• Aucun résultat trouvé

Quantified Loop-mu-calculus for Control under Partial Observation

N/A
N/A
Protected

Academic year: 2021

Partager "Quantified Loop-mu-calculus for Control under Partial Observation"

Copied!
17
0
0

Texte intégral

(1)

HAL Id: inria-00071630

https://hal.inria.fr/inria-00071630

Submitted on 23 May 2006

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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 destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Quantified Loop-mu-calculus for Control under Partial Observation

Stéphane Riedweg, Sophie Pinchinat

To cite this version:

Stéphane Riedweg, Sophie Pinchinat. Quantified Loop-mu-calculus for Control under Partial Obser- vation. [Research Report] RR-4949, INRIA. 2003. �inria-00071630�

(2)

ISSN 0249-6399 ISRN INRIA/RR--4949--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 1

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Quantified Loop-mu-calculus for

Control under Partial Observation

Stéphane Riedweg — Sophie Pinchinat

N° 4949

Septembre 2003

(3)
(4)

Unité de recherche INRIA Rennes

IRISA, Campus universitaire de Beaulieu, 35042 Rennes Cedex (France)

Téléphone : +33 2 99 84 71 00 — Télécopie : +33 2 99 84 71 71

" #

$%&# '()#+*,#&& '.-0/1!2)# 34& '

57698;: <>=@?BADCEFA;G&HIAKJMLN52OP:B<BEQASREQ?BTU<BEF?B=6

V)WXZY\[\]_^a`bdce[dfhgjik[ lmcenjcolpXZY\[dc

qNreshto[ lmuwv

`xfhyzysKr{lm|z[_rp[d}~W[Zr~}~Wz[€@&v;‚ƒvK‚S^ uw[dyjle[dY„zrp['…ƒ†h†K‡^ˆ]d‡‰yfƒŠK[dc

‹kŒ dŽZUz‘hŽh’”“ ,leW•–c1yfhy[drd—_˜™[š}Zshce•–|z[Zr1} sh;lpresK›4yzrpsh„›œ[dY\cgz|j[Zr1yfhr{lp•–fh›_sK„ce[Zrp9fƒle•–sh,•–Mf

›–shŠK•–}dfƒ›ce[ lele•–zŠžBŸš[4[Ziwle[Z|\leWz[4Ygj ¡}Zfh›–}Zgz›–gcN„wnfh|z|z•œzŠles¢£sKreY‰gz›¤fhc¦¥;gfƒle•œ§}Zfƒle•–shc¦sUh[dr¦fUlpshY\•–}

yresKys;c{•œle•–shcfh|š¢£shr~} [\lpWz[ZY¨•–c{•¤|j[›–swshy©Ygz ª}dfƒ›¤} gz›–gc'|j[Z§fh„z›œ[}Z›–fKcece[dc«sh¢¬…U­FžŸS[ksh„jl~fƒ•–f

K[ZrpnS[Zijyzre[®cece•–h[›–shŠK•–}kWf9w•œzŠšf¯z[®} [®cecpfƒrpnšfh|(cegj°} •–[Z;l} sK|j•œle•–sh±sK±leW[cenwKl~fh} le•¤}zsKreYfƒ›

¢£sKreYc_sƒ¢•²l~cm¢£sKreY‰gz›¤fhc4les1|j[d}Z•–|z[lpWz[Z•–rY\sw|z[Z›œ ª}~Wz[®}~³w•œzŠS´µfƒ|¯lpWz[c{nw;leWz[®c{•¤c4sƒ¢™}Zsh;lerpsh›–›–[Zr~cleW[Zn

cey@[d} •œ¢£nz¶ž'·shr«[ izfƒY\yz›–[h—¸fYfUij•œYfh›&y@[ZrpY\•–cpc{•–h[I} sKKlpresK›œ›–[Zr_fƒY\shŠfk}Z›–fKcec4sƒ¢™} sh;lpresK›œ›–[Zr~cxgz|j[Zr

yfƒrele•¤fƒ›Psh„ce[ZrpUfUlp•œsK}dfƒ„['cenwKlpWz[dce•–¹Z[d|7—zfhc)˜™[Z›–›7fhc|j[®} [Z;lprpfh›œ•–¹Z[®|k} sK;lerpsh›–›œ[drpc™•–1ceshY\[}ZfKc{[®cZž

º¼»;½P¾p¿À UÁ h’àgfƒ;le•œ§[®|¼Ä¼gj ¡}Zfƒ›¤} g›œgcd—¥;gfh;le•œ§[d|›–s;sKy¼Ygj ¡}Zfh›–}Zgz›–gcZ—} sh;lpresK›2c{nw;leWc{[®c{•¤cZ—|j[  

}Z[Z;lerp[Z›–•–¹Z[d|} sK;lerpsh›–›œ[drpcd—Kyfƒrele•¤fƒ›¸sh„c{[dreUfUlp•œsK7—wshyjlp•œYfh›œnk}Zre•œle[dre•¤f

(5)

™'‰2#B' # !”!2'!/1!2)# 34« ' 1#&& Z

 ’ fhc>} [)yfhyz•œ[drd—hzshgc>fh„sKrp|zshc ›–[™yresK„z›œXdY\[)|zgI}Zsh;lerh›–[™ceshgcBsh„ce[ZrpUfUle•–sh‰yfƒrele•–[Z›–›–[

|gzy@sh•–;l&|j[Nwgz[¦›œsKŠh•¤¥Kg[hžmshgc2b le[d|jsKc¸›–[NYgj ¡}Zfh›–}Zgz›ƒyfhr&¥;gfhKlp•²§@}ZfUlp•œsKc¸cegzr2›–[dc¸yresKys;c{•œle•–shc

fƒlesKYI•¤¥;gz[dc_} sK;ler~fƒ•–Klp[dc|z[dc«} ›¤fhcpc{[®c4|zb §z•¤cpcefh„z›œ[®c_[ZD›¤f›œsKŠh•¤¥Kg[‰›–swshyj ªYgj ¡}Zfh›–}Zgz›B|j[¬…9­ªžmshgc

sK„jle[dzshcmgz[I›–shŠK•–¥;gz[lerpXdc_[ ijyzrp[dcpce•œK[¥;gz•Bys;ceceXd|j[Igzz[\} sK|j•œle•–shSzbd}Z[dcpcefh•œrp[[Zl«cegj°cefh;le[‰cegzr

›–[dc>¢£shrpYI[®cBzshrpYfƒ›–[dc>|j[®cB¢£shrpYg›œ[®c>ysKgzr¦|jb®} •¤|j[ZrN|z[x›œ[dgzr~cBYIsj|j[d›² ¡}~Wz[®}~³;•–zŠ´¸[Zl¦|j[mcen;;lpWzb lp•–ce[Zr>›–[dc

}Zsh;lerh›–[Zgrpc¥Kg[d›œ›–[dc)|jb®} rp•œK[Z;l~¶ ž q>fhr™[Zij[ZY\yz›–[h—wgz1} sh;lprK›œ[dgzr|j[«y@[ZrpY\•–cpc{•–w•²lpb4Yfƒiw•–Yfƒ›–[4yfƒrpYI•

›–[dc1}Zsh;lerh›–[Zgzr~ckceshgcksK„c{[dreUfƒle•–sh,yfhr{lp•œ[d›œ›–[šy[dgjl lerp[c{nw;leWzbZle•¤c{bK—4fh•œce•¥;gz[š|z[dc1} sKKlprK›œ[dgzr~c

|zbd} [d;ler~fƒ›–•–cebdc)|zfhc} [dr{l~fƒ•–c}ZfKcZž

Ž® ¾ ‘  ’ įgj ª}dfƒ›¤} gz›I¥;gfƒ;le•œ§bK—›œswshyz kYgj ¡}Zfh›–}Zgz›¥;gfhKlp•²§bh—'cen;;lpWzXdce[±|j[(}Zsh;lerh›–[Zgzr~cd—

}Zsh;lerh›–[«|jbd}Z[Z;ler~fƒ›–•–cebh—wsK„c{[dreUfƒle•–shkyfƒrele•–[Z›–›œ[K—z} rp•²lpXZrp[dc|!shyjlp•œYfh›œ•œleb

(6)

: ;

‰2#B¦& '

`x[d} [d;le›–nh—'¬!<d­«•œ;lerpsj|jg}Z[d| lpWz[S¥;gfhKlp•²§[d|.Y‰gj ª}dfƒ›¤} gz›–gc\lps cey[®} •œ¢£n.fƒ|,cenw;leWz[®c{•–¹Z[D}Zsh;lerpsh›–›–[Zr~c

¢£sKrf¼›¤fƒrpŠh[}Z›–fKcecsh¢x} sh;lpresK›Nyzrpsh„z›–[ZYcdžV)Wz•¤c'›–shŠK•–}k} shc{•¤col~c•–±¥Kgfƒ;le•œ§}Zfƒle•–shc'shleW[kfUlpshY\•–}

yresKys;c{•œle•–shcsƒ¢2leWz[«Ygz ª}dfƒ›¤} gz›–gc¢£shrpYgz›¤fhcdž>= ¥;gfh;le•œ§[d|kyzresKys;c{•œle•–shk•œ;leg•²lp•œK[Z›–n˜•²lpz[dcpce[dcleWz[

[Zij•–c{le[d} ['sh¢¦fk}Zsh;lerpsh›–›œ[dr@?™leWz[‰yzrpshy@sKce•²lp•œsK¼•–c4gce[d|1lpskYfƒrp³š´µsh1lpWz[I}ZshY\yzgjl~fUlp•œsK1lerp[Z[sh¢>leWz[

cenjcolp[ZY¶&leW[xY\sUh[dc lpWfUl™fƒrp[xfƒ›–›œsU˜™[d|‰gz|j[dr¦} sK;lerpsh›Fž [d} [d;ler~fƒ›–•œ¹d[d|\} sKKlpresK›}Zfh\Wz[Z}Z[„[4c{y@[d}Z•² 

§[d|fhcfƒsUK[Zrp›–fhy‰sh¢@lo˜™s'yzrpshy@sKce•œle•–shcd—hsKz[)¢£shr[dfh}~W}Zsh;lerpsh›–›–[Zr®ž “l¦•¤c¦fh›–ces'y@sKcpc{•–„z›–[les'rp[d¥;gz[®col¦f

Y\•–z•–Y\fh›2} sK;lerpsh›N´£sKrx[d¥;gz•–Ufƒ›–[Z;le›–nh—zleWz['Yfƒiw•–Yfƒ›¸y@[ZrpY\•–cpc{•–w•²lonsƒ¢ leWz[‰} sh;lpresK›œ›–[Zr¶ž*=m›–ces—}Zsh;lerpsh›

sh¢sKy[d\cenjcolp[ZYc>fhcB•–¬vh­fUlpgzr~fƒ›–›œn'}ZshY\[dcBsKgjl>˜•²lpWz•œ\lpWz[xcp} shy@[)sƒ¢leWz•¤cNce[ l{lp•œŠž>Axj¢£sKr{lpgzfUlp[Z›–nh—

ce•–} [_leW[_yresKy[dr{lp•œ[®csh¢B}Zsh;lerpsh›–›–[Zr~c}dfƒk„@['cey[®} •œ§[d|shz›–nkfƒ¢le[Zr)lpWz[Z•–rxcen;}~WzresKz•–¹dfUlp•œsK˜•œleWleWz[

y›–fhKl®—7lpWz[Zrp[\•–cs˜)f9n1les¼[Zijyzre[®cec_gzsh„ce[ZrpUfƒ„z•–›œ•œlon¯yzrpshy@[ZrelonhžV)Wz•¤c«•¤c«leWz[Yfh•œš›œ•–Y\•²l~fUlp•œsKšsh¢

lpWz[˜WzsK›œ[fƒyyzres;fh}~W7ž

“

šleW•–cyfƒy@[Zr®—7˜™[[ iwle[d|SleWz[¥;gfhKlp•²§[d|šYgj ¡}Zfh›–}Zgz›–gc_les¼sUK[Zr~} shY\[\} sKKlpresK›Bgz|j[dr'yfƒrele•¤fƒ›

sK„ce[Zrp9fƒle•–shB?7leWz[¥;gfƒ;lp•²§}dfUlp•œsKc&fhre[™yzgjl>•–;les_›œswshy‰Ygj ¡}Zfh›–}Zgz›–gc&}Z›–fKcece[dcdž&V)W[›–swshyY‰gj ¡}Zfƒ›¤} g›œgc

˜)fhcBrp[d}Z[Z;le›–n•–;lerpsj|jg} [®|\„wnk¬…®­yzrp[d}Z•–ce[Z›–n'lps'} sKy[x˜•œleWgzsh„ce[ZrpUfƒ„z•–›–•²lon'sh¢7} sh;lpresK›œ›–[Zr~cZž2ŸS[lpWz[Z

sK„jlpfh•œ\f_h[dren[Ziwyre[®cece•œK[›–shŠK•–}™˜rp•²lele[dDCFEHG

I

•œ\˜Wz•¤}~WI[hžŠžƒYfUij•–Yfƒ›wy[dreY\•¤cece•–;•œlon'sh¢Pf«} sK;lerpsh›–›œ[dr

fhY\shzŠ\f\} ›¤fhcpc)sƒ¢>} sh;lpresK›œ›–[Zr~c¦g|j[Zryfƒrele•¤fƒ›¸sh„c{[dreUfUlp•œsK}dfƒ„[«¢£sKreY‰gz›–fƒle[®|¸ž

·shrx} sK;lerpsh›¸c{nw;leW[dce•–c)yzgrey@sKce[dcd—w˜[c{leg|jn|j[d}Z•–|fƒ„z•–›œ•œlon•¤cecegz[®c™¢£shr} sKKlpresK›œ›–[Zr~c™gz|j[dryfƒrele•¤fƒ›

sK„ce[Zrp9fƒle•–sh7—PsK¯leW[I„fhce•–c4sƒ¢)fcenwKl~fh} le•¤}'¢£r~fƒŠKYI[d;l4sƒ¢8CJE GI

}dfƒ›–›œ[®|KCJELMON

I

ž=4c«[ ijy@[d}lp[d| ´o¬…®­£¶—

lpWz[xyfhr{lp•–fh›jsh„ce[ZrpUfUlp•œsK¢£[®fUlegre[xYfh³h[dc lpWz[¢£gz›–›PCFE6LQMON

I

g|j[d}Z•–|fƒ„z›–[h—Kyzre[dh[dKlp•œŠ«¢£resKY } sK;lerpsh›–›œ[dr

cenw;leWz[®c{•¤cZž “'lpWz•–c ˜™shrp³P—®˜™[™•–|j[d;le•œ¢£n«leWz[)[ izfK}l cenw;lpfh} le•¤}N¢£r~fƒŠKY\[Z;l&sh¢PC-E LMON

I

˜Wz•¤}~W•¤cB|j[d}Z•–|fƒ„z›–[hž

“ ;lp[Zrp[dc{le•–zŠh›–nh—leWz[‰[ ij•¤colp[Z}Z[sƒ¢¦fkYfUij•œYfh›2y@[ZrpYI•¤cpc{•–h[}Zsh;lerpsh›–›–[ZrmfhYIsKzŠkf} ›¤fhcpcxsƒ¢¦}Zsh;lerpsh›–›–[Zr~c

g|j[Zryfƒrele•¤fƒ›¸sh„c{[dreUfUlp•œsKlpgzrec)shgjllpsI„@['|j[®} •¤|zfƒ„z›–[hž

V)Wz[|j[®} •¤c{•–shSyzrpsw}Z[d|jgre[1´µfh|¯lpWz[c{nw;leW[dce•–c_sƒ¢)} sKKlpresK›œ›–[Zr~c~¶)re[d›œ•–[dc_sKDlpWz[fƒgjlpshYfUl~f}Zshgzj 

lp[Zrpyfƒrelpcsƒ¢&¢£shrpYgz›¤fhcdž qNresUw•¤|j[d|kc{sKYI[_WwnwyshleWz[®c{•¤c™shlpWz[«gzzsK„c{[dreUfh„z•œ›–•œlon\ce[ l~c)•œkleWz[_¢£sKreY‰gz›–fKcZ—

¥;gzshle•–[Z;lpcNsƒ¢¸fƒgjlpshYfUl~f\¬…®­P}ZfhIW[Z}Z[)lpfƒ³K[yfƒrele•¤fƒ›sh„ce[ZrpUfUle•–shI•–Klps'fh}d} shgKl®ž V)W[W;nwy@sƒlpWz[dce•–cN•–c

ygzre[d›œnc{nw;lpfK}le•¤}_fƒ|•–|z[Z;le•œ§[dcleWz[«„@shr~|j[Zr)sƒ¢>|j[®} •¤|zfƒ„z•–›–•²lonKž>=4cfIY\fƒl{lp[Zrsƒ¢&¢µfh} ld—|j[®} [dKlprpfh›œ•–¹Z[®|

}Zsh;lerpsh›c{nw;leWz[®c{•¤c„@[d}ZshY\[dc‰|j[d}Z•–|zfh„z›–[˜Wz[Z[Zh[dr'lpWz[k} sK;lerpsh›–›œ[drpc¸gzzsK„c{[dreUfh„z•œ›–•œlonšce[ l~c}Zfh©„[

}ZshY\yfhre[®|¸ž

V)Wz[1yfhy[dr•–c\sKreŠ;fƒz•–¹Z[®|±fhc‰¢£sK›œ›–sU˜xcR?¼uj[d}lp•œsK …Dyzrp[dce[Z;lpc‰lpWz[¼›œsKŠh•¤}Zfh›™ce[ lele•–zŠžTS>izfƒY\yz›–[dc

sh¢x} sh;lpresK›>yzresK„z›–[ZYccey[®} •œ§}Zfƒle•–shcfƒrp[Šh•–h[Z±uw[®}lp•œsK±‡zž¼V)Wz[Yfƒ•–©rp[dcegz›²l‰sh©|j[d}Z•–|fƒ„z•–›œ•œlonš•–c

yre[®c{[dKlp[d|1uw[d} le•–shv—zfh|˜[} sK} ›–g|j[«•–¯uw[d} le•–shVUjž

WWYX+Z\[&][&]

(7)

'‰& k ™' ™ !

Ÿš[fhcpc{gYI[«ŠK•œK[Zf‰§•²lp[«ce[ lsh¢&[ZK[Z;lpc—zf‰§z•œle['ce[ lsƒ¢ yzresKys;c{•œle•–shc\—zfƒ|fh•–j§z•œle['ce[ l

sh¢>9fhre•¤fƒ„›œ[®c "!$#%! &'&'&(;ž “ ¼lpWz['¢£sK›œ›–sU˜•œŠ—˜™[|j[Z§z[leWz[I¥Kgfƒ;le•œ§[d|¯›œswshy¼Y‰gj ¡}Zfƒ›¤} g›œgc

´OC-E GI

¶—™leW[¯›œswsKy Y‰gj ª}dfƒ›¤} gz›–gc¼´3E GI

¶\fh| leWz[¯Y‰gj ¡}Zfƒ›¤} g›œgc¼´3E

I

¶I˜Wz[dre[1lonwyz•¤}Zfh›[Z›–[ZY\[Z;lpckfƒrp[

) ! )+*

—re[®c{y7ž-, G !.,

G

*

—fh|re[®c{y7ž+,%!.,0/Fž

1»24345Ž 5µÀ37698$:¸½;3 Ž®=< À?> CJE

G

IA@CB

\ " $*$&+'2+" $@D\E + B+ !#" +

FHGI? A '23& C-E

G

IJ

FLK 6" 3*P 7NMDD\ ',+ ''O

PQR , G &

)TS.U-)+*+S)-*WVX)4Y%S

,

L&' Q Z Q'[

S R \(]G^ , G " $&+'2 $ _R + !#"R% " a` Gb

+c Q +PB+"_d $'+ F ) R C-E GI J

`fe FLK ) *

) Y

+'*$&+'2+" C-E GI J

FLK *

, "@ $&+'2R$ D\1', &#" + FAg

B

\@ + !#"@ 3*Phjik '2" 2" _H' &#" 7NMB. +c DD\ ++4

$&'2 "l

[

$&+' R ORD\B"& $8$&'2 +" $DD\ P@ &#"D FmG? '23&

E GI J

F4K 6" 3*P 7'MD_ ',+D+''O

n

S_opS S l [

S.U , G

*

Sq Nrs,

G

* S, G

* V , G

Y

St

"&, G

* J uK

L&'

o R F R R 8+RD\$&'2 " , G

*

+P , G

Y

', E G

IvJ

FLK gpw ',+9+&'8

', ' O &"' _+F4x.\3"H$&+'2+"y+9+DD+c +" "',E ', D\+0 D$&+'2

t

C&, J

uK &'" .&'K 9 7 ' $ j + 4+ "M+D7+" U , J uK g

B

" H $6$&+'2 " $ D\1' !#" F *P+ E I J

F4K *" D\$&+'2 " $ D\ P

+ !#"@+ F _+1+?M ++4 $&+'2D+!"l

[ g

z { '2p|y\}l

[u~

€

)

)+*")4Y

‚ C&, J

uK QR , G &)

',&" \ 39+&„M@$&+'Dj + n al [

q

€r

U-)

U-)+*VCU-)4Y t

"&

U , JU uK

+P P=Q{R

, G &

U-) gIz

… '2 "

[

†

\}

[

† +P

)+*p‡ˆ)LY

'2" 9 ‰M $&'

q

'r n ~

€

|y

U-)-*+VŠ)4Y

g

Ÿš['}Zfh›œ›

" &P2"

fh›œ›7¥;gfh;le•œ§[d|Ygj ¡}Zfh›–}Zgz›–gc™¢£shrpYgz›¤fhc™˜•œleWzsKgjl¢£rp[Z[Ufƒrp•–fh„z›–[dcdž

» ±jŒ‹ 6 B

\V"M+ O x $ CFE

I 2" +ŽE + 3H&'"V+ ‰M +3"24 Lx#++ &'2 "

\"23'+3" _pE + 3H&'" OB',+c .+9+ ' "& P +&!#" "" &" g

B

pE + 3H!$', $', D& H$ C-E G

I " D\', + !#" g B

6$'' $ CJE GI _+

\"23'++HE& '" " D\E& D+ !#" .46DŠhk g

V)W[m¥;gfƒ;le•œ§[®|Ygj ¡}Zfh›–}Zgz›–gcZ—;fhcfŠh[dz[Zr~fƒ›–•œ¹®fUlp•œsK‰sh¢¸leWz[xY‰gj ¡}Zfƒ›¤} g›œgcd—;•–cfƒ›¤c{sŠK•œK[Zfƒ•–;le[dreyzrp[  

l~fUlp•œsKsUK[Zr|j[ lp[ZrpY\•œz•¤c{le•¤}_ler~fƒce•²lp•œsKc{lerpg}lpgzrp[dc}Zfh›œ›–[d|

'&"" &"

•–¬…U­¦ž

1»24345Ž 5µÀ3‘7’€yzrpsj} [dcpcxsK"FIG“? "R3 P•” q

FW!$–+!$— ˜™!›šœ!,Er AL\ ', – "D\ " &8$

"2+2" — ˜ R – " D\B 4+>"2OON šŸž –¢¡u

† – "B\'234$ 34 + D\ ler~fƒce•²lp•œsK

¢£g}lp•œsK +P E¢ž€–

†¤£

J

FLK D" "+&" "27&"& H$',\+"234\" g

B

P',&""\” " §z•œle[ $ – "16 " } sKYIy›œ[Zle[ $1$&' + J

!$—ŒK R ¥¡]– š J

—™!$=K "

.46 g

¦D§QW;¦_¨

(8)

sKYIy@shg|yzresj}Z[dcpc{[®c™}dfƒ„[„gz•œ›œlxgzy1„wnkcen;}~WzresKzshgcyresj|jg} ldž

1»24345Ž 5µÀ3

B

´Qc{nw}~Wzrpshshgc~¶Nyresj|jg} l $ 6 ', 2"" 2"

” * q F *

!$–

*

!$— ˜

*

!›š

*

!E

* r +P”

Y q F Y

Y

!— ˜

Y

!.š

Y

!,E

Y r + "d 0" &3" F *

+P F Y "D\

P', &""a”

* ¡ ” Y q FW!–

*

¡f–

Y ! J

— ˜

*

!— ˜

Y

Kœ!›šœ!,EIr + FC‘F

* e F Y

"2 D\+O

J— /*

!$— /Y K R š

J.J

— *

!$—

Y

Kœ!›=K

L\ 9 '

— /* R š * J— *

!›=K

— /Y R š Y J — Y

!$cK›K

E J— *

!$—

Y

KA E * J— * K e E Y J— Y K'g

“ šleW[c{[®¥Kg[Z›F—2˜™[\ceWfh›œ›N•–yfƒrele•¤} g›–fhr_Yfh³h[IleWz[yresj|jg} l'sƒ¢fyzresj}Z[dcpc«shHF ˜•²lpWfhzsƒlpWz[Zr

´Q} sKYIy›œ[Zle[®¶Nyzrpsw}Z[dcpc>shf|j•–cµtosh•–;l™ce[ l™sƒ¢¸yzrpshy@sKce•²lp•œsKc ` •–sKrp|j[dr>les‰sh„jl~fƒ•–f'ce•œY\•–›–fhr¦yzrpsj} [dcpc>sK

F e\` ž&V)Wz•¤cN•¤cBlpWz[x˜)f9n•–˜Wz•–}~W CFE

I

˜•œ›–›z„@[mfƒyyz›œ•–[d|Ilps'c{sK›œK[x} sKKlpresK›jyzrpsh„z›–[ZY+´µce[Z[mV)Wz[dshrp[ZY …

uj[d}lp•œsK‡;¶ž

1»24345Ž 5µÀ39’ ›¤fƒ„@[Z›–•œŠyzrpsj} [dcpcNsK ` G¢? " "2 ‰MDD+ O', 2"" ` g -\$&'

M P',&""A” q

FW!$–+!$— ˜ !›šœ!,EIr _ F +"_d J$',+¥`•” ¡ "+R ›¤fƒ„@[Z›–•œzŠ\sh¢ ” ´£„wn

¶™sh ` gXz @ EA .&P+@D\@" &$ .7&!c @', 2"" 2"X` g

msƒlp•–}Z[4lpWfUlx›¤fƒ„@[Z›–•–zŠ ” sh‰fhY\shgz;lpc™lps\gzj¢£sh›¤|yzrpsw}Z[dcpc ” ž

Ÿš['}ZfhsU˜M|j[Z§z[«leW['c{[dY\fh;le•¤}Zc™sƒ¢ leWz[›–shŠK•–}hž

1»24345Ž 5µÀ3 8$:7» ± 3 Ž 5‘K À>CFE

GI @…B

\ O ' '&+ 4+B$6_P$&+'2 "H CFE

GI J

FLK "6'&39+

V P', &""” q

FW!–-!$— ˜™!›šœ!,E r VV9 !+ 4+ ž y€

† £ J

–WK'g

B

"B&' P',&+ 4+

)

[

JG –WK " 3*PD& 9+&‰MR "*$&+>" g

B

\ O ' ', + 4+ P=Q“R , G &

)

[

$ D\sE + 3HR$&'2 PQ R , G &) L\ ',

, G R E G

I J` K "@_@" $ "2OO&" — R – "2$VD\+HD\ ', ›x."24"R+ O P', &""! q

` !#"{!%$˜€!.š/!,E%/;r R EA B9 !+ 4+& '/ ž€y€

†¤£

J

"yK "2 D\+O

J

—€!%$

˜ K R

)() [+*-,.

*0/

+P

$ ˜ R , G

[212

/

L\ ',

J -¡&"yK

J

uKW '

J

uK¡3" $&'+M R g

_@ O ' '&+ 4+ $ D\+D\ '*$&+'2 "@".46 7'M™O

n

[

“–

o4 [

7Œ— R – So R E J

—ŒKœ(

l [ [

mŒ—

R – Sš J

—€!›=KW—(

[

J

uK

U-)(

[5

‘–6

)

[

q

Nr

)

[

mŒ—

R – SP — / žš

J

—€!$cKA‘— /

— / R

)(

[5

(

)+*-VŠ)LY7) [

)-*8 [

e

)LY7) [

t C&

) J uK

[

:9A G – S

);

[=<?>-@#ACB?

GI{(

WWYX+Z\[&][&]

(9)

B

9+++ 4+

.&"D0 _D" 42" $ " &P

) R CFE

I

C6X '2O

” S )

L&P&9+&'_@ 34-"2OO $a” 7 "

)

g

msƒle•¤} [™lpWfUl>lpWz[c{[ZlNsƒ¢Y\sj|j[d›–cBsh¢@fƒwn CFE

GI

¢£shrpYgz›¤f4•–c>}Z›œs;c{[®|gz|j[Zr>„z•¤ce•œY‰gz›–fƒle•–sh7ž&ŸS[x|z[ §z[

sU˜TC-E LMON

I

—UlpWz[)¢£rpfhŠhY\[Z;lBsƒ¢-CJEHG

I

¢£shr>}Zsh;lerpsh›wgz|z[ZrByfƒrele•¤fƒ›wsK„c{[dreUfƒle•–shrp[d¥;gz•–rp[ZY\[Z;lpcd—ƒfhcBceWzsU˜

•–¯uw[d} le•–sh1‡zž

1»24345Ž 5µÀ38$:¸½;3 Ž®=< À> CFE6LQM N

I

@9B

" R $ $&'2 +" F G $ CJELMON

I '23&

CFE6LQMON

I J

FLK "@.46 7'MDD\ ', D+''O

P=QŸR ,



J K &

)TS.U-) * S) * VX) Y S , /

L&' Q Q [ S R (fG " "7&" & $

J

K " D\ $&'2 Š‚ #A&

[

~

™_#



@

L J† ‡ l K , "8$&'2 R $@_P',@+ !#"@+ " &4`

G¢?

+ Oc Q

"_d +$', F ) R CFE6LQM N

I J

`ye FLK ) * +P

) Y

',$&+'2 "8 C-ELMON

I J

F4K > ,0/ "0*$&+'2

$ _P', +&!#"@+ FAg

$%'‰2#B'±)# 1#&& '±'/!&)#&3_«& '

V)W•–c\ce[d}lp•œsK±yzrp[dce[Z;lpc‰Ufƒrp•œsKgc‰[ izfƒY\yz›–[dc‰sƒ¢m}Zsh;lerpsh›™re[®¥;gz•œrp[ZY\[dKl~c•–%C-E6LQM N

I

ž±·&•–r~col®—¦f¼lprpfhco 

¢£sKreYfƒle•–shsƒ¢)¢£shrpYg›–fKc•–c‰|j[ §z[d|¸—B˜Wz•¤}~W©gce[d|šlps¯›œ•–z³ CFE6LMON

I

˜•œleW(}Zsh;lerpsh›–›œ[drpc«g|j[Zr‰yfƒrele•¤fƒ›

sK„ce[Zrp9fƒle•–sh7—Kfhc>ceWzsU˜\„;nIV)Wz[ZsKre[dY …jžNfƒrp•¤fƒ;lpcBsh¢@lpWz[mV)Wz[dshrp[ZY%fhre[™lpWz[Z[Ziwy›œsK•²lp[d|‰les}dfƒyjlpgzrp[

rp[d¥;gz•–rp[ZY\[Z;lpcd—¸c{g}~WšfKc D'x.D+‰M '2 ""29 R+ 3', 'D V_ + 3', '"P ' +'2 4+

.72" '29 34 fƒ| . ',+R+ ',&'" ž

1»24345Ž 5µÀ3 "! +' +?MB" & O

) R CFE LQM N

I J

FLK 8 $&+'D+?M# # [%$ R (WGŠ 8_&#  ª›–•²¢l

sh¢

) "@_*$&'2 )('

# R C-ELMON

I JF e)#

K

6P+ 39+&„M 3* 7NMXO

n ' # n

o*'

# o

' # ¢

J

U-)

K ' # U J

)('

# K

J

)+*+VŠ)LY K ' #

)+*'

#

VŠ)4Y+'

# Jq 'r ,LK

' # # [  q 'r

J, ' # K

Jt

"&,LK

' # t C&

J, ' # K J

P=QR ,

,

J Kœ&

) K ' # P=QR ,



J Kœ&

J

)('

# K

1»24345Ž 5µÀ3-/. 9+& ', 2""p” q

F%!$–-!— ˜™!.šœ!E r " 9+& #[%$ R \(sG F D\0#  

yregz•œzŠ1sh¢ ” "DD\ ', 2""Ÿ” <2143 B q

FW!–-!$— ˜™!›š /!E r "2KD\+Q$&' + — R – R

š /J

—€!$cKW‘š

J

—€!$cK $(#[ R E J— K +' š/J

—€!›=K " P 3*P +D\ 'œ "& g

± 66! ' +?M '&""\”)+ F Q$&'RM7# # [ S R ("GF @$&'RM " &P

) R C-E6LQM N

I J

FLK W6\\+9+ O

)

%8:9 3;

)<'

# g

=  À7À>auwler~fƒ•–ŠhW;l{¢£sKre˜)fƒr~|\„wnk•–|jg} le•–sh1sh

) ž

¦D§QW;¦_¨

(10)

sKKlpresK›cen;;lpWz[dce•–cx¬ —w…j—h‡—9­@•–cf«Šh[dz[Zrp•–}™yzrpsh„›œ[dY%lpWfUl}Zshce•¤col~cZ—ƒŠK•œK[Zf«yzrpsj} [dcpc

”

}dfƒ›–›œ[®|If

y›–fhKl™fƒ|f«yzrpshy@[Zrelon

)

—Kles'[dj¢£shr~} [lpWz•¤cNyzrpshy@[Zrelon„wnIY\fh³w•œzŠ«lpWz[xyzrpsj|jg} l¦sƒ¢PleW[myz›¤fƒ;l¦˜•œleWfh

fK|z|j•œle•–shfh›@}Zsh;lerpsh›@c{njc{le[dY leWfƒl™rp[dc{lerp•–} lpc•²l~c¦„@[ZWf9;•–shr~cdž •¤c™}Zfƒ›–›–[d|kf

+ ',&' +…” $&'

) —

fh|kleWz[ŠKsKfƒ›¸•¤c™lescenw;leWz[®c{•–¹Z[«c{g}~W1}Zsh;lerpsh›–›–[Zr®ž&Ÿš[«¢£sj} gcsKklpWz['} ›¤fhcpc)sƒ¢B} sKKlpresK›Pgz|j[drxyfƒrele•¤fƒ›

sK„ce[Zrp9fƒle•–sh,¬…w—N‡ƒ­ ?¢£shrfSŠh•–h[Z yfƒrele•œle•–sh sƒ¢+•–;lesšgzzsh„c{[dreUfƒ„›œ[[ZK[Z;lpc fh|±sK„ce[Zrp9fh„z›–[

[dh[dKl~c

leW['} sh;lpresK›œ›–[Zr Ygc{lxcefƒle•¤co¢£nlpWz[›–s;sKyj ªYgj ¡}Zfh›–}Zgz›œgc¦¢£sKreY‰gz›¤f

J

»wÀ  » /. 9+& ', 2""” + F 8 "&

G9 0+ 6R" &P2"

) R CFE6LQM N

I J` e FLK

, R E I J` K L\ ',` +P F +',+"_d + H_6$& c "" &'2 4+"+',E&9++ O

B

\ ', "@ + ',&' "&3"$M+c , 

J

K "2 D\+0” ¡

S )

” S P=Q R , ' Q



J Kœ&

)('

Q L&', Q • Q [

S R \( "D"& 1 $ $',&"›P', "2 4+\" g

=  À7À> · •–rpc{ld—7cegzyzy@sKce['lpWfUl«lpWz[Zrp[‰[Zij•–c{lpc«fyzresj}Z[dcpc sK

`

cpfUle•¤c{¢£n;•–zŠ

J

K  , fƒ|Sc{g}~W

lpWfUl

”

¡

S )

ž_•–h[d Q Q

[+$

R \("G 6 J

`se

F4K— ˜[}dfƒ[dfKc{•–›–nS|j[Z§z[f1›¤fƒ„@[Z›–•œzŠ

yresj} [®cec

R EA- cefƒle•¤co¢£nw•–zŠ

J

K‰fh|±ceg}~W©lpWfUl

J K <21B •–c‰leW[kyzrpsj} [dcpc ˜•²lpWzshgjl Q ž

msU˜—

J” ¡ K

<21 B

shr[®¥;gz•œUfh›œ[dKlp›œn ” ¡ J K <1 B

cefƒle•¤co§[dc

)

—ce•œ}Z[ •¤c

J K

<21 B

˜•²lpWzshgjl Q fh| Q

|zs;[®c)zsƒlxsj}d} gzr)•–

)

ž6A4c{•–zŠ7[dYIYf]h—j˜™['} sK} ›–g|j[_lpWfUl

J” ¡ K S )<'

Q fh|leWfƒl S

‘,

' Q ž

ujgzyzy@sKce[zsU˜,lpWfUl ” S P=QR , ' Q 

J KN&

)('

Q ž™n [ §z•œle•–sh Uz—jleW[Zrp[_•¤cxf\›¤fƒ„@[Z›–•œŠIyresj} [®cec

R EA-mceg}~W1lpWfUl S

J

K  , ' Q fƒ| ” ¡ S

)('

Q ž_V)Wz[Z

<21 B S

J

K4fƒ|1„wn

2[ZY\Yf]K—

<21 B

cefƒle•¤co§[dc4,fh|

J” ¡ K

<21 B

cefƒle•¤co§[®c

)

ž>V&fh³h[ fKc

J K <1 B

les} sK} ›–g|j[Kž

Ÿš[•–›œ›–gc{ler~fUle[\lpWz[gce[sƒ¢ CFE6LMON

I

les¯cey@[d} •œ¢£nS}Zsh;lerpsh›>rp[d¥;gz•–rp[ZY\[Z;lpcdžkV)Wz[

#

 F›–•œ¢l‰•¤c[ iwle[d|j[®|

lpskfƒwn1} sKUtogz} le•–shcm„;n

) ' J#

*+

# Y

Ka

J

)('

# * K ' # Y

žŸš[›–[ l

J

,4Ka

‚

#%&

[

~

€_#



,(fƒ|

!#"$&%('

J

,4K%

t

"&*)

[

q

Nrs

V

,Nž

‘ »c3 ŽZU 5,+K» Á ‘ À3 ŽZ À » U >ªÀ 

)-

V)Wz[¢£sKreY‰gz›¤f©´o]®¶'[ ijyzrp[dcpc{[®c«leW[[Ziw•¤c{le[Z} [sh¢lo˜sD}Zshj 

lpresK›œ›–[Zr~c.

*

fh|

Y

sh ” ceg}~WlpWfUl ” ¡/

*

¡/

YS

)

fƒ|¢£shr)[dfK}~W0421™!43j—5 }dfƒsh„ce[Zrph[®c

sKz›–nleW[c{[Zlmsƒ¢B[ZK[Z;lpc

5 G‘'—@fƒ|¼}dfƒ¼|j•¤cefh„z›œ[®csh›œnkler~fƒce•²lp•œsKc›¤fƒ„@[Z›–[d|„wnf¼´Q} sKKlpresK›œ›¤fƒ„›œ[9¶

[dh[dKl•–76 5 G

” S P=Q

* R

J

* K  J 8

J @:9<;

†

K.K

' Q * &

P=Q

Y R

J Y K  J

8

J @:9>= † K.K

' Q Y &

)('

JQ *  Q Y K

´o]®¶

=< 5±½?¦»  5® 5,@» ‘ À3 ŽZ À » _ À3BA Ž » ‘ À?3 Žd À » U 3 Á »  ? j9Ž 5 À ŒB »  @ jŽ 5QÀ?3

>ªÀ 

)C-

= } sK;lerpsh›–›œ[dr Q ¢£shr

)

•–c D'x.D+‰M &'2""9+ •œ¢sshleWz[dr_}Zsh;lerpsh›–›œ[dr Q /2¢£sKr

)

}Zfƒš} gjl

c{lerp•¤}le›–n«›œ[®cec&lprpfhc{•œle•–shc¸leWfƒ Q ž&Ÿšre•œle•–zŠ QD Q /KleW[™Y‰gj ¡}Zfƒ›¤} g›œgc7¢£shrpYgz›¤f

J J [

Q [ ‡ Q /[

K›K

'

Q  J

!#"$&%('

J) [

U Q [  Q /[

K.K

' Q /

leW•–crp[d¥;gz•–rp[ZY\[Z;l•–c)[Zijyzre[®cece[d|„wnleW[«¢£shrpYgz›¤f´F…h¶

” S P=QR

J KN&

)('

Q  ~ƒ Q / R

J

Kœ&QD Q / ‡ U J

)(' Q /

K ´Q…h¶

WWYX+Z\[&][&]

(11)

™ Z

t

V)W•–c\ce[d}lp•œsK(•¤c\|j[d|j•¤}Zfƒle[®|±lpsDlpWz[1YIsj|j[d›² ¡}~Wz[®}~³;•–zŠ¯sh¢ CFE6LQM N

I

ž(Ÿ.•²lpW V)W[Zshrp[ZY‡

˜™[yzre[®c{[d;l

f1cegj°} •–[Z;l'fƒ|šz[®} [®cecpfƒrpn¼} sh|j•²lp•œsKšles¼|j[d}Z•–|z[\leWz[\Y\sj|j[Z›œ ª}~W[d}~³w•œŠfh|š˜Wz[ZSleWz[}Zsh|z•²lp•œsK

Wsh›¤|zcZ—@V)Wz[ZsKre[dY •–cmgc{[®|1lesc{nw;leWz[®c{•–¹Z[‰} sh;lpresK›œ›–[Zr~cZž)V)Wz•¤c4}Zsh|z•²lp•œsK¼•–cm„fhce[d|1sK1leWz[‰shr~|j[Zrxsh¢

¢£sKreY‰gz›¤fhc

J

Ksw}d} gzrprp•œzŠI•–shrpY\fh›@¢£shrpYsƒ¢>c{[d;le[Z} [dc´ [Z§z•œle•–sh¼‚K¶ ž

1»24345Ž 5µÀ3 8¼À  ± >FÀ 

@TB

\ " $0" &P&" $

CFE LQMON

I J

FLK

zshrpYfƒ›¢£shrpY

" _ " &

$ " & O &"NO

* Q * R ,

*+

J

* K 

&'&'& Q R ,



J Kœ&,

L&'$&' 0 R 1€!'& &'&œ!-( O 5-R P !ƒ ( , 5+R E I J` 5 K > , R E I JF e`

*

e e` K'gw ',+9+&'

$&' + 0 R 1€!'&'& &N! 1(

5 }

5

* }: +P 5 5

* „MB_+

5

* " +1 P', '

"27&" &$ 5 g

“ l'}ZfhDsK„;w•–shgce›–n¼„[ceWzsU˜¼leWfUl[dfK}~WSce[Z;le[d} [Isƒ¢8C-ELMON

I

}dfƒD„@[\˜re•œl{lp[Zš•œšzsKreYfƒ›&¢£shrpY

´Q¥;gfƒ;le•œ§[drpc)sh¢2leW['cefhYI[_lonwy@['} shY\Y‰gjle[9¶ž

»wÀ  » ¥8  53  »  £Ž

@ g F

) R CFE LQM N

I J

FLK 7_ " & O

KP+'2D+#$&+'2 * Q * R

, * 

J

*

KN&'& &'&'& Q R , 

J

Kœ&,Ag

B

',.7&& O \ L\ _&' '&""p” "&3"H2"

) " . 4 .7&V$ +„MD$ J

6\K O D\

"27&" $E $

J 5K

*

5 $ P+P& M"& 3" " P&',+"c g

! "$#&%(') *,+.-/#1023*54768#&9;:

Ÿš[\gce[‰lpWz[Ilo˜sYfƒ•–š}Zshc{lerpg} le•–shc_sƒ¢_¬…9­>les¼c{WsU˜ leWfƒl«leWz[} sK|j•œle•–sh

J

6\K_sh¢™V)Wz[dshrp[ZY ‡•–c

cegj°} •–[Z;l®ž\V)W[dce[dc} sKc{lerpg}lp•œsKc«fƒrp[IŠK•œK[ZS•œ ¬…9­¦fhcfhgjlesKYfUlpf} sKc{lerpg}lp•œsKcZž\qNrpshy@sKce•œle•–shDv

fh|1qNrpshy@sKce•²lp•œsK U\Šh•–h[«lpWz[Z•–r_›œsKŠh•¤}Zfƒ›2h[Zr~ce•œsK1ce•–} [lpWz[Y‰gj ¡}Zfƒ›¤} g›œgcmfƒ|1lpWz[yfƒrp•²lonfƒgjlpshYfUl~f

fKc)˜[d›œ›7fKc™leWz[›–swshyj ªYgz ª}dfƒ›¤} gz›–gc)fƒ|kleW[yfƒrp•²lonk›œswshy1fƒgzleshYfƒlpfIfƒrp[_[®¥;gz•œUfh›œ[dKl¬…j—¸]Z­Fž

=  À?¦À  5Ž 5µÀ3 ¥8< À Ž 5µ»c3 Ž À> E G

I

À@»  ?  À ‘ » d

@

&?” 7 H'&""0 F F , G R E G

IJ F e

` K

7 " & O +B"2 " _+-` +P

F

+',+"_d + H"& 3"

g

B

&',D$x."24" " &P

, G>=

” $@D\ P &#"+

F

1+RD\{E + 4& 1$

, G +9+ '

”• " D\$&' +?M ', 2""

£

+

F

W6Ÿ\+9+ O

£ S

‘, G = ” $@+P ‰MD$ £ ¡ ” S

‘, G

=  À?¦À  5Ž 5µÀ3 8< À Ž 5µ»c3 Ž À> E G

I

À@»  E I @ , R E I J` K +PV& , G R E G

IJ

F eC` K 7

" &P2"@+P "2 " _+-` +P F +',+"_d + H"& 3" g

B

&', ›x."24" " &P , G?= , $ _+ !#" F 8R_{E&34 $ , G +9+ '

, " D\$&' +?M ', 2""

£

+ F W6Ÿ\+9+ O

£ S , G = , $@+P + ‰MB$ PA@ " gg @ S

‘, +P

£ ¡ @ S

‘,

G

¦D§QW;¦_¨

Références

Documents relatifs

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 -

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP

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 -

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 -

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 -

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP

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 -

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 -