How to Verify and Exploit a Refinement of Component ...

How to Verify and Exploit a Refinement of Component-based Systems Olga Kouchnarenko, Arnaud Lanoix

To cite this version: Olga Kouchnarenko, Arnaud Lanoix. How to Verify and Exploit a Refinement of Componentbased Systems. [Research Report] RR-5898, 2006.

HAL Id: inria-00071369 https://hal.inria.fr/inria-00071369 Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destin´ee au d´epˆot et `a la diffusion de documents scientifiques de niveau recherche, publi´es ou non, ´emanant des ´etablissements d’enseignement et de recherche fran¸cais ou ´etrangers, des laboratoires publics ou priv´es.

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

How to Verify and Exploit a Refinement of Component-based Systems Olga Kouchnarenko — Arnaud Lanoix

N° 5898 Avril 2006

N 0249-6399

ISRN INRIA/RR--5898--FR+ENG

Thème SYM

apport de recherche


 
   
! "#%$& '  )( *+,
! -
(./
 *02143 !56 *87+529 )(5 : ;=@?ACBDFEGH>JILKMG9NOA PQ IRGH>OBSUT)>OGACVXW Y Z4[R\^]!_M`badz|ce{lp~}k_g]fijhkjla[m\^pxh€]R]mh)q hnfg\oOprqtsvuxwy]Rh )‚rƒ4ƒOpr{€j)„i]b{l]R…"Z4]m{"…"Z4]‡†‰ˆ~Šx‹rŠ/cŒ6Žg{€stq6rrx‘’c”“F•!ƒy‚r–r]Rh ∗

†

OƒƒO—™pMpM˜9hkhnjl]*šRƒO›mp~œFpr¦C4†4j€žr]¨Z4›rst]Ÿ¡†0¢¨  ƒy†pr‚~{l{€pr°!j{"„i¢©j€Zy]m]*{H]©ƒ4j€ƒ4{lp{€]Rpx„ih€o4]m]R†Mqt‚r]mj|qy\ª¢*st†^p~s£jlj€¦JZ^Z4…msvprjlh|Z4\ƒy])o4‚rŽrstƒO†y]R]m‚F{€{|s¥jl¤JprsvhH…m{l‚~stj€‚rj€p’qxstpr]L]†^«ihkj"ƒ4p~‚~qt¦Co4pxqvqth€‚rsvsthnpr{€Z–x†]*‚!„ihnw4…fipr{lhnst\^j€†4]R–¬ƒO\^px\^hRh€§Ms¥pij€…„istprpr]R\^†yqM‚r]mƒOqg«ipx¦Xƒ4h€{lq¥s¥‚rpxj€\^st{lpr‚~]m†yj€¢¨‚rstprq4px†®{€‚r°!­ƒ4Y ƒ4st†{lZ4px]©¢*‚rƒ4Z4…"Z4w4sv…l]R{€Z h¯ jloyZ4‚x] hn]LŽr„™]RŽr{€s¥]R¤J{€…Rs¥¤J‚Fj€…Rst‚Fprj€†±stpr†¸\‚r‚Lf²ƒ4ƒ4ƒ4{lpx{€p4‚r……"]mZ4]L]L„³hm­ j€Z4{lprwy–rZ´‚µ{€]m¤y†4]m\^]R†xjn¯¶oy‚xhn]L„±hnƒO]L…s¥¤J…m‚~j€stpr†´‚r†y„´‚·…pr\^ƒOpx†4]m†xj€¯ Žrpr]R¦{€‚0s¥Œ¹¤J¢*…m‚~…Z4pxj€pxst†yq¥pr]bhk†jlh€{lfgp~‚r¦hns¥j€†M‚^]Rjb\Á…mhnprfg¦X\^†y{lpr…"ƒOZ4\,pr{l†4prqt]Rpi†4†x…msvj€hn‚r¯½]LqCoy„hl‚x‚Fƒ4hn¦X]L{l]„pijkf„ihnwyfiÀi…ºhnY*jbj€]mÀ/pr\¾ƒOƒy]R{l{€{l]px‚~¤yƒOj€†4]mpr]R{n{¬jl\+st]m]R]R†yh¨†x‚rp~j)o4¦qt{l]Rs¥]mjlh¬qvh*‚FwJj€…sth+prpr\^†6“L»¬§iƒO‚~‚~px†J†/†4„ ]m‚r†xrwij"»©j€hmpx­Hhl\‚FY ¦X‚F]mZ4j¿j€]bf]L„¼\ÀiY*‚~…prstÀµ†\^‚xƒ4ƒO„i{lpMprŽFhnƒO‚rs¥j€†x]Rst{nj"prjl‚~†ys¥–r]L‚r]h q pr…mpr¦O\^prw4ƒO{|pxh€h€ƒOs£jl]Rs¥px…s¥†¸¤J…R‚~‚F†yj€„stpr{l†^]¤J‚~†4†J]m„0\^Žr]m]m†M{ls£jR¤O­ …m‚Fjls¥px†+‚~ƒ4ƒy{€pM‚r…"Z4]Rh9sthj€Zy‚~jÃÀiY*À¸ƒ4{lprƒO]R{njls¥]Lh‚~{l]©ƒy{€]Lhn]R{€Žr]L„’jlZ4{€pxw4–rZ jlÄs¥pxÅM†®ÆC§4ǀŽrÈ0]mÉ{ls£œ¤JÊ9…Ršx‚FŸjls¥px† …mpr\^ƒOpr†4]R†xj€¯½oy‚xhn]L„µh€fihkjl]m\hm§9\^pi„iw4qt]RhR§9{€]m¤y†4]m\^]R†xjR§HÀiY*À˃4{€pxƒO]m{€j€st]RhR§9…pr\^ƒOpxh€s¥¯

∗ †

ÌgÌ4Ívê9Î4Ñ6ÏÍìÐëÃÎyÐFÑ6íbҙæºß½ÓºÝkÔºîgÔLïµÕ¨Ï2ð9Ö9ÝkñFÑã"×*îòÝØOØCÙÏ®ÚÖÛ ÜÑ9ÝkÞà×~ß½ÐFÛ á=Ö9â|ã"ÎFÚÞ=ä¿ã"óÚÐFä¶ÎFåÞ=Ý¿ã"ØàÚÏCäkæºÝ ç¨á=âºÐ~è2Ýkß½ã"ÚékæºÚMÐ~ÎFÞ=ã"ÚäkÝ

Unité de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois, Campus scientifique,

-
(%( *+
 9 º' ¾ ©Á © 
! " &  ¸ )( *0¹ *5¹5652)( *5 
(./
!5!05 q h€fi]m hk«ijlš ƒ4[mq¥\^pM hn]Rsth pr†0„4Ÿ ]ÅÀ®pr–x]R\{lh‚roy†y‚~s¥ƒ4„i†yƒy]|‚~{€j€jlpipx‚~…"s¥st{lqtZ4q¥]Ã]x]Rh9st­hl…Àh€prw0 px\^„iogƒO]©}k]LpMq …ºhn]jls¥«ij€s£¦Jstƒ4pr„iqt†4prw!†y{"jl‚F]m{lqtjlq¥‚s¥]LpxŽFh6†!‚rƒOs¥„4qx]mw0ƒ4{l\^{\+R]h€p4jn]mjl„i†x]m[mj†M¨qt]©jH„4„i„i‚rw+]©†y{hh€Rfg…mh€hn]prj€jwy[R‚r\^„i{n{l]rjl]Ãst§F…m]mqtq¥pr†0]Ã{"]Rƒyhhn‚~„ij{€]©„j€stqv]¨‚jlq¥Ž]©‚rmo4ƒ4{lqts¥{ls¥¤J{9pr…mo4w4‚~qt†0j€[mst\^…mpr‚x†]©„i„i„i{€]]] „i‚Žr] ]R…¡Ž mw4{l†4s£¤J]!…R‚F‚~jlƒ4s¥ƒypx{€†²pi…"…Z4px]b\^„iƒO]’pxh€Ž s£mjl{ls¥pxs¥¤J†4…m†4‚~]mj€qtstqtpr]¸†…prƒy\‚~{)oy…ms¥†ypr‚r\^†xƒOj^pxw4hl‚~†y†M] jlhR‚~­ ƒ4ƒy{€pi…"Z4]¸„i] h€ƒ R…ms£¤J…R‚Fjls¥px†²ƒy‚~{^{l‚ ^†y]m\^]m†Mj \x»’‚~„i†y)]Ls¥[R†^h{€] prƒ4ƒ{l…prpxRƒ4{l\+‚~{lƒOj€"s m]RpM#j w4hnL{Hs¥h!j€„istÀipr]*†yY*ƒ4†4À {l]mpiqt„iq¥„i]¡]w4“Ls¥#h j|»$4h€w4{lfg†4]†y#j ] …"{€Z4]R„i{€qtw²px‚~†4j€h€stsvprhfi †0hkjl…m„4[mpr\^]©†x{"jl]™‚{l‚r…ms¥pr†M†4\^jH]Rƒ4\+ƒOqt]m]R]{l%j†x\^j|ƒ]‚~jÃw0J„i‚~†4]*{€stj€ŽrŽst]R{0m‚r{l„is£w0¤y]™„4]R{Hƒ4w+{l‚rh€prwifiƒ4j€hn{lpxj€"s \[mm\^#j ‚FL!]j€h’sv¡…uMqtpiw4m…R]mpr‚~\^\^qt]R]RƒOh†xpxj|„ihl]]‚~jÆMhnjl„i]LhR]h § …muMprw4\^]bƒOqt]Rpxh*hl‚~ƒ4†x{lprj"hmƒ4­0{l"s Àm#j  L‚h ŽF‚rÀi†xY*j"‚~À·–rh€]0prƒ4†M{lj*s¥†yƒ4…m#{ s¥LƒJhn‚~]R{€qHŽ „4R]+]Rh †y‚~p~wij€{lj"]^‚~†M‚~j*ƒyƒyƒ4‚~{€p4{*…lqvZ4‚^]^…px„i\+]ƒOhnƒpMLhns¥…jls¥s¥¤Jpx…m†¸‚~j€uxstwypr†/]bƒy]‚rj!{ „iqt]^]bŽ{"‚R{€s¥¤J†4…m]R‚~\^j€st]mpr†x†¼jL­ ]Lhkj Ž&‡R{€É s¥¤J›Rš…mǂ~(ž j€'stpr †š Ÿ hnfihnj€[m\^]L)h …mpr\^ƒOpxhl‚~†MjlhR§y\+p4„iw4qt]RhR§J{"‚ †4]m\^]m†MjR§Oƒ4{lprƒ4{l"s m#j Lh)ÀiY*ÀçJ…mpr\^ƒOpxh€s¥j€stpr†®§





 
 "!# %$& '(

)*
+-,&
.' /
.0

)21+345 476859 '5

:

+ 69
>=  
! jlst?†4p¼p–¸¢©jlhn]m‚xsCqtBm„4]R]+…‚pxfg‚~\+hR†y§g\„¼¦Xprwy…{l†4px\\+st…R‚~‚Fƒyq®jlq¥\^s¥]mpx«g†®]s¥j€jk§HZyf j€pi{"p~„4‚~¦Hh†Jjlhn‚~Z4ƒO{l]Rpr]bh€{€]+wyj"‚Fhnh€]Lj€fgst„¸hnprj€†²st]R† \‚~ŽFh†y‚r„³\{€stpr\‚~wJ°r‚rh ]†M‚rj€wiZ4{€¦à]L‚x]R‚rs¥…º{bhRj€§Mwyh€¦X{€ƒO{lst]Rpr†4…m\#–µs£¤Jh€‚L…RfgŽg‚Fhnjlstj€prs¥]Rpx†4\† sv…mhm‚rh*­A†y‚r„ @†y„ pŽr¢¨]m‚~{l]Rw4s£Žr¤Jj€]mpr…R{L‚F\§jl‚~s¥j€pxj€Zy† sv]¸…b„ih€s¥†Js"fi…hkjl{l…]m]Rw4\‚rq¥hnjRh¯ ­ D px\+ƒOpMhns¥j€stpr†J‚~qC{l]R‚rh€pr†4st†4–0svh)‚+¢©‚LfÂj€p^\‚xhkjl]m{ j€Z4svh*ƒ4{lpro4qt]m\ ­ ¢¨\^pxpi{€„iY °¼]RZ4q¨s¥]b†@]m«gƒ4¢*ƒywyZ4q¥{€pxsvƒO…l{lpMZ³‚~hnj€]¡st‚~pr†@pr†@¦‚rp~j€q¥Z4¦–x]’pr…{lpx¢¨s£\^jlpxZ4{€ƒO\^°Âprst†4¢¨…+]m]bŽr†M]Rjlƒyh0{€{€s¥]L¤J…Rhn…m‚~]R‚~†³†xj€j*stoOprst]¸†µ†‚rj€p~Zyhl¦*hnstpi‚h*…mƒy{lst‚~]‚rj€¤yƒO]L†y]m„µ{*]m\^¢*sth ]ms¥j€j€†MZ³p^j!]Lj€prhkZ4¦*j"]¸‚~…o4prŽr\^qt]msvhn{lƒOZs£¤Jpx…R‚^†4‚F]m…jl†Mpxs¥jnpx\+¯¶†@oyƒO‚rp~pMh€¦ hn]Rs¥„µj€stprh€†Jfghn‚~j€qOƒ4]R¦X\{l{lpr‚rhbƒO\^]mog]{€f ¯¯ jljls¥Z4]L{€hmpx­ w4  –r†ZJpxI w4{*…px\+ƒOpMhns¥j€stpx†y‚~qJ¦X{"‚~\^]m¢¨px{€°C§M¢©]¡–xs¥Žr]¡¢©‚fiFh E=hn]RH] G9st–y­2“L»Ãj€p^ƒ4{€]Lhn]R{€Žr] LTLTL ƒ4L{lprƒO]R{njls¥]Lh “r­*]RY {nZ4jkf¼]¡…mh€pr‚~\^j€svhnƒO¤ypx]Rh€„¼s£jlstogprf¼†‚ pxƒOh€]m]m{"ƒy‚F‚rj€{lpx‚~{Hj€]+¦Xpx…m{¨prƒ4\^{l]RƒOh€pr]m†y{lŽg]m†xs¥†yj’–!svh’hl‚F‚~¦X]mqvh€j¿p¸f hlLT‚FjlstLhn¤yƒ4]L„¼{lproMƒOf/]R{njl‚¸s¥]L¢*hm§MZ4\+prqt]L]^‚~…m†4prst†4\^–!ƒOjlprZy†y‚F]mj †xj€‚¯½oJƒ4‚r{lprh€]Rƒi„ ¯ h€fihkjl]m\ ­ i­*stY †4Z4–·])j€{lZJ]‚F¤yj†y]m‚·\^ƒ4]m{€†xpxjÃO{l]m]m{€qvj¿‚Ff²jlstpr]R†0hnjl¦X‚~pxoy{|q¥svƒ4h€Z4{l]R]Rh€„±]m{l¦XŽMprst{™†4–b‚~†ËoOp~‚rjloyZhkhljl‚F{l‚x¦X]m…ºj¿jÂf+h€‚rfi†yhk„+jl]mqt\ stŽr]m\^†4]Rpihl„4h ]mLTq)svhL]Rƒy†y{€h€pxw4ƒJ{€]L]R„±{njlst¢*]RhmZ4§r]R\^†´]Rjl‚~Z4†4] ¯ h€fihkjl]m\¾sth*{€]m¤y†4]R„™jlp‚+{lst…"Z4]R{ qt]mŽr]Rq2p~¦„4]jl‚rs¥qvhR­ ;


ºÚFãlá=Ýkßá=åÝ ßàá¶ãlá=Ý|#æ @ A B ëÚ¬ÝkîòÝnç*ÝkÚá6#æ @ Ûòß6ã"îòß=æ ñLÝnÚLæ"á½Ýkñ A 0

0

0

1

V

0

0

2

V

V

V1 ∪V2

V1

ith

(σ, i)

T

ÑÑ&Ú.N)OPQP

1

V2

σ e q → q0

2

1

2

‘


  
 0.  
  ) 
 

j€stpr†/¢©]Zy‚LŽr]s¥†Mj€{lpi„iwy…m]R„ ts †òˆ   †µ½­j€À®Z4‚^]msvj h’\^SAƒypr‚r{lƒO]’=]m„i{L]m(Q§®jl¢©‚rs¥]qt,]R]Q„ «iƒ4À4qtY)pr, _Cs¥Ejb­JjlY ,Z4Z4T]Â]!h€,h€fiVfghk†xjl]mj",\‚rl…ºjl)j€stpx…¡oOƒi¦X]*¯¶]L„4‚~‚Fpjl†^¢*w4‚r{€†µoy]Lh*hk{ljl]p~{l¤y¦9‚x†y…ºjl]mjHZ4\^]!ÀiY)]m{€†x]m_0j¤y‚r†4{€†y]m]R\^„qt‚~SR = (Q , Q , E , T , R ] x † ¬ j V ,l ) {l„i]wy¤y…m†4]R]R„\^‚r]m†y†x„j¬jls¥Z4†Mj€]¡{lprpiqv„i„Âwypx…]L†4h*]Rh †4]R‚~¢ {l]‚x{l…º]mjl†ys¥‚rpx\^†yhm]R§J„ hnI p E ⊆ E ­*_g­H]L…Y prZ4†Jst„2{"„2§J§ih€pr‚\^ƒy]’{€px†4ƒO]Rpx¢ h€s£jlŽ~s¥‚~px‚~{l†y{lst‚~‚r]’qOo4‚x…mq¥]Lh©‚rh*qt¦X…mpx…mw4q¥‚~qtqtpF†wy¢)hÃoOhm¦X]!­*px{€s¥G9†M\stj€{lw4{lhnp~qvjR‚ ¯ § pFŽr]m{ V ∪ V §y…m‚rq¥qt]R„     0  4.V9 §J∩qtst†4V°ih =ŽF‚r{€∅sv‚~o4qt]Rh©p~¦9oOprj€ZÀiY)_ihm­ gp ¸ Å  › àÉ 
 '  Å Êšm›Li› Å š 

 gp ∈ SP 3 H      4. 4. *  / 5. q ∈ Q  5*–rqtw4]L„ 9
q ∈ Q  *C *C9* gp "& 2

 q µ q "& 6 (q , q ) |= gp * Y Z4]’{l]¤y†4]R\+]R†xj){€]Rqt‚~j€stpr†¢*s¥j€Z¸jlZ4]!hn]R\^‚r†xjlst…¡¦ì]L‚Fjlw4{€]Lh oO]mqtp¢Usvh*‚+{l]Rhnj€{lst…j€stpr†pr¦ µ ­ “r­*‚rY {€Z4]b]°rjl]m{lƒi‚rjL†y­ h€s£jls¥px†yh|pr¦ SR §xjlZ4]¬qv‚~oO]mqvhÃpr¦®¢*Z4sv…"Z‚~{l])st† E EXs½­ ]r­qv‚~oO]Rq¥qt]R„ÂoMf+jlZ4] prqv„  qv‚~oO]mqvh"» i­ ?‚r{€]m]¢ …pxj€†y{"‚~h€st†J„ihn]Rs¥{€j€]Lstpr„^†y‚xhbh|st†xoOjl]m{€stp4†4„i–wy†4…pr]L†4„µ¯½px„4oyw4hn{€]Rst†4{€Ž~– ‚Foyj€Z4q¥] ]Â~j€{lZ4]¤y]Rf†4]R‚~\+{l]*]R†xqv‚~j0oO„i]R]Rq¥qth€]Rs¥„–x† oMf Eàs=­ ]r­™‚~†Jqv‚~„oO…R]R‚~q¥qtqt]Rq¥]L„µ„ st† ¯=jlE{l‚r†yrh€s£jlEs¥px†y»h τ τ st†jlZ4]!h€fghnj€]R\ SR ­ :4­*apx{€]RpŽr]R{R§g†4]m¢ jl{l‚r†yh€s£jls¥px†yh h€Z4prw4qv„†4prj*st†xjl{€pi„4wy…]b†4]R¢ „i]R‚x„iqtpi…l°ihR­ ­ jlG9{lst‚r†y†y‚~hnqts¥q¥j€fxst§6pr†y†J]mhm¢§is=­ j€]r{"­ ‚~†y¯¿h€…s£fijlst…prqt†y]RhRh’§ih€‚rZ4{€pr]¡w4¦Xqvpr„/{lo4†4stpr„yj’„i]mjl†®‚r°r­ ]…px†xjl{€pxq9¦Xpx{€]RŽr]m{L­Â_gpJ§6st†i¤y†4s¥j€]™h€]RuMw4]R†y…]Lhbp~¦ τ ¯ τ ¸ Å  › àÉ  2™ Å Å  Å  › Å 'ài› = 6859 ' 5   6 F§®“L/ˆ En“L» Iò•  Š4§ xw4qtf “L‹r‹r‹y­  :!Œ!­Œ{€†4pxqt„® ­ 685 '( 5K >% ./5%
./5 $& 5 F5 '-.)2 ) K 5 0
 55  54
.''    5º­ D prqtqt]R…ºjls¥px† Hjlwy„i]Lh*]mj))]R…lZy]m{"…lZ4]Lh ]m†   †4¦Xpr{l\^‚~j€svuMw4]RhR­4a ‚rhlhnpx†®§gzH‚~{lsthR§2“R‹x‹xi­  !Œ!­JŒ{€†4pxqt„ ‚r†y„ a@­ ?)stŽF‚FjL­ D pr\^ƒOpr{€j€]R\^]m†xj"h*„i]!ƒ4{lpi…]Lh€h€wyhR­   †   9 5"A,
.
)  ,   1  5  '-.% ) 5K "  
 '(2 ) m§yƒJ‚~–r]Lh :xˆ g‘xŠi§C“R‹xŠxi­ òˆ  G©­ ©]Rq¥qt]m–M‚~{"„i]r§ J­ xw4qtq¥sv‚~†y„®§Ã‚r†y„ ­ prwJ…lZ4†J‚~{l]m†4°rpy­³*]R‚x„ifM¯¶h€s¥\wyqt‚~j€stpr†³svh+†yp~j^{l]R‚x„ifµj€p ]«i ƒ4H{l6 ]R hlh9‚¬\+F§yp4Žr„iprw4qtw4qv‚~\^{9]^{l]“¤y•~†yŠ ]m:\^pr]m¦ †Mj7{l]m,qv‚F6Jjl§4s¥pxƒy†®‚~­ –r  ]L† h*r‘r0‘ 4K~Š.'(:4 ­4_gƒ4.{l s¥F†4–x52)]m{

 LH 5 ]m{l
qt ‚r–y6 §i
Œ2 ƒ4{ls¥ q ~ xry ­  4

  ‘  )­Ja@­ D qt‚r{€°r]x§ %­ /¡{lw4\oO]R{€–J§i‚r†y„ ­JŒ!­4z]mqt]R„2­


*, /

 .r­ÃY Zy]’a   YUz|{€]Lh€hR§y~xry­  •  y­ ¯¿a@­ D proyq¥]Rs¥–xZ®§ 0­ /¡sv‚~†y†y‚~°rpxƒOprw4qtprw®§x‚~†y„ D ­xzH‚rhl‚~{l]R‚~†gw®­OÀ6]R‚~{l†4st†4–!‚rhlh€w4\+ƒ4j€stpr†yhH¦Xpr{¨…mpr\+¯ ƒO,&px
h€)s£jl5st2pr †y‚r qi% Žr
.]m {ls£¤O…m‚Fjls¥ px†®­  8† 5 .5>
(608

5 90 .'%5
.0. ,&,
F 6



  §H HŽrpx
.q¥wy\+)
]
r5H‘4“L0‹¸ p~ ¦   
( , 6J '§ µ5 ‚r
.{l #hl‚ ¢’ ) § z9pxqt‚r†y„2§yŒ)ƒ4{lstq6rr :y­i_iƒ4{€st†4–x]m{n¯ LH]R{€qv‚~–y­  Š  D ­ ¬‚r{€qtp~jL§ J­ rw4qtqtst‚r†y„2§C‚r†y„ ­ ¬prwy…"Z4†y‚~{l]m†y°rpy­)*]m¤y†4]m\^]R†xjbƒ4{€]Lhn]R{€Žr]Lh*z¨ÀiY*À³ƒy{€pxƒO]m{€j€st]RhR­   † )0& ­ C
4©0]R'({njL

§ )JM­gz ­ ¨.pF0¢¨ ]R†®§4_C­ D ­  bst†4 –y§i§iŽr‚rpr†yqt„w4\^a@] ­ ~·‘Mˆi‚~“ qv„ip~]R¦ †® §i(]R„i, s¥jl6Jpr§x{"Yhm§ w4{l°M
.w6 '-§KG9.s¥ †y6.qt‚r)†y

 „2 $§ xw42†4 ]¬
 ~x :4 ­ _gƒ4{ls¥†y–r]m{ LH]m{lqv‚~–y­  ‹  ­ ¬prwy…"Z4†y‚r{€]R†4°rp‚~†J„/Œ!­6À6‚r†4prs¥«2­ ! # I’‚{€]m¤y†4]m\^]R†xj!‚~†J‚~qtfgh€svhj€piprq9¦Xpx{’h€fM†J…lZ4{lpr†4Cs Bm]L„ …px\+ƒOpx†4]m†Mjn¯¶oy‚rh€]R„h€fihkjl]m\hm­   † )

.&  32
 
9 5#"
 '-.

 /
5  F­ t“R  ­ prwJ…lZ4†y‚r{€]R†4°rp ‚r†y„dŒ!­À6‚~†yprs¥«C­ *]¤y†y]m\^]m†xjª‚r†y„ Žr]m{ls£¤J…R‚Fjls¥px† p~¦±h€fM†J…lZ4{lpr†4Cs Bm]L„ …px\+ƒOpx†4]m†Mjn¯¶oy‚rh€]R„µh€fihkjl]m\hR­   † ^­Œ){"‚~°gs½§_C­ /¡†4]Lhns½§‚~†y„@a ‚~†y„4{€stprqts ­t§9]L„is¥j€pr{"hR§
 '(






   









!




















 )

 





%$















 !"!#

!







'&









+ 

 , .-

4)



+   -/ !0! )




*$




/

2




( ! )

,

*$

'$

3

1

  4 ! 5

%

6

7





ÍìÖ9Ñ®Íìë

“Lˆ





 
 "!# %$& '(

)*
+-,&
.' /
.0

)21+345 476859 '5

g_ ƒ4 {l s¥ /†y
–r8]m5 { LH]m{lqv‚~–y­ F§2Žrprqtw4\^]0~Šxxˆ0pr¦  7, 6J§Jƒy‚~–x]Rh#: “  :xˆ~Š4§4z|svh€‚y§   jl‚rq¥fx§C_g]mƒ4j€]m\oO]R{b~x :4­ t“r“  ­ pxwy…"Z4†y‚~{l]m†4°rp@‚r†y„ËŒ­©À6‚r†4prs¥«2­ LH]m{ls£¦Xfgst†4–³st†MŽF‚r{€sv‚~†Mjlhpr¦!…px\+ƒOpx†4]m†Mjn¯¶oy‚rh€]R„h€fihkjl]m\h j€Zy{€ px /w4
–r Z¼
C{l
].¤y7†4.]R\+0"]R†x6 jL
­ %  † . D ­®))

‚F j€j€{"
‚Lfx
§®._C­®a ‚rZyH‚~6 {"‚F }§2‚~†y§g„ ŽrDpxq¥­®wy\+_gZJ] ‚~:4†4“r°g“Lqv‚~‘¡†ypr„2¦ §2 ]L7„is¥,j€pr6O{"§xhRƒy§ ‚~ –x ]R hà 3 r Šr‹   :rK :4§4_Mjls¥{lqts¥†4–J§y_i…mp~jlqt‚r†y„2§ xw4qtfrr ­i_iƒ4{€st†4–x]m{ L|]m{€qv‚r–y­ t“L ­ bwyƒi¦X]m{l\‚~†‚~†y„a@­~`+­ L|‚r{l„is½­Oa pi„iw4qt])…"Z4]R…"°gs¥†y–y­   †Â)‚F}k]R]mŽ+Œ)qtw4{¨‚~†J„^Yb­ Œ!K­ @]m/† BRs¥†4–x]m{L§ ]R“r„i“Ls¥xjlipr§g{"hmƒy§ ‚~ –x]R  h •~ ˆiŠr‘4§

? 0]m.¢ %
.¨0{lw4 †y,&h€¢*
sv…l°C  §/

?  J § ¡
 _iŒ!,&§6
“R'#‹x ‹r ‘y9­i _g ƒ4 {lst †4 –r ]R { )LH

]R {€ $ qv‚~–J­ 2
 , "6§xŽrprqtw4\^] t“ : ­ bw4ƒi¦X]m{l\‚~†U‚r†y„ a³­©`+­ L|‚r{l„is½­ a pi„iw4qt]µ…"Z4]L…l°gst†4–±{l]mŽgsvhns¥j€]L„2­   † . )9

02
 ,&
'#  9    K 4 )

 $ 2
 ,&
  

  m§yƒy‚~–x]Rh :r‘  •g§O“L‹r‹g•g­ t“ ’Œ!­xÀ6‚r†4prs¥«2­ 685 '( 5 4
'#/
5 )%5 5  +
.) 5 5  4
.)2  3  2
)5  &  $ 2
 
'#/
1 _g5]m%ƒ4
.j€)]m\0

oy{€C (]~xx ˆi ­  '( 

  5 
40   5º­)z|Z ‡jlZ4]Rh€svhm§†4s¥Žr]R{lh€s¥j#„i]"Gy{"‚~†y…"Z4]¯¿…px\!j#r§ t“Lˆ  9­9a¼‚~†4†y‚¼‚r†y„³Œ!­9z|†gw4]mqts½

­  / )

' /
.  
  
 !# " %5& .0 ,&
   

)F6859 '5  6.)

  $ 2
/5º­©_gƒ4{€st†4–x]m{ L|]m{lqt‚r–y§C“L‹r‹x4­ t“R‘  ^­ À|­2a …Rastq¥qv‚~†®­+Œ¹\^]jlZ4pi„ipxq¥px–rf¸¦Xpx{bZy‚~{"„i¢©‚r{€]0Žr]m{ls£¤O…m‚Fjls¥px†¼wyhnst†4– …mpr\^ƒOpxh€s¥j€stpr†y‚rq9\+p4„i]mq¥¯ …lZy]R…"°Mst†4–J­ 6    4
(,&
'#  9


 ' '~0§ :M• I M•F‹  :rr‹4§irrx4­ t“•  ?!­6_gZy‚~†y°F‚~{L­’À6‚ Bmf…mpr\^ƒOpxh€s£jls¥px†y‚~q9Žr]m{ls£¤J…R‚Fjls¥px†®­   † ,

6 "!# )&  5 4 

   5 
' ˆ ) “ iˆ~‘ 

§g0À6.pr%†y
.„i0px†® § 6 ^'#§2/“L
8‹r5‹r  Šy' ­i_gƒ4
{lst†4,&–r
.]R{n' ¯LH)
8]m5{lqv%‚~ –4
.­  2  / 6  ) $4.)  6#



  §Ãƒy‚r–r]Rh t“RŠ ’’­ y­JŽ~‚~† /¡qt‚ro4oO]m]R°C­)Y Z4]q¥st†4]R‚r{)j€st\^]¯¶o4{l‚r†y…lZys¥†4–jls¥\^]0h€ƒO]R…j€{lw4\ EX]m«Mjl]m†y„i]L„ ‚~oJhkjl{l‚x…ºj"»­   † ,
(, ! F§yƒy‚~–x]Rh*M•FŠi~‹M•M­4_gƒ4{ls¥†y–r]m{€¯LH]m{lqt‚~–y§C“R‹r‹x4­  



 ! 5

%$














!

0



%$

0




)










&






 










)












)





$  ' &1!

ÑÑ&Ú.N)OPQP




 &



“L‘


  
 0.  
  ) 
 



^
!F5


        

rpÀ6¦]j „i]M¤y†479=s¥8;j€: stpx{(q†ˆ0,prq¦ ρ)­ | q ∈ Q ∪ Q ∧ q ∈ Q ∪ Q } ­ /])h€Z4pF¢³j€ZJ‚Fj M Žr]R{€s¥¤y]RhH‚~qtq4…mpr†y„is¥j€stpr†Jh “rF­ 60%    %$& '(

)/*hnwyƒ4ƒOpxh€] (q , e, q ) ∈ T ‚~†J„ e ∈ E ­ µ]\wyhnj¬ƒ4{lpŽr]’j€Z4]R{€]!]m«isthnjlh q ∈ Q ‚~†y„ q ∈ Q hnwy…"Z jlZy‚Fj (q , e, q ) ∈ T § q ρq ‚r†y„ q ρq ­ ©f„i]¤y†4s¥j€stpr† E = E ∪ E ­Hz|{lpgp~¦„i]mƒO]R†y„4h*p~¦ e ­ e∈E ‚r†y„ q ­ ∈µQ]!Zyh€‚wyŽr­ …"] Z™(qjlZy‚F,j e,qq ρq) ∈§ T(q ­, _ge,st†yq…] ) M∈ TR v‚~†J„ qMρqA §y­ j€qZy]m{l‚~]!†y]„ «istqhnjlh oOq]Rq¥px∈†4–0Qj€p Q =Q ∪Q e∈E ‚r†y„ q ­ ∈µQ]!Zyh€‚wyŽr­ …"] Z™(qjlZy‚F,j e,qq ρq) ∈§ T(q ­, _ge,st†yq…] ) M∈ TR v‚~†J„ qMρqA §y­ j€qZy]m{l‚~]!†y]„ «istqhnjlh oOq]Rq¥px∈†4–0Qj€p Q =Q ∪Q iF­ 5  %9  - %$& '(

)/*h€w4ƒ4ƒOpxh€] (q , τ, q ) ∈ T ­ µ]’\wyhnj)ƒ4{€pFŽr]¬jlZ4]m{l]b]«isvhkj"h q ∈ Q h€wy…lZ¸j€Zy‚~j q ρq ‚r†y„ q ρq ­ ©f„i]¤y†4s¥j€stpr† E = E ∪ E ‚r†y„ E = E ∪ E ­Hz|{lpgp~¦„i]mƒO]R†y„4h*p~¦ e\τ ∈ E r E ­ ­ j€ZJµ‚F]+j Zy‚Žr] (q‚~†y,„ e\τ, q ) ­ ∈ T oO­0]mqt_gpr†4s¥†J–M…h¨] j€Mp R v M A §C­ j€Z4]R{€]+]«isvhkj"h e ∈ E rE € h y w " …  Z q ∈Q q ρq q ρq q Q =Q ∪Q ­ /  + ] y Z  ‚ r Ž ]  ­ g _ t s y †  … ] §C­j€Zy]m{l]0]«isvhkjl] e ∈ E rE (q , e\τ, q ) ∈ T MR v MA € h y w " …  Z € j J Z F ‚ j ~ ‚ y † „ ­ O o m ] t q r p 4 † M – ¨ h € j p q ∈Q q ρq q ρq q Q =Q ∪Q :4­
.
 0

 K 4K 
 85 ¡h€w4ƒ4ƒOpxh€] q 9∈ T ­ µ]’\wyhnj)ƒ4{€pFŽr]¬jlZ4]m{l]b]«isvhkj"h q ∈ Q h€wy…lZ¸j€Zy‚~j q ρ q ‚~†y„ q 9 px{ q ∈ D 4D ­ pr©¦ fq„4]­ ¤y†4s¥j€stpr† Q = Q ∪ Q = (Q ∩ Q ) ∪ (Q r Q ) ∪ (Q r Q ) ­z|{lpgp~¦„i]mƒO]R†y„4h ­¸­ _gst†y…]oO]mMqtpr†yR–xh¨vjlp M4Apr{ §]ms¥j€Z4]R{’j€Z4‚r]R†y{€„]]«isvhkj"h ­|q Œ¬h ∈ Q hnwJ…lZµj€ZJ‚Fj q ∈ Q rQ ~ ‚ J † „ q 9 ‚r†y„ q ∈ Dq § ρqq ∈qD r Q ­ q QoO]Rq¥px†4–xh©qj€p ∈DD4D ­ q ρq q ∈ Q r Q ­¸­ _gst†y…]oO]mMqtpr†yR–xh¨vjlp M4Apr{ §]ms¥j€Z4]R{’j€Z4‚r]R†y{€„]]«isvhkj"h ­|q Œ¬h ∈ Q hnwJ…lZµj€ZJ‚Fj q ∈ Q rQ ~ ‚ J † „ q 9 ‚r†y„ q ∈ Dq § ρqq ∈qD r Q ­ q QoO]Rq¥px†4–xh©qj€p ∈DD4D ­ q ρq q ∈ Q r Q q ∈ Q ∩Q ‚r†y„ ­ _gEk“st†y»¨…px] { M R v‚r†y„ M A §6jlZ4]mE={lx]»º­]m«gsvhnjlh q ∈ Q hnwJ…lZµjlZy‚Fj]Rs£jlZ4]m{ R

A

1 R

R

2 R

1 A

A

0 R

R

A

0 R



2 A

R

A

0 A

1 A

A

1 A 0 A

A

2 A 0 A

A

1 A

1 A

1 R

0 R

R

A

2 R

0 R

R

A

R

0 R

A

A

A

R

0 R

A

2 A

R

R

0 R

A

R

A

1 R

R

0 A

2 A

R

R



A

R

A

D2 ρ

1

0 R

0 A

A

0 A

A

0 A

2

0 R

0 A

A

1 A

A

2 A

1

A

D1 ρ

A

2 R

0 R

A

A

R

1 R

0 R

A

0 R

A

2 R

1 R

2

A

D2 ρ

A

A

1

1 A

2 A 2

1 A

2 A

R

A

2 R

D1 ρ

R

R

1 A

1 A

A

2 A

1 A

0 A

2

A

A

1 R 1 A 2 R 2 A

A

2 A



0 A

A

1

2 A

R

2 A

A

2 A

R

1 A

A

0 A

R w A

1 R

2 R

1 R

A

2 R

1

R

1 R

2

2 R

R

1 R

R A

R 1

R

2 R

R A

qA 9

A

1

R

A

R

2 R

qR ρqA

1

D1 ρ

A

1 R

2

1 R

1

A

R

R

R

2 R

2 R 2

A

1

2

qR ρqA

A

R

2

R

1

1

1 R

2 R

2 R

1 R

2

2 A

A

R

D1 ρ

R

2

A

1 R 1

1

R

R

D2 ρ

1 A

A

R

A

R

2

A

1 A

qR ∈ D 1

ÍìÖ9Ñ®Íìë

“•





 
 "!# %$& '(

)*
+-,&
.' /
.0

)21+345 476859 '5

xp_i{s¥†yq…m] ρqM R‚r†yv„ q M∈ AD §FjlE Z4]m»º{l­ ]©]m«isthnjlh q ∈ Q hnwJ…lZ0j€Zy‚~jH]ms¥j€Z4]R{ q 9 ‚r†y„   ¦ Ek“»©‚r†y„ E%:x»º§ijlZ4]m{l]b]«isvhnjlh q ∈ Q ∪ Q h€wy…"ZjlZy‚Fj q 9 ­   ¦ Ek“»©‚r†y„ E »º§ijlZ4]m{l]b]«isvhnjlh q ∈ Q h€wy…lZ¸j€Zy‚~j q 9 ­   ¦ E=x»©‚r†y„ E%:x»º§ijlZ4]m{l]b]«isvhnjlh q ∈ Q h€wy…lZ¸j€Zy‚~j q 9 ­   ¦ E=x»©‚r†y„ E »º§ q ∈ D ∩ D I q oO]mqtpr†4–Mh¨jlp D 4D ­ 2

R

2

D2 ρ

A

2 A

A

qR ρqA

A

E%:x»

2

R

1 A 1 A 2 A

A A A

1

R


        

2

2 A

A

A A

1

R

2

§ oMfY Z4]Rpr{l]m\&i­ ‚~†y„z|{lprƒO]R{njkf •+st\^ƒ4qtf (SR , SR , syn ) v ‚r†y„ RD = ∅ st\+ƒyq¥f D 4D = ∅ ­ RD = ∅ ‚~†yŠ4„ ­ ¬ div (SR , SA ) st\^ƒ4q¥f ¬ div ((SR , SR , syn ), (SA , SA , ¬ div (SR , SA ) M o ™ f | z l { r p O ƒ R ] n { k j ™ f syn )) Y Z4]R†¸¢¨]’Zy‚Žr] (SR , SR , syn ) v (SA , SA , syn ) oMf™z|{lprƒO]m{€jkf “~­


1 2 2 D2 1 [SR1 ] vD ρ [SA ] [SR ] vρ [SA ] 1 2 (SA , SA , synA ) 1

2

τ

1

1

1

τ

1

ÑÑ&Ú.N)OPQP

2

R

η

R

D1 4D2 ρ

2

2

2

τ

A

1

2

1

2

A

1

2

R

1

2

Unité de recherche INRIA Lorraine LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod - 91893 ORSAY Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (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)

Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)



        

ISSN 0249-6399