HAL Id: inria-00072640
https://hal.inria.fr/inria-00072640
Submitted on 24 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.
About Folding-Unfolding Cuts and Cuts Modulo
Gilles Dowek
To cite this version:
Gilles Dowek. About Folding-Unfolding Cuts and Cuts Modulo. RR-4004, INRIA. 2000. �inria- 00072640�
ISSN 0249-6399 ISRN INRIA/RR--4004--FR+ENG
a p p o r t
d e r e c h e r c h e
THÈME 2
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
About folding-unfolding cuts and cuts modulo
Gilles Dowek
N 4004
Septembre 2000
Unité de recherche INRIA Rocquencourt
Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France)
Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30
#%$'&(&')+*-,.0/1)3254
6879;:=<>@?BADC;EFG<IHKJML+FKNOFG<;H
<OPQNORSHTNUVHXWZY[:@\]JMHKFT^+UV<
_a`bJScd<OPQeXJML+FgfRMH
hiRSjVjkJ+`ZPil0<I`b<;Nm7<;`bNm7V<nE]oqp+r+rSp?Bs[<;j0Pt<;:@\`b<@>Sr+rMrI?vu;r=jVRMLM<;W
wxqy;zO{|}MzM~% <"WZ7JFGEPt7FTWIEJMPt<=Pt7kRPNU0PI<;HGFK:-FGEVRSPtFKJMEFGEl0<l0UVN PbFGJ+E:-J0l0UHKJ%WZUV\VWZUV:=<WINU0P
<;HGFK:-FGEkRPtFKJMEFKEl<;l0UVNPtFKJMEFGPt7Pt7<J+HKlFGEL"RMEVlUE0JMHTl0FKEL-`tUVHG<WO
%3b@
{qyM~
fUPi<OHKFG:-FKEVRPbFGJ+EXVl0<l0UVN PbFGJ+E:=J0l0UVHGJk[JMHTl0FGEVLVUEJMHTl0FGEVLV
at mO3K;tObO ¢¡£Z¤0T¥ £§¦¨ ¨m©Vª« « b¬bb¤['¢¡£Z¤0T¥ £;«;®O¯OtOb
y ~ E :=J+E3Pt`b<^3U<HC;HGFK:-FGEkRPtFKJMEnl0<W§NJ+UjU`b<;W<;E lC;l0UVNPtFKJME :-J0l0UHKJ LMCOEVCO`mRSHKFKWt<
HCOHKFG:-FKEVRPbFGJ+El0<WiNJ+UjU`b<;W8R+<;NDl0<;W`b9OL+HG<W8l0<IjHKFKRMLM<<Pil0<l0C;jHKFKRMLM<M
zy
}
y~
HKFG:-FKEVRPbFGJ+E%l0<;WNOJMUjUV`t<WOl0Cl0UVNPtFKJME:=J0l0UVHGJkjHKFKRMLM<MlCOjHKFKRMLM<+
'¢E)(k`bWZP+*JM`ml0<O`IEkRPtUV`bRMH l0<;lUVN PbFGJ+EX R§NU0P=FKW1R%WZ<^3U<OEVNO<"JM`b:-<;lFGPt7 RMEFGE3Pt`bJ0l0UVNPtFKJME `bUHG<
J+HGHKJ<;l\[YRSE<;HGFK:-FGEkRPtFKJME1`tUVHG<+687FTWqEJSPbFGJ+E@NORME@\]<<-,3Pb<OEVl<;lIPbJ l0<RSH[FGPt7Pb7<8R,0FGJ+:"W5JMkWZJ+:-<
Pb7<OJ+`tFK<;W;<MLVDPdY[jk<1Pt7V<OJM`bYM Wt<PDPt7<;JM`bYM]Pt7V<"s[Pt`mRPtF(V<l/.J+UEVlRSPtFKJMEVW; GKI_a`mRFP1032Ku;r54 j`bJMj]J+Wt<;W
R `bRSPt7<;`UEVFJ+`t: 8RY PbJ <-,3Pb<OEVl Pt7<EJSPbFGJ+E JM@NU0P\[Y6(k`bWZP<-,3Pb<OEVlFGEL7(V`mWdP8*'J+`bl0<;`EVRSPtU`mRSH
l<;l0UVNPtFKJME FPb7 PdJ`bUHK<;W9 RSEVl ;:N;RSHKHG<l=< >@?8%>-A RMEVl6< +BC#DE@A
\[Y§_ `bRFGP805FRSEVlPt7V<OE NJ+EVWtFKl0<;`tFKELR%WZ<^3U<OEVNO<-JSWZUVNm7`bUHK<;WIRMW@REV<O JM`b: JSNUP :Wt<O< RSHTWtJ
2>[G0Vu+ & IH4JF <i7VR+<`t<N<OE3PbHGY=j`bJMj]J+Wt<;l-RSEJMPt7<;`a8RYPtJ@<,[Pt<;EVl"l0<;lUVN PbFGJ+E NORSHKHK<;l B>@A
#K% 2ML0N4X7<;`t<Dj`bJMj]J+WtFPbFGJ+EVW<;^3UFRSHK<OE3P:-J0l0UHKJ"R1NOJMEL+`tUV<OEVNO< RS`b<DFKl0<;E3PtF(V<;l O'dl<OE3PtFGY[FGEL
jV`tJ+jkJ3WZFGPtFKJMEVW5Pt7VFKWq8RYD<-,[Pt<OEklWqPt7<EJMPtFKJME@JMkNUP; <8Wt7J FKEPt7VFKW EJMPt<Pt7VRSPal0<;l0UkN PtFKJME1:-J[lUHGJ
WtU\VWtU:-<;Wl<;l0UVNPtFKJMEFGPt7Pt7< RMEVl `tUHK<;W;
P Q SRmUTUT WVYX[Z]\_^a`cb ed `fVYXgZ]\O^a`cb
'¢EPb7FKW8EVJSPt<+<NOJMEVWtFTl0<O`iRD(h,0<;lPt7V<OJM`bY3i JM`b:-<;lFPb7%R,0FGJ+:"W
j!kmlDnnnjEko/pqUrUsutvr@w
7V<O`b< Pb7< q r WRS`b<IRPtJ+:-FKNDj`bJMj]J+WtFPbFGJ+EVW8RSEVlPt7V< t r WRS`b\FGPt`mRS`bY-j`tJ+jkJ3WZFGPtFKJMEkWO
<iWtRYPb7VRPaRDj`bJMj]J+WtFPbFGJ+EDx 5 PtJIRSE-RPbJM:-FKNj`bJMj]J+WtFGPtFKJMEDy :`t<WZj5XPt7kRP[y 5 PtJvxDF
FGOy{z}| qUr RSEVl~xz| t r JM`iWtJM:-<R,0FKJM: jEkElDnnnhj!k!opqUrUstvrw JSOi RSEVlWtJM:-<IWZUV\VWdPbFPbU0PtFKJME
|V < R+WtWtU:-<8Pt7kRPi FTW WtUVNm7-Pb7VRP8RSE RPbJM:-FTN8j`tJ+jkJ3WZFGPtFKJME"UVE0JMHTlW PbJ@RP:=J3WdPJME<ijV`tJ+jkJ3WZFGPtFKJMEX
687<R,0FGJ+:-WJM_i NORME\]<I`b<OjHTRMNO<;l\3Y Pt7<l0<l0UVNPtFKJME`tUVHG<W
/K| t r
/3| qr
/3| qr
/K|
t r
RMEVl1FP FTW <;R+WZYPbJNm7V<;NPt7kRP RDWZ<^+UV<OE3Pgi8 "y NORME1\]<8j`bJ M<;l@FGED(V`mWdP8*'J+`bl<O`qEVRPbU`bRMHl0<;lUVN PbFGJ+E
FGaRSEVlJ+EHGYF Pb7<@Wt<;^3U<OE3P ~ynNORME%\k<jV`tJ +<;lFGE%l0<l0UVN PbFGJ+E%FPb7Pt7< RSEVl %
`bUHK<;W;'¢Ekl0<O<l RSE[YFGEkWdPmRSEVNO<JSQPt7< RMEVl `tUVHG<W1N;RSE \]<WtFK:@UHTRPb<;l UVWtFKELRME
R,0FGJ+: JSi RSEkl Pb7<IR,0FGJ+:"WJMi NORME\]<j`tJ +<;l"FKEl0<;l0UkN PtFKJMEFGPt7Pt7< RSEkl
%
`bUHK<;W;
s[FKEVNO<RMERPbJM:-FTNQj`bJMj]J+WtFGPtFKJME UEJMHTlWPtJ-RP8:=J3WdP8JMEV< j`bJMj]J+WtFGPtFKJMEX[R1Wt<;^3U<;EVN< JM`b:-<;l"FGPt7
R RSEklRME
%
`tUHK<7VRMWPb7<JM`b:
nnn
/"x
"y
%
/"x
RMEVlNORSE\]<I`t<l0UVN<l PbJ
nnn
/"x
Uha
s0UVNm7R"Wt<;^3U<;EVN<FTWPt7[UVWiN;RSHKHG<lR % -
fU0Pi<;HGFK:-FGEkRPtFKJMEPb<O`b:=FKEVRSPt<;WJ+`iWZJ+:-< Pb7<OJ+`tFK<;W i\U0PiFGPQl0J[<;W8EJMP8JM`iJMPt7<;`bW;
Q !"
'¢E l0<;l0UkN PtFKJME :=J0l0UVHGJkaRPt7<;JM`bYFTW@JM`b:=<l FGPt7 RWZ<OP1JSQR,0FGJ+:-WD !;?a>BE1B J+E
jV`tJ+jkJ3WZFGPtFKJMEVW;Q<O`b<MMPt7<QNOJMEL+`tUV<OEVNO<FTW Pb7<QWt:-RMHGHK<;WZP NJ+ELM`bU<;EVN<8FTl0<OE3PtFGY[FGEVL qUr RSEVl tvr J+`a<;R+Nm7
687<1l0<;l0UkN PtFKJME`bUHK<;W8PmRM<Pb7FTWiNJ+ELM`bU<OEkN<FGE3PbJ RMNONOJMUE3P .J+`FGEkWdPmRSEVNO<MPb7< #Kh!B-h
FTW8EJMPQWdPmRPt<lR+W8UVWtUVRSH
/3y x /"y
/"x
R+WPt7< (V`mWdPj`b<O:-FTWZ<E<;<;lEJSP\]<<,RMNPtHKYy x \U0P:"RY \]<DJ+EHKYNJ+ELM`bU<OE3PPtJ=Pt7FTW8j`bJMj]J+WtFC*
PbFGJ+EX07<;EVN<IFGPiFKWWZPbRSPt<l
/ /3y
FG y x
/"x
HGHXPt7<I`bUHK<;W8JSqFGE3PbUFPbFGJ+EFTWdPbFKNEVRSPtU`mRSHXl<;l0UVNPtFKJME:"RY \]<WZPbRSPt<lFKE%R-WZFK:-FGHTRS`8RY :(VL+U`b<=u5F
- FGEl0<;lUVN PbFGJ+E:-J0l0UHKJFTW;XHGFM<-FKE/(k`bWZP+* JM`ml0<O` HGJ+LMFTNSXRWt<;^3U<OEkN<1JM`b:=<l§FGPt7 RFGE3Pt`bJ*
lUVN PbFGJ+ERMEVl RME<;HGFK:-FGEVRSPtFKJME`tUVHG<+-fU0PI<OHKFK:=FKEVRSPtFKJMEPt<O`b:-FGEkRPt<W JM`WtJM:-<=NOJMEL+`tUV<OEVNO<;W;]\VU0PFP
lJ3<W8EJSPJ+`JSPb7<O`mWO
q"@ ;[ !
<UVWt<R=UEVNPtFKJMEVRMH5EJMPbRSPtFKJMEJM`ijV`tJ[JSW;6J <RMNm7j`tJ[JMqJM R"Wt<;^3U<;E3Py l nnn8y o ~x FKE~(V`bWZP+*
J+`bl<O`qEVRPbU`bRMH0l0<;l0UkN PtFKJME5<RMWbWtJ[NOFKRSPt<RQj`bJ[JS*'Pt<;`t: 7J3WZ<`t<;< RM`tFTRS\HK<;WqRS`b<8RS:-JMEL l nnn o
_ `tJ[JSWQ\UFKHPDFPb7%Pt7V<1R,0FKJM: `tUVHG<WiUVWtFGEVL"Pt7<=R,0FKJM: y r RS`b<I`tFGPZPb<OE r ]j`bJ[JSWi\VUFGHGP FGPt7 Pt7<
FKE3Pt`bJ0l0UVN PbFGJ+ERSEVl<;HGFK:-FGEkRPtFKJME@`tUHK<;W5JS[Pt7V< FK:=jVHGFTNORSPtFKJME@RM`t< `bFPtPt<;ED< RMEVl pw jV`tJ[JSWX\UFKHP
FGPt7@Pt7<8FKE3Pt`bJ[lUVN PbFGJ+E1RSEkl<OHKFK:=FKEVRSPtFKJME@`bUHK<;WqJSVPt7<NJ+EcdUEVNPtFKJME=RS`b< `bFPtPt<OE !O" !# %$'&)( p w RSEkl
&)*+ p w j`tJ[JMW \UFKHPqFPb7IPt7V< FKE+Pb`tJ0l0UkN PtFKJMERMEVl<OHKFK:=FKEVRSPtFKJMEI`bUHG<WXJM3Pb7<lFKWcdUEVNPtFKJMERS`b<a`bFPtPt<;E
p w , p w RMEVl p.- El/0214365 w 3j`bJ[JSW \UVFGHGPFGPt7-Pb7<i<OHKFK:=FKEVRSPtFKJME-`tUVHG<iJMPt7<QNOJME3Pt`mRMlFKNPtFKJME-RS`b<
`bFGPZPt<;E p87:9 (<;>=@? w jV`tJ[JSW \UFKHPFPb7"Pt7V< FKE3Pt`bJ[lUVN PbFGJ+E RSEVl"<;HGFK:-FGEkRPtFKJME"`bUHK<;W JS Pt7V<QUEFM<;`bWbRSH
^3UVRME3PtF(V<O`RS`b<`tFGPZPb<OE"< k §RMEVl pA(w RSEkl"j`bJ3JMW \UFKHGPFGPt7"Pb7<QFKE3Pt`bJ0l0UVN PbFGJ+ERSEkl-<OHKFG:-FKEVRPbFGJ+E
`bUHK<;W"JMPt7<§<-,0FKWZPt<;E+PbFKRMHQ^3UVRSE3PbFC(V<;`RM`t<`bFPtPt<OEB .(aC # RSEVl p; k
;D=@? k
w
BE
B>@A
J+E
jV`tJ[JS*'Pt<;`t:"WFTWl0<(VEV<;l\[Y Pt7V<JMHKHGJFKEL=`tUHK<;WPb7VRPi<;HGFK:-FGEkRPt<NOU0PbWWZPt<;j\3YWdPb<OjX
p
<6F l
15wHGJI
1K
0LM
l
$'&)( p .ElC61 #wNG ml
&)*+ p !ElC21 # wOG 21
p.- p lw 1 3 5%wNGJI l)K 0LM 1
P!QPSR
! #"
%$&'(*)+-,. 0/
1)
1) 2 #"4365
7*)+-,. 0/
8 !9 "
%$&':);-,.
9
(/
1)
1) 9 "<3=5
7);>,.
9
0/
1
1) 9 "<3=5
)+>,.
9
0/
?;@ "
%$&'();>,.
@
0/
1)
!@ "
%$&':);>,.
@
0/
)
1A ?) B!) @ "4365
7AC-,.
@
(/
1)
!D
"<3=5
E>
D
?
?
,GFH'I/J
"
K$&'0>>,LJMF?N/OFP?QSRT,.H/
,GFH'SVU'/J
"<3=5
E!-,LJWFN/OW);-XU'YZF[\
)
1)
,GFH'SVU'/^]
"
%$&'(!>,4]FI/OW)>XU'YZF[\
?) '8
,GFHN/^]
"4365
7);-,4]KF?I/OWF_PQSR:,.H0/
.qFGL+U`t<-uI9O` RPbU`bRMHXl0<;lUVN PbFGJ+E:-J0l0UHKJ
Uha
p8- , p l-w 1 30 55wOGJI l)K 3'L 5
p< k ( wHGJI( K5k LM
p; k
;>=@? .(aCml # k 21
wOG I( K5k
"El
K
0LM21
_a`bJ[JS WqFKE1l0<l0UVN PbFGJ+E@:-J0l0UHKJVSRM`t<`tFGPZPb<OE1R+W5FKE (k`bWZP+*JM`ml0<O`EVRPbU`mRSH[l0<;l0UkN PtFKJME5MRSEVlIPt7<8j`bJ[JS
`b<;lUVN PbFGJ+E`bUHK<;WRS`b< Pt7<WbRS:-<M
_a`bJ[JS W \UFKHP FGPt7 Pt7< `bUHK<QRS`b<`bFPtPt<OE@RMEVl=j`tJ[JMW \UFKHPFPb7-Pt7< `bUHG<
RM`t<D`bFPtPt<OE Va687<NOJM`b`t<WZj]JMEkl0FGEVL@`b<;l0UVNPtFKJME`tUVHG<FTW
G
q@@
WRSHK`b<;RMlY WtRMFKlXRWZ<^3U<OE3P }6y FKWIjV`tJ RS\VHG<-FKEl<;l0UVNPtFKJME FGPt7Pt7V< RSEVl
%
`bUHK<;WDFGRSEkl§JMEVHGY%FGPb7<-WZ<^+UV<OE3P iK8 y FTWDj`bJ RM\HK<1FKE/(k`bWZP+*JM`ml0<O` EVRPbU`mRSH l0<;lUVN PbFGJ+EX1687<
Wt<;^3U<;E3P 6y FKWRSHTWtJj`bJ RS\HK<@FKEl0<l0UVNPtFKJME :-J0l0UHKJFRMEVl§J+EHKYFG Pb7<-WZ<^+UV<OE3Pi8 7y FKW
jV`tJ RS\VHG< FGE"(V`mWdP8*'J+`bl0<;` EVRSPtU`mRSH l0<l0UVNPtFKJMEXNi<;EVN<DPb7<IWZ<^+UV<OE3P /"y FTWjV`tJ RS\VHG< FGEl0<;lUVN PbFGJ+E
FGPt7Pb7< RSEVl `bUHG<W8F RSEklJ+EHKY F FGPiFKW8j`bJ RS\HK<FGE%l0<l0UVNPtFKJME:=J0l0UVHGJk
687FTWDNORSE§RMHKWtJ \]<@j`bJ M<ll0FG`b<;NPtHKYMc' FTWQRj`bJ3JM JSg y FKE l0<l0UVNPtFKJME§FGPt7%Pb7<
RMEVl `bUHK<;W;0Pt7<;E%Pt7V<j`bJ3JMH J+\0PbRMFGEV<;l\[Y<O`mRMWtFKEL=Pt7<IJ+HKlFGELRSEVlUE0J+HKlFGELWdPb<OjVW
FKE Y3FK<OHTlWiR-jV`tJ[JSqFGE§l<;l0UVNPtFKJME:-J0l0UHKJV687V<NJME +<O`mWZ<DFKWQR-HGFGPZPbHG<\VFPQ:-J+`t<Il0F NUHGP;'¢EVl0<;<;l
FG8Pt7<j`tJ+jkJ3WZFGPtFKJME;x UEJMHTlWIPbJx Pt7<;EPt7< jV`tJ+jkJ3WZFGPtFKJME;y/x RSEVl;yx RM`t<"NOJMEL+`tU<;E3P
RMEVl7<;EVN<IPb7<@Wt<;^3U<;E+P y~x y x 7VRMW R=Pt`bF[FTRSHj`tJ[JMqFKE§l0<;lUVN PbFGJ+E%:=J0l0UVHGJkkUVWtFKEL"JMEHKY
Pb7< IA# `bUHG<+UPPb7< % `bUHK< l0J[<;WEJMP8RMjjHKY1PbJPb7< jV`tJ+jkJ3WZFGPtFKJME3y x 0\U0P8JMEVHGY=PtJ
Pb7<j`bJMj]J+WtFPbFGJ+E3x Q<OEVNO<QPb7<j`tJ[JM5JSUy3x :@UkWdP8\k<`bFPtPt<;E $'& ( p w &)*+ p w # 7<;`t< Pt7<
RM`tFTRS\HK< FKWI*<,0jVRMEVl0<lWtJPt7VRSPIPt7< % `bUHK<"NORSERSjVjHGY PtJPt7V<"j`tJ+jkJ3WZFGPtFKJME;x 687VFKW
cdUkWdPbFC(V<W8Pt7V<IE<O<lJMPb7<JMHKHGJFKEL=HG<;:-:-RV
| >B !?8h@A y "!B?a x EB aB#@[B O?aB$$ y&%
l
x('*) >B
$hB-?8B"?8BE?8%] 13]
y x
!
x y
7B>-@A O$$hB
JMHTl0FGEVL
!
UE0JMHTl0FKEL
?aB+
,
>Ba-B/!?8%] 1 ?8B$1B
NOJME M<;`bWtFKJMEWZPt<;jVW
+
- ?1+ .+ Y FGEkl0UVN PbFGJ+EJ M<O`Pb7<WdPb`tUkN PtUV`t<JS_y
/ ' _y FTWiRSPtJM:-FTNS[Pb7<OEy UE0J+HKlVWPtJx RSEkl<DPbR+<DPt7<Ij`bJ[JS Wc<6F0"RMEVl <6 0V
/ ' cy z y l y 1 Pt7<;E x z x l x 1 < 7VRM<-<;FPb7<O` y l z x l RMEVl)y 1 % l x 1 JM`
y l % l x l RSEVl;y 1 z x 1 YFGEkl0UVN PbFGJ+E 7[Y3j]JSPb7<;WtFTWI<"7kRM<-FGE \]JSPb7 NOR+WZ<Wj`tJ[JMW(1 l
RSEVl21 l
JSgy l x l RMEVl~x l y l RSEVl1 1 RSEkl31 1
JM_y 1 ux 1 RMEVlx 1 y 1 <IPmR+<
<=< 3 p1 1Kp p1 l 3 w8w+w RMEVl <66<3 p1 1 p p1 l 3 w+w8w
/ ' y z y l ;y 1 Pt7<;E x z x l ;x 1 <7VRM<<OFGPt7<;`y l z x l RSEVl=y 1 % l x 1 JM`
y l % l x l RSEVl;y 1 z x 1 YFGEkl0UVN PbFGJ+E 7[Y3j]JSPb7<;WtFTWI<"7kRM<-FGE \]JSPb7 NOR+WZ<Wj`tJ[JMW(1 l
RSEVl21 l JSgy l x l RMEVl~x l y l RSEVl1 1 RSEkl31 1 JM_y 1 ux 1 RMEVlx 1 y 1 <IPmR+<
< p1l $'&)( p w+w p1 1 &)*+ p w+w # RSEVl<6 p1 l $'& ( p w8w p1 1 & *+ p w+w#
P!QPSR
/ ' y z y l y 1 Pt7<;E x z x l x 1 <7VRM<<OFGPt7<;`y l z x l RSEVl=y 1 % l x 1 JM`
y l % l x l RSEVl;y 1 z x 1 YFGEkl0UVN PbFGJ+E 7[Y3j]JSPb7<;WtFTWI<"7kRM<-FGE \]JSPb7 NOR+WZ<Wj`tJ[JMW(1 l
RSEVl21 l
JSgy l x l RMEVl~x l y l RSEVl1 1 RSEkl31 1
JM_y 1 ux 1 RMEVlx 1 y 1 <IPmR+<
< p.- 3 p1hl3 w , p1 1 Yw+w RSEkl <6 p8- 3 p1 l 3 w , p1 1 Yw+w
/
s[FKEVN< y FTW8`t<l0UVNOFG\HK<MVFPQN;RSEEJMP\k<
/ ' yz j!k y l Pt7<;ExWz j!k x l RSEVl%<=7VRM< y l % l x l YFKEVl0UVNPtFKJME 7[Y[j]JSPt7V<;WtFKW <
7VRM<j`tJ[JM W1 l RSEkl 1 l JS y l Sx l RSEVl)x l Sy l < PmRM< < < k p1 l/p k!w+w RSEkl
<=<
k p1 l p kEw+w
/ ' yz k y l Pt7<;ExWz k x l RSEVl%<=7VRM< y l % l x l YFKEVl0UVNPtFKJME 7[Y[j]JSPt7V<;WtFKW <
7VRM<8j`bJ[JSW1 l RSEVl 1 l JSmy l x l RSEklDx l y l <8PmRM<c<6 p; k ;D=@? k 3 k p1 l 3 w # w
RSEVl<6 p; k ;>=@? k 3 k p1 l 3 w # w
{ |0{ y y ! E?8+ "+ { y() >B= >B?1B "E?8%] ~] =y +
$hB ?a# p1 o{nnnp1 l wannnMw >B?8B 1 l nnn 1 o ?8B"!B?aA}B+> !?8%] B
Pb`bRMEVWZJM`b:-RSPtFKJME ] +
{
yz
,
hB E?8>5a@A> !?1 !-B; B>-@A O $hB
JMHTl0FKEL
!
UE0J+HKlFGEL
?aBa E$hB!?8h@A>4!?1 !-BDB>@A#$% ?1B $hB -#KB +
-
?1+ .+
`mRMWtFGEVLPb7<%J+HKl0FKEL RMEVl UE0J+HKl0FKEL WZPt<;jVW-Pb`bRMEVWdJ+`t:"W"RME3Y j`bJ3JM FKE l0<;lUVN PbFGJ+E
FGPt7Pb7< RSEkl
%
`tUHK<;WiFGE3PtJ R=j`bJ[JSO FGE§l0<l0UVNPtFKJME:-J[lUHGJkfJME +<O`mWZ<;HGHKYM0FG R
Wt<;^3U<;E3P7VR+W RjV`tJ[JS'§FKEl0<l0UVNPtFKJME:-J0l0UHKJV+Pt7V<OE<QN;RSE \UVFGHTlR@j`tJ[JM FGEl<;l0UVNPtFKJME FGPt7
Pb7< RSEVl
%
`bUHK<;W8FKEVWZ<;`ZPbFGEVL-Wt<;^3U<;EVN<W8JS NJME +<O`mWZFKJMEWdPb<OjVW87<;EE<O<l0<;l
ODV O q !"
<@EVJ j`bJ M<Pb7VRPNU0PD<OHKFG:-FKEVRPbFGJ+EPt<;`t:-FKEVRPb<;WiFKE l0<l0UVNPtFKJME%FGPt7%Pb7< RSEVl %
`bUHK<;W8FG RMEVlJMEHKY FG NOU0Pi<OHKFG:-FKEVRPbFGJ+EPt<O`b:-FGEkRPt<W8FGE%l0<l0UVNPtFKJME:-J[lUHGJk
|
B-
y
!
y
B~[ !?1>5aAh~a> >
y y
!
y
!
y
?8B/B-
$hB-? 8[$# !CA1Ah
) [aE@A>
) [K aE@A>
) [K?8IA-Ah
) [ > !B?-
!
>"@Ah$? [~BIaBA
!
>"@Ah+
B
B !?8%] $]
/y
!
!?1+
+
/"y
> ?1hJ ?a#$A;+
+
>@?8>@A
)
$hB-
?8B>Bac ~>@?8%>-A
ED >B> !?1+ 1 ]
?8B
@?8hJ ?a#K@A> + a>@E?8%] 1 ]
+
- ?1+ .+ YFGEVlUVN PbFGJ+E JME Pb7<HG<;ELSPb7 JSPb7<Pb`bRMEVWZJM`b:-RSPtFKJME`bJM: PtJ 687<`t<WZUVHP=FKW
J+\ 3FKJMUkW FG z Pt7<;`tFTWt<M Pb7<"j`bJ3JM 7kRMWDPt7V<-JM`b: p1 o l p1 oEnnnp1 l wannnMw+w 687<"j`bJ[JS
p1 oEnnnp1 l wannnMw FTW"R j`bJ[JSQJMDRj`bJMj]J+WtFPbFGJ+E6y Pb7VRP"FTW-<OFGPt7<;` RPbJM:-FKNJM`-7VR+W=Pt7<%WbRS:-<7<;R+l
WtY[:@\]JMHXR+W y
Uha
' vy
FTW"RPbJM:-FKNPt7V<OE *zRSEkl p1 o
l5nnnp1 l wannnMw FKW"Rj`bJ[JSQJM y 1 o z <6B RSEkl
1 o l z <6 V2YFGEkl0UVN PbFGJ+E 7[Y3j]JSPb7<;WtFTWO p1 o
lnnnp1 l wnnnw `t<l0UVNO<;WPbJ§RSE FKE+Pb`tJ0l0UkN PtFKJME
RMEVlPt7<WtU\j`bJ[JSW8JM RM`t<DPt`mRSEkWdJ+`t:"RPbFGJ+EVWJSqPt7<WtU\j`bJ[JSWJSO a687<Ij`bJ[JSH `t<l0UVNO<;WPtJ
-RSEVlPt7V<OEPtJ
Pb7<O`bFKWt</y 7VR+W"Pb7<WbRS:-<§7V<;RMl WZY[:@\]JMHDRMW"y Y FKEVl0UVNPtFKJME 7[Y[j]JSPt7V<;WtFKW;8Pt7<j`bJ[JS
p1o nnnp1l: wannnMw `b<;lUVN<W8PtJ"<6F
l :`b<;WtjX !
l
C
1 # p l w , p
l w E< k
l
H !(a"
l # F kPt7<@WtU\jV`tJ[JSW
RM`t< Pt`mRSEkWdJ+`t:"RPbFGJ+EVWJSPt7<WtU\j`bJ[JSW8JMHRSEVl31o
l 7VR+WPt7<J+`t:
<7<3 p)1Kp p
l 3 w8w+w
:
`t<WZj5
<6 p l $'&)( p w+w p 1 &)*+ p w8w#
<6 p.-
3
p%l
3 w , p)1 mw8w
<6 p.-
3
p%l
3 w , p)1 mw8w
<=< k p%l$p k!w8w
<6 p; k ;>=
@?
k 3 k p%l 3 w # w
F@687[UVW;Pt7<-j`bJ[JS `b<;l0UVNO<;W PtJRSEFGE3Pb`tJ0l0UVNPtFKJME RMEVl%Pb7<"WZUV\j`bJ3JMWQJM RS`b<@Pb`bRMEVWZJM`8*
:"RSPtFKJMEVW8JMqWtU\j`bJ[JSW8JSO
{ yz
cBC#DE@A)B?a# !B ;IB>-A O$
JMHTl0FKEL
!
UE0J+HKlFGEL
?aB
! 3] -B-# !@A;#KKB-?a#DEBa+
-
?1+ .+
WbWZUV:=<1Pt7kRPNU0PI<;HGFK:-FGEVRSPtFKJME:-J[lUHGJPt<;`t:-FKEVRPb<;WRSEkl NJ+EVWtFKl0<;`IRNOU0PI<OHKFG:-FKEVRPbFGJ+E
Wt<;^3U<;EVN< ml5C21
nnn
FKEl<;l0UVNPtFKJMEFGPt7Pt7< RMEVl
%
`tUHK<;W;
RMNm7r
lDFTW8JM\PbRSFKE<l
`bJM: r <OFGPt7V<O`\[Y `b<;l0UVNOFGEVL-R=HKJML+FKN;RSH NU0PiJ+`R@J+HKlFGEL*'UVE0JMHTl0FKEL"NU0PiRSEklJ+EHKY R (kEFPb<IE[U:@\]<O`
J+HKlFGEL*'UVE0JMHTl0FKELQNUP`b<;l0UVNPtFKJMEVWNORME\k<jk<;`ZJ+`t:-<lINJMEkWZ<NU0PbF+<OHKY
:
RMW Pb7<WtF0O< JMj`bJ3JMWX`b<;l0UkN<;W
7V<OE<"`b<;lUVN<"WtUVNm7 R%NU0PaF 7V<OE r
l=FTWIJM\PbRSFKE<l\[Y`t<l0UVNFKEL%RHKJMLMFTNORMHaNU0PFKE r8Pb7<OE
r l
FKWQJM\0PmRSFKE<;l\[Y`b<;lUVNFKEL R"NUPQFGEA
r
RMEVl7<OE !r
lIFKWiJM\0PmRSFKE<;l\[Y`b<;lUVNFKEL"R=JMHTl0FKEL*
UVE0JMHTl0FKEL NU0PiFKEAr Pt7<;EA
r l z
r Q<OEVNO<IFGEPb7<Wt<;^3U<OEkN<
l
"
1 nnn FGE§l0<l0UVNPtFKJME:-J[lUHGJ
<RMNm7%j`bJ[JS FKW <OFGPt7V<O`QJ+\0PbRMFGEV<;l%\[Y`t<l0UVNFKELRNU0PDFGE%Pb7<@jV`t<3FKJMUkWOkJM`iFTWQ<^3UVRSH5PtJ"Pb7<@j`b< [FKJMUVW
RMEVlJMEHKYRD(VEFGPt<IE[U:1\k<;`JS NJMEkWZ<NU0PbF+< j`bJ[JSWN;RSE\k<<;^3UVRMH( WNU0Pi<;HGFK:-FGEkRPtFKJMEPt<;`t:-FKEVRPb<;W
FKE%l0<;lUVN PbFGJ+E:-J0l0UHKJV0Pb7FTWiWZ<^3U<OEVNO<FKW(VEVFPb<RSEVlWtJ=FTWPt7V<WZ<^+UV<OEVNO< l " 1 nnn
fJ+E +<O`mWZ<;HGHKYMXR+WtWtU:-<1Pt7VRSP@NU0P<;HGFK:-FGEVRSPtFKJMEPt<;`t:-FKEVRPb<;WFKEl<;l0UVNPtFKJMEFPb7Pt7< RSEkl
% `bUHG<W RMEVl"NJ+EVWZFTl0<;`RjV`tJ[JS*`b<;l0UVNPtFKJME"Wt<;^3U<OEkN</ l " 1 nnn FGEl<;l0UVNPtFKJME :-J0l0UHKJV 687<
jV`tJ[JS l NOJME3PbRMFGEkWIR`b<;l<, K'¢E l
R%NOJME M<;`bWtFKJME WZPt<;jVWI:"RY§7VR+<=\]<O<;EFGEVWt<O`tPt<l\k<OPd<;<OEPt7<
FKE3Pt`bJ0l0UVN PbFGJ+E `bUHK<-RMEVl%Pb7<-<OHKFG:-FKEVRPbFGJ+E`bUHK<-JSaPb7FKW`b<;l<, U0PX\[Y§HG<;:-:-R~L0KuM5`t<l0UVNFKELPt7<
NOJME M<;`bWtFKJMEWdPb<OjVWDRSjjVHGFK<;lPbJ"Pt7V<1FKE3Pt`bJ[lUVN PbFGJ+E%jk<;`t:1U0Pt<WiPb7<@FKE3Pt`bJ[lUVN PbFGJ+E§RSEVlPb7<1NOJME M<;`bWtFKJME
WZPt<;jVW;i6873UkWO6
l
`t<l0UVN<W8PtJ 1
FGE RSPQHK<;RMWZPQJ+E<WdPb<OjX/Q<OEVNO<Pb7<@Wt<;^3U<OEkN< l
"
1 nnn
FKWc(VEFGPt<
RMEVlWZJ-FTW8Pt7<IWt<;^3U<;EVN< ElC21
nnn
P!QPSR
@ qO !" RmUTUT
VYXgZ]\O^a`cb d `YVYX[Z]\_^a`cb
'¢E2N%4X<D7kRM<WZ7JEPt7kRPiR1Pt7<;JM`bY":=J0l0UVHGJ-7VR+lPb7<INU0Pi<;HGFK:-FGEkRPtFKJMEj`bJMj]<O`tPdY"F FGP87kRMlWZJ+:-<
[FKEVl=JS :"RSE[Y*RMHGU<l@:-J0l0<;H :7J3WZ<8Pb`tUPt7 RSHKU<;W RS`b<iWZ<OPbW JM]j`bJ[JSWF NORMHGHK<;l-R !?1B#KIB RMEVl=<
7kRM<Wt7JEPt7VRSP HTRS`bLM<NOHKR+WtWt<;WJM Pt7<;JM`bFG<W:-J0l0UHKJ7VR+lPt7<=NU0PD<OHKFG:-FKEVRPbFGJ+E%j`tJ+jk<;`ZPdY+ WDNU0P
<;HGFK:-FGEkRPtFKJMEFKW<^+UVFRSHK<OE3PFGEl0<;lUVN PbFGJ+EFGPt7Pt7< RMEVl
%
`bUHK<;W8RSEVl FGEl0<;lUVN PbFGJ+E
:-J0l0UHKJVkPt7V<;Wt<PtJ[J+HKWDNORME%\k<-RMlRMj0Pt<lPtJj`tJ +<@NUP <OHKFK:=FKEVRSPtFKJME§FKE l0<l0UVN PbFGJ+E§FGPt7§Pb7<
RMEVl `tUVHG<WO
HPb<O`bEVRPbF+<OHKYMPt7V<OJM`bFK<;WXUkWZUVRMHGHKYj`b<;Wt<OE3Pt<lIFGE=l0<;lUVN PbFGJ+EFPb7Pb7< RSEkl `tUHK<;W
N;RSE<;^3UFRSHK<OE3PtHKY \]<Ij`t<WZ<;E3Pt<;lFKEl<;l0UVNPtFKJME:-J[lUHGJk
_a`bJ[JS WFGEl0<;lUVN PbFGJ+E :-J[lUHGJ=RS`b<Q:-JM`b<QNOJM:-jVRMNP Pb7VRSE jV`tJ[JSWFKEl0<;lUVN PbFGJ+EFGPt7"Pb7<
RMEVl % `bUHK<;W;W.qFG`mWdP \]<;N;RSUVWt<Pt7<JMHTl0FKEL RMEVl UE0JMHTl0FKEL WdPb<OjVWRS`b<HG<OgPFK:-jHKFKNOFP%FKE
l<;l0UVNPtFKJME:-J0l0UHKJVq\U0P1RMHKWtJ\k<NORSUkWZ<"FKE l0<;lUVN PbFGJ+E FPb7Pb7< RMEVl `bUHG<WO RMW
Wt7J RM\kJ +<M j`bJ[JSWDE<O<l%PbJ\]< *'<-,[jkRSEVl0<l WtJPb7VRPPb7< RMEVl % `bUHK<;WD:"RY%\k<
RMjjHKFG<l 687FKW * <,0jVRSEkWZFKJME§NOJMUHTl%\]<=R+JMFTl0<;lFG <1<-,3Pb<OEVl<;lPb7< RSEkl `tUHK<;W
PbJ"R "!B?aA `tUHK<
/"y
!5B-?aA FGOy x
/"x
FGPt7§RM`t\VFPb`bRM`tY jV`tJ+jkJ3WZFGPtFKJMEVW y RSEVlx UP;kPt7<;E§HG<;:-:-R"L0KuIJ+UHKlEJMPQ7J+HKl%RME3Y[:-JM`b<RSEklR
NOJME M<;`bWtFKJME WZPt<OjFKEVWZ<;`ZPb<;l\k<OPd<;<OER@FKE3Pt`bJ0l0UVN PbFGJ+ERMEVlRSE<;HGFK:-FGEkRPtFKJMENJ+UHTl\HKJ0N R-NU0PRMW8FKE
Pb7<Ij`bJ[JS :FPb7x x F y 8x "y y 1x"x
U*'FKE3Pt`bJ
y 8x "y x
"!B?aA
y 8x3y x
U* <OHKFK:
y 8x "y
<J+UHTlPt7<;E EV<O<;l PtJ<,[Pb<OEVl Pt7<EJMPtFKJME JMDNOU0P-RMEVl l0<(VEV<RNU0P R+W=RWt<;^3U<OEkN< J+`t:-<l
FGPt7 RSE FKE3Pt`bJ[lUVN PbFGJ+E `tUHK<MaR WZ<^+UV<OEVNO<JSQNOJME M<;`bWtFKJME`tUHK<;W1RSEVl RSE <;HGFK:-FGEVRSPtFKJME `bUHK<2p4 F(<M
<WtWt<OE3PbFKRMHGHKY RMWR-NUPi:=J0l0UVHGJk
P HTRMWZP;[l0<l0UVN PbFGJ+E":-J[lUHGJ@FKW :-J+`t<L+<OE<;`bRMH0Pt7VRME l0<;lUVN PbFGJ+E"FPb7-Pt7< RSEVl
%
`bUHK<;W;RMW8FGPQl0J[<;WEVJSPi`b<;^3UFK`t< Pt7VRSPQRSE%RPbJM:-FTN jV`tJ+jkJ3WZFGPtFKJMEUE0J+HKlWPtJ"RPi:-J+WZPiJMEV<DjV`tJ+jkJ3WZFGPtFKJMEX
687VFKW"j]<O`b:-FPmW"FKE jkRS`tPtFTNUHTRS`-PbJ FGEVNOHGUkl0<M8\]<;WtFKl0<W"<;^3UFRSHK<OEVNO<;W-\]<Pd<O<;E jV`tJ+jkJ3WZFGPtFKJMEVW"WtUVNm7 RMW
k
z k z
z V+<^+UVFRSHK<OEkN<\]<Pd<O<OE-Pb<O`b:"W WtUVNm7 RMW k
k 687<;E RIj`bJMj]J+WtFPbFGJ+E
WtUVNm7"RMW pJk w3p w z :"RY=\k<i<^3UFRMHG<;E+P PbJ:"RSE[Y@jV`tJ+jkJ3WZFGPtFKJMEVW WtUVNm7 R+W k"p w z
pAk w z k z z k z z VVKG 7FKHK<8Pt7FTWaFTW EJMPajkJ3WtWtFG\VHG<FKE l0<;lUVN PbFGJ+E
FGPt7Pb7< RSEVl `bUHG<WO
5
&'
5
! Z$' $#"
3
&
3 L&$
3
$T$#"
3
&&%
')(+*-,/.10/2436587.:9<;>=1?A@#.87BCDE.#F1BCGHJIKGI
L:NM8O1OQP R+R)>S8T1I
"
S1MQH
Uha
6
&B
"
&'
5$'Z$'
3 5
$#" 6&' 3
3M3
&' 365
G +$ &' R $
L: M8T P4
IJ E
&B
$&BZ$' $'$
"4365
T$'
')(>*,/.J0K23517 .:9;>=1?A@#.87BC DE.#F1BCG HJS L:NM8MK PR+R)
I "
1S/
1
3
G
&'% &'
5!6$'L& &#$V$
"
&B
3
&O&'
5
Z$'"1"
3 &"
&B 3 & 5
G !
%$
3 &
W #"
365
$% L$
K$
P4G)')(>*#.J2#*4?&% 2&.' B3/F B3)( BF (+*42+*.12, *42 D).#F1BC.-#G/!
3
=$0 &3 $
3 $ 16
R
$
3
&E$=
3 3
8T1H/G /R
&' 3 &
"32*3
&5
L: M1MQT1P4G6R+R 4QH
"
8 M/65 5+7/7K.1298:, *;5*&C (+*42:C (+*I1I8O8I/G=<>;?@<: L: M8M1OQP4
H A
3
GCB
"
ED
&B G W
AF
&0 "
3 &
AB
" 3
&
3 R
&'#G_TK 5
H? R+R
&$ 3 ? 3 4"
3
&04"
3
I/44KG<
+$V$'$0$
Z$'W
5 3 ? 3 " 3
&04"
3 3 <
&' $'JIK
3 3 $ 3
$' Z$'LIK
3
L:NM8M1OQP4B
: R+R
3
Z&^
,/.J0K2 3517 .:9NM08.1?58*0,O5*&5-4.13>B3/F1
SJ :
# 3
G1B
"
:D
&B
G
W :F
&04" 3 &
:DQPQ!
"SR T
+%$
3
%$'W
5
&#$V$
"
&B
3 & 3 R & 3
$&$;
" 8"
3 & "
&B 3 & 5
J E<
U
Z&
3
WK&B W
V?
+$ $04"
G 3 $'&#$
G:5 *+W 24BX8B3/F ' *&C (/3>B$Y0+*+-513,
MZ7/77BC 58B.13 -#G
3 &
NS8IK
! 3 $0 &3
$
3
$^
R $3 &
6
3 3
G+KR
&' 3 &
"323
& 5
G L:NM8M8M8PR+R)
I/ T "
I1IK1
T
3
W [
3
&' 3 &
=
&'% &'
5
! Z$'1TK
5
=<
'>=+7>*+- 9N.12:762:. .:9+- 513=,)762:.#F12&51?\-O]^1G
G
5
3 SQH1T
D *&C.80/2#*Q_ .8*+- B3a`.1?;7608*42 ;CB*43C *4G /R
&' 3 &
"32*3
& 5
G)L: M8M1MQP4G+R+R)>S
"
T8T
OJcbKH
GV
&' 5 R
&'%$: $3
$0$#"
3
&&%
GZd. C.8.12:587C8(+*+-#BX-#G
"W 5 3
&#$\eI JG
3
&#$$%
B 3 " 5
J%
WeIG
3
&#$$%8
;f
$3 M&0
L:NM8MQP4
MJc! D 55
=gJ$
GhP
&'
5
! Z$'
R
&'%$ $ 3 $ $#"
3
&&%
Gd. C.8.12:587i8(+*+-#BX-#G
e G
3
&#$$%8
$'K " 5
L:NM8O1IQP4
4Jj
&B $#!
=
$0 &B 5 3
=$'
G R
&'%
"
$#"
3
&
3
$'J6
5
$V$0K%
=M7?jY.' Bk-98mlonBLp-*77+L: M8S8H8P4
P!QPSR
Unité de recherche INRIA Rocquencourt
Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)
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 Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (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