy�7s�egi«|}�I¤¥i2|§{Qs�iv��³�~�¤Uxy¢®s�|¥h0i2©Ri�s¯°%ivi�{

suegi�s�i��uh0|¥{+~<su|§xY{�xy¢S′′

~�{+�$suegi8©Ri��Y|§{+{g|§{+�«x�¢"suegi8{gi��qs �Ri��u|§xg��ª?\iv{+w�iY²ke�~yr�s�x�©Ri�w�xyh0�gzIsuiv�

~Yr�!

k =

T − fhglade(S′′;hbeat)

THB

⌉ · �Y¸

�¢ s�i�� suegiM|§{Qs��uxI�Iz+w�s�|¥xy{�x�¢qe+iv~���s�©RiP~<s*r�²�s�egiM�Riv��|¥xI� ©Ri�s¯°%ivi�{�wue+ivwu­q�RxY|§{Qsur�°�|¥¤¥¤�©RiT ′

CP

(

1 + hTHB−h

) ²-/. 01.32

TCPªMlaxy�uiG�+��iPw�|}r¡iv¤§pY²q|§s�¢£xy¤¥¤¥x´°r®¢£�uxyh �¬��xY�Ri���s¯p ��sue+~<s6!

& � Ê � Æ �´� Ç�� $�&�'"6�+ ��';> +0/ � & /21�+.&�) �*'*>F�∆CP

'*1��∆HB

-(&�+ � &�&�1 +�� :@����6�6(&�� �,/"�*&6($#&(6(8�9#:;/21 +8�';1 � $#&�';),+%-(&�';+8� '*)7& ����6($ +2$�'*+ '

TCP − ε ≤ ∆CP < TCP + ε + h'*1��

THB − ε ≤ ∆HB < THB + ε

)7:�: �('?d!e+i �g�uxqx�¢ |}r%©+~yr�iv�«xy{��M�uxy�Ri���stp��+ª"�¢ s�i��®su�u~y{+r¡¢£xy�uh0~�s�|¥xy{ IT ′

CP

checkpt(S, T ′CP )

²q�¬��xY��iv�¡stp ��Y|§³yiPr�!

T′

CP − ε ≤ ∆′

CP < T′

CP + ε·c�Q¸

� r�r�zgh0|§{+�9suegi f glade x�¢Gs�e+iah0xYr¡s«i��I�Ri�{+r�|¥³yia~�s�xyh0|}w�w�xyh0h2~�{+� x�¢checkpt

|}r2¤§iPr�r0sue+~�{ºxY�iP¾Yz�~�¤Rsux

ε²g~<¢ sui��!s�egi�r¡iPw�xy{+�«su�u~y{+r¡¢£xy�uh0~�s�|¥xy{ ² ITHB

hbeat(hbeat; S′, THB)²g�M�uxy�Ri���stp �¦ru~<su|¥r¡Ã+ivr%suegi

w�xy{+�I|§s�|¥xy{THB − ε ≤ ∆HB < THB + ε

ª®d!egi�r�ivw�xy{+��s��*~�{+r¡¢£xy�uh2~<su|§xY{ ²Iegx´°®i�³yi��P²+wue+~y{g�yiPr∆

CP�Y|§³yiv{2|§{ E¬¾Qz+~�s�|¥xy{�·£�Q¸"sux∆CP

r¡z+w*e2s�e+~�s%iv~yw*e2�RxY�¡su|§xY{0°�|§s�e2suegisu|§h0i?|¥{Qs�i��u³<~y¤T

CP

|¥{2s�egi?Ã+{+~�¤�g�uxy�Y�u~�hn°�|§¤¥¤M©Ri«|§{�r¡iv�¡suiv� T

CP

THB−hhbeat

w�xYh0h0~y{+�grvª�d!egi��ui�¢£xY��iY² ©qpZ¢£xY¤§¤¥x´°�|¥{g�+E%¾Yz�~<s�|¥xY{º·^�y¸�²

T′

CP +T

CP

THB−h.h¤§iP~y�gr%s�x

TCP²9-/. 01.32Is�e+i��Iivr�|¥��iP�«³<~�¤¥zgi xy¢�w*egivw*­q�Rxy|¥{Ysu|§{g�¦|§{Qs�iv��³�~�¤^ª"�¤§s�e+xyzg�Ye�°%i

s*~�­yi®|¥{Ysux7~yw�w�xyzg{Qs"egiv~y�¡su©Riv~�s�|¥{g� |¥{�s�egi!Ã+�ur¡s"s��*~�{+r¡¢£xy�uh2~<s�|¥xy{ ²´s�egi�egiv~y�¡su©Riv~<su|§{+�Gw�xYh0h0~y{+�hbeat|}r�|¥{Q³q|}r¡|¥©g¤¥i�sux0suegi7Ã+�*rts�su�*~�{+r¡¢£xy�uh0~�s�|¥xy{ ª¬d!e+i7°®xy�*rts?w�~yr�i7xIw�w�zg�ur�|§{�s�e+i�©Rxyzg{+�g~y��p�w�xY{+�I|§s�|¥xy{�xy¢

E¬¾Qz+~�s�|¥xy{E·^�y¸!°�egi�{Z~0egiv~y�¡su©Riv~<s|}r�|¥{+r¡iv�¡suiv���tz+r¡s?©Ri�¢£xY��i�~2w*egivw*­q�Rxy|¥{Ys?w�xyh0h2~�{+��ª �¯{$sueg|¥r?wv~yr�iy²TCP

|}rUr¡eg|§¢ s�iP��zg�Q°!~��*�gr:©QphªB�¯{�suegi!©Rivr¡sMw�~yr�iy²<s�eg|}rUr�eg|§¢}sM|}r �vi��ux+ª�d!egiv��i�¢£xy�uiy²<©Qp�r¡eg|§¢ s�|¥{g��zg��suegi

¤¥x´°®i���~�{+��z+�g�Ri���©Rxyzg{��gr xy¢∆CP

°�|§s�e[0, h]

²´°%iMÃ+{+~�¤¥¤¥pG�Iiv��|¥³yiTCP −ε ≤ ∆CP < TCP +ε+h

ª�

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�?r��Rxy|¥{Ys�iP�9xyzgs8~�©Rx<³yiy² suegi2s��*~�{+r¡¢£xy�uh0~�s�|¥xy{ I ��iP¾Yz+|§�uivrGsuegi2�Riv��|¥xI��suxa©Ri2©g|¥�y�Yi���sue+~�{9suegiw�xYr¡s!xy¢�s�egi�w�xyh0h2~�{+��ªM¹gxY�wuegiPwu­q�Rxy|¥{Qs�|¥{g�2~�{+��egiP~���s�©RiP~<s�|¥{g�¦°®i�h8z+r¡si�{+r�zg�ui sue+~<s6!

T ′CP >

fhglade(hbeat)

~�{+�THB >

fhglade(checkpt)

dµx0|¥¤§¤¥z+r¡s��*~<suiGs�egiPr¡i�s��*~�{+r¡¢£xy�uh2~<s�|¥xy{�r%xy{$xYzg���g��iv³q|§xYz+r®i��g~yh¦�+¤§iY²I°%iGs*~�­yi !aic3ade

(hbeat) = 3adc ade

(checkpt) = 10 TCP = 80 THB = 10

oqx+²g°%i��Yi�sT ′

CP = 80 − 3∗T ′

CP

10−3

-/. 01.32T ′

CP = 56~y{+� I56

checkpt(Fac1, 56)�g��xg�Iz+w�iPr�!

Fac2 = read(i);if i > 10 then i := 10; o := 1; else o := 1; skip3;for l = 1 to 6 do

if l <= i then o := o ∗ l; else skip3;l := 7; if l <= i then checkpt; o := o ∗ l; else checkpt; skip3;for l = 8 to 10 do

if l <= i then o := o ∗ l; else skip3;write(o);

�±r�|§{g�Y¤§icheckpt

|}r |§{+r�i���s�iP�Z~�¢ s�i��56su|§h0i¦zg{g|§surv²�°�e+|¥w*e�e+~��+�Ri�{+r °�|§s�eg|¥{Zsuegi¦w�xy{+�g|¨su|§xY{+~�¤

xy¢µs�e+i�<Psue�|§s�iv�u~�s�|¥xy{�x�¢�s�e+ifor

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checkpt~�s8s�e+i���|¥�yeQs��g¤¥~Yw�iPr�ª��"xy��s�|¥xy{+r8xy¢�s�egi$w�xI�Ii�~��ui2suegi�{½¢£xY¤¥�IiP��sux�h2~�­yi«st°%x

for¤¥xqxy�+rvª¹gxy�Ms�egi?{gi��qs%r¡s�iv� ²Y°®i?r¡zg�g�RxQr¡iY²�¢£xY�Usuegi?ru~�­yi�x�¢�s�e+ii��g~yh¦�+¤§iY²�sue+~<s

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zg{g|§s*r®x�¢ su|§h0iy²gwv~�{�©Ri�r¡�g¤¥|§s®|¥{�s¯°®x��+~y�¡s*rcheckpt = checkpt1;checkpt2

°�egi��uicheckpt1

~y{+�checkpt2

su~y­yi���iPr¡�Rivw�s�|¥³yi�¤¥p <�~y{+�@��s�|¥h¦i7zg{+|¨s*r�i��g~yw�s�¤¥pyª �¯{$xys�egiv��°%xy�*�grv²Qs�egi7¤}~��u�yiPrts fhgAaieZxy¢~y{¦~<s�xYh0|¥w%|¥{+r¡s��uz+w�su|§xY{���ivh2~�|¥{+r �¦·£|§sU°%xYzg¤}�8©Ri��v� |¨¢

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~�s�s�e+iiv{+��ªUd!egisu�u~y{+r¡¢£xy�uh0~�s�|¥xy{ I10

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Fac3 = hbeat; read(i);if i > 10 then i := 10; hbeat; o := 1; else o := 1; hbeat; skip3;for l = 1 to 6 do

if l <= i then hbeat; o := o ∗ l; else hbeat; skip3;l := 7;if l <= i then hbeat; checkpt1; hbeat; checkpt2; o := o ∗ l;

else hbeat; checkpt1; hbeat; checkpt2; skip3;

for l = 8 to 10 do

hbeat; if l <= i then o := o ∗ l; else skip3;write(o); hbeat; skip5; hbeat;

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TCP

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200

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400

500

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f(TCP , THB)

¹:|§�Yzg��i�� !f(TCP , THB)

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T ?CP =

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T ?HB =

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T ?CP =

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T ?HB =

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# �<� � �%$ Ê �*�½�g��� Ê � �* B Æ � �

1. Traces(a) = {a} / �a/ � '*+.:�� / 6

2. Traces(S1;S2) = {T1;T2 | T1 ∈ Traces(S1), T2 ∈ Traces(S2)}3. Traces(if b then S1 else S2) = {skip;T | T ∈ Traces(S1) ∪ Traces(S2)}4. Traces(for l = n1 to n2 do S) = Traces(Unfold(for l = n1 to n2 do S))

d!egi®|§{+r¡s��uz+w�s�|¥xy{skip

|§{���zg¤¥i � ~�©Rx<³yi¬��iv�g��iPr¡iv{Ysurµs�egi%s�|¥h0i¬s*~�­yiv{�©qpGsuegi%s�ivr¡sP²�-/. 01.32yxy{gi¬s�|¥h0i®zg{g|¨sPª¹+xy�"~�{Qp7|§{g|§s�|}~�¤Ir¡su~<suiy²´suegi��ui%|¥r"~�¤¥°®~PpIr�~s��*~yw�i

τ|§{

Traces(S)�ui��g�uivr�i�{Qs�|¥{g�?i��g~yw�s�¤¥pGs�egi®i��Iivw�zIs�|¥xy{

xy¢Sª7d!egi�|¥h0�Rxy��su~y{YsG�Rxy|¥{YsG|}r?sue+~<s7r¡z+w*e�i��IiPw�zIsu|§xY{asu�u~Yw�ivr

τe+~´³yi�~ 6�:;1��,+.'*1�+?i��Iivw�zIs�|¥xy{Zs�|¥h0i

·8-�.F0�.&2ljkgAade(τ) =

fhglade(τ) =

adc3ade(τ)¸�²�~�{+��h0xy�ui�x<³yi��!°%i�e+~´³yiG¢£xy�~y{Qp

τ!

τ ∈ Traces(S) =⇒{ jkglade

(S) ≤ adc ade(τ) ≤ fhglade

(S)~y{+�

jkglade(S) =

fhglade(S) ⇒ adc3aie

(τ) =adc3aie

(S)

·B�y�Y¸

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� i!|¥{Ys��uxI�Iz+w�i%s�egi!iv¾Qzg|¥³<~y¤§iv{+w�i®��iv¤¥~�s�|¥xy{ .=suxG{gxY��h2~�¤¥| ��i®~�{+��w�xyh0�+~��ui%i��Iivw�zIs�|¥xy{8su�u~Yw�ivrvª�d!egi

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void²Rr�|¥h0|§¤}~��s�x

skipi��gw�i��Is?s�e+~�s adc3ade

(void) = 0ª

d!egi7�ui�¤}~<s�|¥xy{ .=|}r�r�z+wue$s�e+~�s!

(τ1;τ2);τ3.= τ1;(τ2;τ3) τ

.= (void;τ)

.= (τ ;void)

� i7�yiv{gi��*~�¤¥| ��iG�M�uxy�Riv�¡stp ��sux0su~�­yi7|§{Qs�x«~ywvw�xYzg{Ys~y{Qp�|¥{g|¨su|¥~y¤�s�|¥h0i7�uivr�|¥�gz+~�¤�©Ri�¢ xY��i7|§{+r�i���s�|¥{g�¦suegiÃ+�*r¡sw�xyh0h2~�{��

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& � Ê � Æ �´� Ç�� &�+S�c�t� '*1��

T-�& ����6�$ +2$#';+ '

(0)jkgAaie

(S) =fhglade

(S) (1)adc3aie

(c) + ε < T

(2) t ≤ adc3ade(S) (3) −ε < t ≤ T

$#&�1 ∀τ ∈ Traces(ITc (S, t))

�τ

.= S1;c;S2 . . . c;Sn (1 ≤ n)

';1 � � &�),/ � &�� '

t ≤ adc3ade(S1) < t + ε (Init)

T − ε <adc3ade

(c;Si) ≤ T + ε (1 < i < n) (Period)r − ε <

aic3ade(Sn) ≤ r

�8� adc3ade(S) = t + q(T − adc3ade

(c)) + r (End)� /2+<$0 ≤ q

'*1��0 ≤ r < T − adc3aie

(c)

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t·£h0xI�Izg¤¥x

ε¸�s�|¥h0i9zg{+|¨s*r$©Ri�¢£xy�ui�|¥{+r¡iv�¡su|§{+�¿s�egi9Ã+�*rtsaw�xYh0h0~y{+�

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r©Riv|§{+��s�e+i0��ivh0~y|§{+|§{g�$x�¢%s�egi2�I|¥³q|¥r�|¥xy{�x�¢

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0 < t ≤ T²g©qp�rtsu��z+w�s�zg�*~�¤�|¥{+�Iz+w�s�|¥xy{�xy{

� � ��a S = a! ?®p�eQpq�Rxys�egiPr¡|}r�²

0 < t ≤ aic3ade(a)²qr�x IT

c (a, t) = a;cªUd!egixY{g¤¥p�i��Iivw�zIs�|¥xy{0su�u~Yw�i

|}ra;c

.= a;c;void

²g°�eg|}w*e�ru~<su|¥r¡Ã+ivr®s�e+i7�g��xY�Ri���s¯pYª �¯{+�gi�iv�4!

�?®p��gi�Ã+{g|§s�|¥xy{�x�¢

ε² aic3ade

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0 < t~y{+�

t ≤ aic3ade(a)²Isuegi��ui�¢£xY��iD!

t ≤ adc3ade(a) < t + ε (Init)

�¹+��xYh adc3ade

(a) = t + r°�|§s�e

0 ≤ r < ε~�{+� aic3ade

(void) = 0²g|§s�¢£xy¤¥¤§x<°r®s�e+~�s!

r − ε <aic3ade

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�yª adc ade(S1) < t

!8d!egi��ui�¢£xY��i ITc (S1;S2, t) = S1;IT

c (S2, t −adc3ade

(S1))©�iPw�~yz+r¡i2x�¢!��zg¤¥ivr2�

~y{+���+ªj%xY{+�I|§s�|¥xy{Ì·B�Y¸�i�{I¢£xy�*w�iPr�s�e+~�s

t <adc3ade

(S1;S2) =adc3ade

(S1) +aic3ade

(S2)²�suegi��ui�¢£xY��i

t − adc3ade(S1) <

adc3ade(S2)

ª)j%xy{+�I|§su|§xY{ · �Y¸�i�{I¢£xY�uw�ivr«sue+~<st ≤ T

~�{+��² ©Qp»eQpq�Rx�suegi��r�|¥rv² adc3aie

(S1) < tsuegi��ui�¢£xY��i

0 < t − adc3aie(S1) ≤ T − adc3ade

(S1) ≤ Tª \?i�{+w�iy²

S2ru~<su|¥r¡Ã+ivr�s�e+i9|§{+�gz+w�su|§xY{`eqpq�Rx�s�e+ivr¡|}rv² ~�{+� ∀τ2 ∈ Traces(ITc (S2, t −

adc3aie(S1))), τ2

.=

S2,1;c;S2,2 . . . c;S2,n (1 ≤ n)~y{+��³yi��u|§Ã+ivr!

t − adc ade(S1) ≤

aic3ade(S2,1) < t − aic3ade

(S1) + ε (Init)T − ε <

aic3ade(c;S2,i) ≤ T + ε (1 < i < n) (Period)

r − ε <adc3aie

(S2,n) ≤ r�[¢Radc3ade

(S2) = t − adc3ade(S1) + q(T − adc ade

(c)) + r°�|§s�e0 ≤ q

~�{��0 ≤ r < T − adc ade

(c) (End)

�?{Qp�i��Iivw�zIs�|¥xy{Es��*~yw�iτxy¢ IT

c (S1;S2, t)|¥r8h2~y�gi�x�¢?~�{½i��Iivw�zgs�|¥xy{Es��*~yw�i

τ1x�¢ IT

c (S1, t)¢£xY¤§¤¥x´°®iv�`©Qp`~�{ i��IiPw�zIsu|§xY{ su�u~Yw�iτ2x�¢ IT

c (S2, t −adc3ade

(S1))ª �¯{ x�s�e+i���°%xy�*�grv²

τ.=

τ1;S2,1;c;S2,2 . . . c;S2,nª:d!e+i®�g�uxy�Riv�¡stp7|¥rUr�~�s�|}r¡Ã+iv�8|§¢

t ≤ adc3aie(τ1;S2,1) < t+ε

²�°�eg|}w*e8¢£xY¤¨�¤¥x´°r�¢£�uxyhKs�e+i®¢c~yw�s"sue+~<s"s�e+i �)7'"6(&��!¢£zg{+w�s�|¥xy{0ru~<su|¥r¡Ã+ivr�suegi!�M�uxy�Ri���stp«·B�y�Y¸�²1-/. 01.328adc3ade

(τ1) =adc ade(S1)

²+~y{+��s�e+i7eQpq�Rx�suegivr�|¥r aic3ade(S1) < t

ª�Iª

t ≤ aic3ade(S1)

! �¯{$s�eg|}rwv~yr�iy²qsuegi��ui7°�|§¤¥¤ ©Ri�~�s¤§iP~yr¡s�xy{+i�|¥{+r¡iv�¡su|§xY{$x�¢c|¥{

S1²+~�¢ s�iv�

ts�|¥h0i

zg{+|¨s*r�²+~y{+���RxYrur�|§©g¤¥p«x�suegi���|¥{+r�i���s�|¥xy{+r!i�³yiv��pTsu|§h0i7zg{g|§s*r�!

ITc (S1;S2, t) = IT

c (S1, t);ITc (S2, t1)°�|¨sue adc3aie

(S1) = t + q(T − adc ade(c)) + r, 0 ≤ q, 0 ≤ r < T − adc ade

(c))t1 = T − adc ade

(c) − r

oq|¥{+w�it ≤ adc3ade

(S1)²S1r�~�s�|}r¡Ã+ivr"suegi|¥{+�Iz�w�s�|¥xy{«eQpq�Rx�suegivr�|¥r¬~�{�� ∀τ1 ∈ Traces(IT

c (S1, t))²

τ1.= S1,1;c;S1,2 . . . c;S1,m (1 ≤ m)

~�{+��³yiv��|§Ã+ivr!

t ≤ adc3ade(S1,1) < t + ε (Init1)

T − ε <adc3ade

(c;S1,i) ≤ T + ε (1 < i < m) (Period1)r − ε <

adc3ade(S1,m) ≤ r (End1)

d!egi¬su�u~y{+r¡¢£xy�uh0~�s�|¥xy{7|¥rµsuegi�{¦~��g�g¤¥|¥iv��xy{S2°�|§s�e�suegi¬su|§h0i%�uivr�|}�Iz+~�¤

t1 = T− adc ade (c;S1,m)ª

d!egiv��i7~y��iGs¯°%x«r¡zg©g�[w�~Yr¡iPr!�Ii��Riv{+�I|¥{g�2xy{�s�e+i�i��Iivw�zgs�|¥xy{�s�|¥h0i7x�¢S2ª

·B~Y¸T − adc3aie

(c) − r ≤ adc3ade(S2)

!d!e+|¥rG|}r�w�xY{+�I|¨su|¥xy{¼·^�y¸sux$~y�g�g¤¥p�s�egi0|§{+�gz+w�su|§xY{�eQpq�Rx�suegivr�|¥rGxy{

S2ª0j%xy{+�I|§s�|¥xy{ ·8�Q¸?|¥r

−ε < T − adc3ade(c) − r ≤ T

²�°�eg|}w*e�¢£xY¤§¤¥x´°r�¢£�uxyh)s�egi�¢c~Yw�s?s�e+~�s aic3ade(c)

~�{��r~��ui

�RxQr¡|§s�|¥³yi®~�{+�r < T− adc3aie

(c))ª ?®p�|¥{+�Iz�w�s�|¥xy{�eQpq�Rx�suegivr�|}r�² ∀τ2 ∈ Traces(IT

c (S2, T−(adc3aie

(c) + r))), τ2.= S2,1;c;S2,2 . . . c;S2,n (1 ≤ n)

~�{+��³yiv��|§Ã+iPr�!

T − aic3ade(c) − r ≤ adc3aie

(S2,1) < T − adc3ade(c) − r + ε (Init2)

T − ε <adc3ade

(c;S2,i) ≤ T + ε (1 < i < m) (Period2)r2 − ε <

adc ade(S2,n) ≤ r2

�[¢ aic3ade(S, 2) = T − adc3ade

(c) − r + q2(T − adc3ade(c)) + r2°�|¨sue

0 ≤ q2~�{+�

0 ≤ r2 < T − adc ade(c) (End2)

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��� 9 >�&�� &�1�+0/2143�� '���> +���:*>?&�)7'*1 6�&@-��� )7:73")7'����)7';1�����:*)�� '*+0/ :*1�� �+�

�?{Qp�i��Iivw�zIsu|¥xy{�s��*~yw�iτ ∈ Traces(IT

c (S1;S2, t))|¥r�x�¢�suegi�¢£xy�uh@!

τ.= S1,1;c;S1,2 . . . c;S1,m;S2,1;c;S2,2 . . . c;S2,n

� i"�tz+r¡sMe+~´³yi®s�x7w*egiPwu­�s�e+~�sT −ε <

adc3ade(c;S1,m;S2,1) ≤ T +ε

°�e+|¥w*e�¢£xY¤§¤¥x<°r�¢£�uxyh+!

(End1) r − ε <adc3aie

(S1,m) ≤ r

(Init2) T − adc3ade(c) − r ≤ adc3aie

(S2,1) < T − adc3ade(c) − r + ε

� i7�yi�sT − ε <

adc3ade(c) +

aic3ade(S1,m) +

aic3ade(S2,1) < T + ε

²+~y{+��s�egi�w�xyh8©g|¥{giv�su�u~Yw�i

τr�~�s�|}rtÃ+iPr®s�egi7�g�uxy�Riv�¡stpyª

·£©�¸ aic3ade(S2) < T − adc ade

(c) − r!

�?{Qp�i��Iivw�zIsu|¥xy{�s��*~yw�iτ ∈ Traces(IT

c (S1;S2, t))|¥r�x�¢�suegi�¢£xy�uh@!

τ.= S1,1;c;S1,2 . . . c;S1,m;τ2

oI|§{+w�isadc3ade(S1) = t + q(T − adc3ade

(c)) + r²exet(S1;S2) = t + q(T − adc ade

(c)) +r +

adc3ade(S2)

~y{+�0 ≤ r +

adc3ade(S2) < T − exect(c)

ª � i�e�~P³yiGs�x«wue+ivw*­2sue+~<s

r +adc3ade

(S2) − ε ≤ adc3aie(S1,m;τ2) < r +

adc3ade(S2)

°�e+|¥w*e0¢£xY¤§¤¥x´°r¬�I|¥�uivw�su¤§p�¢£��xYh(End1)

~y{+�¦s�e+i�¢c~Yw�s¬s�e+~�sMsuegi+�)7'"6(&��?¢£zg{+w�s�|¥xy{�ru~<s�|}r¡Ã+ivrsuegi7�M�uxy�Ri���stp�·B�y�Y¸�²�-�.F0�.&2ladc3ade

(τ2) =adc ade

(S2)ª

� � ��a S = if b then S1 else S2ª �ivw�~y¤§¤�s�e+~�s!

ITc (if b then S1 else S2, t) = if b then IT

c (S1, t − 1) else ITc (S2, t − 1)

\?i�{+w�iY²Ps��*~yw�ivrµ~��uiUxy¢qs�egiM¢£xY��hτ = skip;S1,1;c;S1,2 . . . c;S1,m

xy�τ = skip;S2,1;c;S2,2 . . . c;S2,m

ªoq|¥{+w�i

t > 0~�{+�

ε ≤ 1²I°%i�e+~´³yi

t − 1 > −ε²gr�x�suegi�|¥{+�Iz+w�s�|¥xy{$eqpQ�Rxys�egiPr¡|}r!~y�g�g¤¥|§iPr®xY{

S1·c��iPr¡�µª

S2¸�ª¬d!egi��ui�¢£xy�uiy² ∀τ1 ∈ Traces(IT

c (S1, t)), τ1.= S1,1;c;S1,2 . . . c;S1,m (1 ≤ m)

~y{+��³yi��u|§Ã+ivr!

t − 1 ≤ adc ade(S1,1) < t − 1 + ε (Init1)

T − ε <adc3ade

(c;S1,i) ≤ T + ε (1 < i < m) (Period1)r − ε <

adc3ade(S1,m) ≤ r

�[¢ adc ade(S) = t − 1 + q(T − adc3ade

(c)) + r (End)°�|§s�e0 ≤ q

~�{��0 ≤ r < T − adc ade

(c)

�[sG¢£xy¤¥¤¥x´°r?sue+~<st ≤ adc3aie

(skip;S1,1) < t + ε~y{+�asuegi0w�xYh8©g|§{+iv�asu�u~Yw�i�ru~<su|}rtÃ+iPrsuegi¦�g�uxy�Ri���stpyª

d!egi7�uiv~yr�xy{g|¥{g��|}r!s�egi�ru~�h0i�°�|¨sueS2ª

� � ��a S = for l = n1 to n2 do S! �ivw�~y¤§¤�sue+~<s6!

ITc (for l = n1 to n2 do S, t) = Fold(IT

c (Unfold(for l = n1 to n2 do S), t))

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�q� � �"'���� � )7'��"&�+�� /2)7'���> +

�[s�¢£xy¤¥¤¥x´°r®s�e�~<s!

Traces(Fold(ITc (Unfold(for l = n1 to n2 do S), t)))

= Traces(ITc (Unfold(for l = n1 to n2 do S), t))

= Traces(ITc (l := n1; S; . . . , t))

d!egi�xY�Ri��*~<s�xY�Unfold

�ui��g¤}~yw�ivr!¢ xY�¡�[¤¥xqxy�$©Qp�r¡iP¾Qzgi�{+w�ivr�xy¢Uw�xyh0h2~�{+�grvª®d!eg|}r?w�~Yr¡i7©RxY|§¤}r?�Ix´°�{�s�xsuegi�~�¤¥��iP~y�Ip0su��iP~<suiv�$w�~yr�i

S = S1;S2ª

& � Ê Ê $����:�I� & � Ê � Æ �´� Ç � � Ê BcË:� $ Ê � ��Æ � �I����� Æ ��� � Æ � Æ ���cË� ���Bc��� ¹gxY� −ε < t ≤ 0°®i�e+~´³yi

ITc (S, t) = c;IT

c (S, T − aic3ade(c) + t)

ª�oq|¥{+w�i¦�M��xY�Ri���s¯pa��egxy¤}�gr?¢£xy�G�RxYr�|§s�|¥³yi�s�|¥h0i¦��iPr¡|}�Iz+~�¤}rv²R|¨s¢£xY¤§¤¥x´°rG¢£�uxyh adc3aie

(c) + ε < T~�{�� −ε < t

s�e+~�sT − adc3ade

(c) + t|}r7�RxYr�|¨su|¥³yi2~�{+��s�egi��ui�¢£xY��i

∀τ ∈ Traces(ITc (S, T − adc3aie

(c) + t)), τ.= S1;c;S2 . . . c;Sn (1 ≤ n)

~y{+��³yi��u|§Ã+ivr!

T − adc3ade(c) + t ≤ adc3ade

(S1) < T − adc3ade(c) + t + ε (Init)

T − ε <adc3aie

(c;Si) ≤ T + ε (1 < i < n) (Period)r − ε <

adc3ade(Sn) ≤ r

�[¢nadc3ade(S) = T − adc ade

(c) + t + q(T − aic3ade(c)) + r (End)°�|§s�e

0 ≤ q~�{+�

0 ≤ r < T − adc3ade(c)

d!egi�s��*~yw�ivr®|¥{Traces(IT

c (S, T − adc3ade(c) + t))

~y��i�x�¢�s�e+i�¢£xy�uh@!

c;S1;c;S2 . . . c;Sn.= void;c;S1;c;S2 . . . c;Sn

oq|¥{+w�i adc ade(void) = 0

~y{+��²g©Qp�eQpq�Rx�suegivr�|}r�² −ε < t ≤ 0²g°®i�e+~´³yiD!

t ≤ adc3ade(void) < t + ε (Init)

�[s!��ivh2~�|¥{+rMsux¦r¡e+x´°ºsue+~<s®s�e+i(Period)

w�xY{+�I|§s�|¥xy{�egxY¤¥�grv²=-/.F0�.&2T − ε <

adc3aie(c;S1) < T + ε

ª � ie+~´³yiD!

T − adc3ade(c) + t ≤ adc3ade

(S1) < T − adc3ade(c) + t + ε

oq|¥{+w�i adc ade(c;S1) =

adc3aie(c) +

adc ade(S1)

~y{+� −ε < t ≤ 0²g°%i7w�xy{+w�¤§z+�giD!

T − ε < T + t ≤ adc ade(c;S1) < T + t + ε ≤ T + ε

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Unité de recherche INRIA Rhône-Alpes655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Unité de recherche INRIA Sophia Antipolis : 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)������������� ����� ������� ��� ���

ISSN 0249-6399

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