HAL Id: inria-00074577
https://hal.inria.fr/inria-00074577
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.
Fast and efficient Generation of Loop Bounds
Zbigniew Chamski
To cite this version:
Zbigniew Chamski. Fast and efficient Generation of Loop Bounds. [Research Report] RR-2095, INRIA.
1993. �inria-00074577�
ISSN 0249-6399
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
a p p o r t
d e r e c h e r c h e
1 9 9 3
PROGRAMME 1
Architectures parallèles, bases de données, réseaux et systèmes distribués
Fast and Efficient Generation of Loop Bounds
Zbigniew Chamski
N˚ 2095
Octobre 1993
+-,/.1032/.1465879;:=<>@?.
A;BDCFEGBIHKJJLMONQPRBISUTWVYXDLZS[XD\WBUL@]^_H`B[H`aYacbL@aYL@]d6e3H`]DLf]g6Lg6CGhWhjiL@L@]Zd3BWiL@]ULH`\6k
L@Xl]Umn]UXbL@J%L@]lg6VY]DXUBDVYeW\jiLf]
A;BDCo[LfXlPA;p
qlHK^W^3CFBDXjg6LBULSUTWL@B[SDT=L!hr6sFtGuFvNQwSxXDCFe=BDLMuFuGyRNzMZt^3H`EFLf]
{|-}f~`_K~F\=BDBDLfhnX aC6CG^*EFL@h=L@BIHKXDVYCFhXDLSUTWh=VG\WLf]!H`BUL!e_HK]DLgCFh=CF\WBUVLfBDCFXU@6Vh
^_H`VYBUHKV]UL!LfaVYJ%Vh3H`XDVYCGhd3jTWVSDTVY]6hWC`jhXUC*e3LFLfBDmJLfJCFBUmGH`h_gS[CFJ%^W\WX[H`XDVYCGh6
VYhnXUL@hW]UVGLF phXUTWVY]^_HK^_L@B L L[kW^WaYCFBULH`hH`aYXDL@BUh_H`XUVYFL HZmXDT=L!\W]ULCK^_H`B[H`J%L@XDBUVS
aYVh=LH`B^WBDCGEFB[H`JJ%VYhWE_dZjTWVSUTH`aYaYC`j]XDC]DLf^_H`B[H`XDL XDTWL!SxCFJ^=\WXIHKXDVYCFhCKgV]UXDVYh_S[XjaYC6CF^
e_CG\Wh_g6]lHKh_g%aYLHFg6]jXUCH ^3H`BIHKaYaLfaH`aYEFCGBDVYXDTWJ 1CGBaYC6CF^%EFL@h=L@BIHKXDVYCFh
¡%¢n£¤U¥¦3F§¨}KaC6CG^EFL@h=L@BIHKXDVYCFhdGS[CFJ%^WVYaHKXDVYCFhdD^WBUCFEGBIH`J©XUBIHKhW]1CGBDJH`XDVYCGhd@aYVYhWLHKB^WBUC`
EFB[H`JJVYhWE
ª«¬
c®¯°
¬
/±/²Y®³c´xµ
¶
·¸F¹[º¹[»¼¾½¿ºcÀÁ`¹[ÂFÃ=ľÀDù@ÅU½1ÀUÃÁ`À;ÆÇ[È1ÉÊ ËÌ[ÍfÎKϨºcÀÐÂFÑ[Ò`ľÀ@Î[ÓZºc¹xÂFÅDÀ@Ô
Unit´e de recherche INRIA Rennes
IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France) T´el´ephone : (33) 99 84 71 00 – T´el´ecopie : (33) 99 38 38 32
- z* * )
¢W} ¢L@]jJiL@XDT=CgL@]gLlE-iLfhiL@BIHKXDVYCFhg6Lle3CFBUhWL@]lg6LRe_CG\_S[aYL@]]DCGhnX^3CF\WBaH^WaY\6
^_H`BUX!e3H`]WiLfL@]]U\WBaDiLfaVYJ%V¾h_HKXDVYCFhgL =CF\=BDVYL@BDCGXD@6VYh/L@^_Lfh_gWH`h6XZdS[L@XUXDL*XULSUTWhWVG\WL
L@]DXjS[CGhWh6\WLl^3CF\WB]IH EFBIHKh_g6LlS[CGJ^WaYL[kWVYXWiLjXUL@J^3CFBUL@aYaYLL@X]D^3H`XDVHKaLGH`h=]S[LgCSx\WJL@h6X
hWCF\=]lhWCG\W]lVYh6XWiL@BUL@]D]UCFhW]bH \WhWL!HK^W^WBDCWSUTWLH`aYXDLfBDh_HKXDVYFL!G\WV/S[CFh=]DVY]DXUL bH \WXDVYaYVY]DL@BRaH!
"
$#&%%'#!(*)+&,.-/)0,21
3
#&)4
3
5#%1
3
(7)98:
3;
L@XDXULH`^=^WBDCWSUTWLl^3L@BUJL@XgLj]WiL@^_HKBDL@BaYL@]S@HKaS[\WaY]
g6Le_CGBDhWLf]g6VY]DXUVYh_S[XDLf]LfXlS[CFh_g\WVYX!bH \WhH`aYEGCFBDVYXUTWJLj^_HKBIHKaabL@aYLR^_CG\WBjaH E-iL@hjiLfBIH`XUVYCFh*g6L
e_CG\_S[aYL@]Z
< ¦_~`}f¤I&=>¢ !E-iL@hjiL@B[H`XDVYCGhg6Le3CF\_S[aYL@]ZdFS[CGJ^WVYaH`XUVCGhdUXDBIHKhW] CFBDJH`XUVCGhg6L^WBUCFEFB[H`J%JL@]Zd
^WBDCGEFB[H`JJHZXDVYCFh aYVYhiLH`VYBUL
3 3 3
--)j
jTWLlh=L@Lg 1CFBaC6CG^EFLfhWL@BIHKXDVYCFh%XDLSUTWh=VG\WLf]jT_H`]jg6BIHKJ HKXDVS@H`aYaYmEFBUC`jhjVYXDT%XDTWLg6L@FLx
aYCF^WJ%L@h6XCK;^3H`BIHKaYaLfaVY@VYhWES[CGJ^WVYaYL@BD]ZjTWL aC6CG^ EGL@hWL@B[H`XUVCGh*^WBUCFeWaYL@JS@HKh*e3L!]UXIH`XULg
H`]R1CGaaYC`j]ZDEGVYFL@hHKhVYXUL@BIHKXDVYCFhgCFJ HKVYhHS[CFh6FLxk^_CGaYmnT=Lg6BDCGhdS[CGJ^W\WXUL!XUTWL
e_CG\Wh_g6] C`HaYC6CF^hWL@]UXjTWCG]DL%L[kWLS[\WXUVCGh jVaYa-][S@H`hHKaaVYh6XDL@EGL@B!^3CFVYh6XD] C`VYh H
EFVYFLfhCFB[g6L@BZjTWVY]j^WBDCGeWaYL@J H`^W^3LH`BU]lCK1XDL@hVYh^_H`B[H`aYaYL@aYVYH`XUVCGhd`jTWLfh*XDTWL VXUL@BIHKXDVYCFh
]D^_HGS[LC`XUTWLVh=^W\WX^WBDCGEFBIHKJ(jH`]JCWg6V _Lg \W]UVYhWE!]UCFJ%LjEFL@CGJL@XUBDVSjXUBIH`h=]1CGBDJ HKXDVYCFhd
H`h_g HRhWL@ aYC6CF^ ]DXDBU\_S[XU\WBDLJ\W]UX¨e3LjEFLfhWL@BIHKXDLg VYh!CGBIg6L@BXDC^WBUCWg6\_S[LRXDTWLj^_HKBIHKaaYL@aYVY@Lg
^WBDCGEFB[H`J!#"YM`d%$'&D(jpXVY]HKaY]DCHFS[LgjT=L@hEGL@hWL@B[H`XUVh=E VYJ%^_L@B[H`XUVGL^WBDCGEFBIHKJ])1L@VYXDT=L@B
^_H`B[H`aYaYL@a3CFBj]DLZF\=L@h6XDVH`a*1BUCFJ 1CGBDJH`a3]D^3LS[V S@HKXDVYCFhW])#"s+&D(
\WBDBUL@h6X]DCGaY\WXDVYCFh=],#"YM`d-$&.HKBDLe_HK]DLg CGhHL@aYanh=C`jh aYVYhWLH`B^=BDCFEGBIHKJJVYhWE
H`aYEFCGBDVYXDT=J6XDT=L =CF\WBUVLfBDCFXU@6Vh^_HKVBUjVY]DLL@aYVYJVYh_HKXDVYCFh5R\WLXUC*VYXD]^=BDVYh_S[VY^WaYLFdXUTWL
S[CFJ%^W\WX[H`XDVYCGh*C`jaYCnCG^e3CF\=h_g6]BULG\WVYBDL@]HKaa/XUTWLe3CF\Wh3g6]XDCe_LS[CGJ^W\=XDLgHKXCGh_S[LF
jTWVY]jVY]lH ]DXDBUCFhWE%aYVJ%VYXIH`XUV¾CGh1CGBlg6CFJH`VYhW]jVYXDT*S[CFJ%^WaYL[k g6L/3hWVYXDVYCFhW]ZdW]UVh3S[LXDT=LJL[
JCGBDmBULG\WVYBDL@J%L@h6XD]C`XDT=V]jH`aYEGCFBDVYXUTWJ HKBDLle3CF\=h_g6Lg enmHKhL[kW^_CGhWL@h6XDVHKa31\Wh_S[XUVYCFhC`
XDTWLh6\WJe3L@B0CKOS[CGhW]DXUBIHKVh6XU]!g6L/_h=Vh=E*XDTWLgCFJ HKVYh HKh_gXDT=Lg6VYJL@hW]UVYCFh1CKXUTWL
VYXDL@B[H`XDVYCGh]D^3HFS[LF
jTWL@BDLx1CFBULFd`VYXe3LS[CGJL@]VJ%^_CGBDX[H`h6X_XUC2_h3g!HOHZmRC` g6VY6Vg6VYhWEXDTWLaYCnCG^EFL@hWLfBIH`XUVYCFh
^WBDCGeWaYL@J(VYh6XDC]UJ HKaaYL@BH`h_g%^3CF]D]UVYeWaYmVh3g6L@^3L@h_g6L@h6XR]D\WeW^=BDCFe=aLfJ]GP©]DCGaY\WXDVYCFhXUCXDTWVY]
HK]-CF\=XDaYVYhWLg e6mWLZH`\WXUBDVYL@BVh"v&ÐdKjTWL@BDLXUTWLH`\WXUTWCFB-^=BDCF^3CF]ULg%XDC \W]DLH 5#&$#&%
3
(7)3
aYVh=LH`B^WBDCGEFB[H`JJ%V¾hWEH`aYEFCGBDVYXDT=J XUCS[CFJ%^W\WXULlaYC6CF^e3CF\Wh_g]4OC`LfFL@BZdWXDT=LLxk^=aCGVYXIHZ
XDVYCFhC`XUTWLBDL@]U\WaYXD]^=BDCWg6\_S[LZgenmXDTWLH`aYEFCGBDVYXDT=J]jTWL ^WBDCG^_CG]DLgVY]\WhW^=BIHFSxXDVS@H`ag6\WL
XDCXUTWLl]UVY@LC`L[kW^WBUL@]D]UVCGhW]jXDC e_LL@FHKa\3H`XDLZgH`X-BU\Wh6XDVYJLF5jTWVY]JHZme_LRVYJ^WBUC`FLg XDC
CFeWX[H`VYhHKhCG^WXDVYJ HKaL[kW^WBDLf]D]DVYCGh ]DVY@LGdH`]]U\WEFEGL@]DXULgVYh6"s+&DCFBUL@C`FLfBdVXRVY]^3CF]U]DVYeWaYL
XDC CFeWX[H`VYh H ^3H`BIHKaYaLfaH`aYEFCGBDVYXDTWJ 1CGBaYC6CF^EGL@hWL@B[H`XDVYCGh
jTWLBUL@J HKVh3g6L@BCKlXDTWL%^_H`^3L@BVY]CFBUEnHKhWVY@Lg HK]CFaYaYC`j]3L_BD]UXVYaaY\W]UXDB[H`XDLXUTWL
^WBDCGeWaYL@JCGh$HKhL[k_H`J%^WaYL!H`h_gBULS@HKaa]DLfFL@B[H`ae_H`]UVSSxCFh_S[Lf^WXD]Z76LS[XUVCGh98*XDT=L@h^WBDLx
]DL@h6XU]!XUTWL^_H`B[H`JLfXDBDVS aYVh=LH`B ^WBDCGEFB[H`JJ%V¾hWEH`aYEFCGBDVYXDT=JA/pDA H`h_gXUTWL]UXDBD\3S[XD\WBUL*C`
VYXD]BDL@]U\WaYXD]Zph:76LS[XUVYCFhvLg6L@]ISxBDVYe_L%XDTWLHK^W^WaYVS@H`XUVYCFhW]RC`jXDTWVY]H`aYEFCGBDVYXDT=J XUC aYC6CF^
EFL@h=L@BIHKXDVYCFhdWHKh_gEGVGLHKh L<; S[VYL@h6Xl^_HKBIH`aYaYL@aHKaEGCFBUVXUTWJ 1CGBjEFL@h=L@BIHKXDVYhWEaYC6CF^e_CG\Wh_g6]Z
=LXDT=L@h g6VY]IS[\=]D]XUTWL^_LfB1CGBDJ HKh_S[L@]HKh_g*XUTWLBDCGeW\W]UXDhWL@]U]!CKXDT=V] JL@XUTWCWgd_HKh_g*L
S[CFh3S[aY\_g6LjVYXDT*g6VYBDLSxXDVYCFhW]CFB1\WXU\WBDLCFBU
> ?
zA@ ' & x
/ClVYaYa\=]DXDB[H`XULXUTWLj^WBDCGeWaYL@J]CFhWLHFS[L@]Vh aC6CG^!EGL@hWL@B[H`XDVYCGhd`aYL@X/\W]S[CGhW]DVg6LfBHRTWL[k_H`EGC`
h_H`a6XUVaYLjCGeWXIHKVh=LgjT=L@h^_H`BUXDVYXDVYCGhWVYhWEHR^WBUCFEFB[H`JB*_E3WM*#K(CjTWLjVYhWLG\_HKaVYXUVLf];gL/_hWVYhWE
XDTWVY]XDVYaYLH`BDLEDGF9HIs,J tWdKFLD%I6HIsMJ#t=dNDLJ tWdOHJ t=dPDAQR8_dH`h_gSHQT83_p
CFhWLjH`h6XD]jXUC!][S@H`hXDTWVY]lg6CGJ H`VYh e6mJ HK6Vh=EUHFH`BUme_Lx1CFBULVDDd=XDTWLfh XUTWLe3CF\Wh3g6]jC`D
J\W]DX/e3LS[CFhW]UXIHKhnXjH`h3gXDT=CF]DLCK%H!J HZm!gL@^_Lfh_g%CFhWDUXjTWLS[CFBUBDL@]U^_CGh_g6VYhWE aC6CG^hWL@]UX
VY]XUTWLCFh=L]UTWC`jhVYhE_EG\WBDLMYf
" 3
.
!"#$%&(')*+-,"./$-0&('
123254+2567859:<;=54>?78-7A@B=C6
1-,D-,E(F
1-,D-,E(F
H XDTWLVYhW^=\WXgCFJ HKVYh eXUTWLS[CFBUBDL@]U^_CGh_g6VYhWE%aC6CG^hWL@]UX
VYEF\WBULMF7WHKJ^WaYLgCFJ HKVYhFH TWL[k_H`EGCFh_HKaXDVYaYL
G
'( l l&-
CFhW]UVg6L@B H91KDg6VYJL@h=]DVYCFh_HKaH;hWL]D^_HGS[LHl-P 6,JI
3K 5&-ML
3
&$&, VY]H]DLfXC`
^_CGVYhnXU]ON C`/H][H`XUV]Ð1m6VYhWE!HL_hWVYXDLR]DL@XC`S[CGhW]DXUBIH`VYh6XD]QP(RSNJUTRV#DW MVZZXV#0 =BUVXUXDL@h
VYh!JH`XDBUVk1CFBUJdZXDT=L@]DLS[CGhW]DXUBIH`VYh6XD]1CGBDJ XDTWLRaYVh=L@]-C`¨HKhVYhWLG\_H`aYVYXm]Umn]UXDL@JZY[NJUT`d
TWL@h_SxLXDTWLhWCGXIHKXDVYCFh,\W]^N_^Y[N,JUT-`Aa
CFhW]UVgL@BXCFLSxXDCFBU]NbW .NcVdNfeVZZVdNhg %H`h_gjiUW .i(c^Vie(VZXVdi-kfml-LSI
XDCGBnNVY]]IHKVg -
3CK )
"
#$
)3$#6-4-oL,7%.#&-4-
3 (
#&,Ui 1hWCGXDLgpNjqri VXUTWL@BDLL[kWV]UXD]H
^WBDL<Wk SN c VdN e VZZVdN(s S1jVYXDTtRQJ%VYh0Vdu% S[CGJJ%CFhXDCpNH`h3gUi HKh_g L@VYXDT=L@B
tvW JVYhO0Vu%jCGB[N(sw cyx isw c a)zjE3d6e3L@XL@L@hXDTWL FL@BUXDVS[L@]C`-XUTWLTWL[k_HKEFCFh3H`aXUVYaL
1BDCGJ 3E_/M*#%T=CFag6]lXUTWLBDLfaHKXDVYCFh ctJV[t {q cthVIs Qq cshVIt *qTsJV 8 *q 38fVIs Qq 38fV 8Xa
jTWL-aYL[kWVS[CFEGBIHK^WTWVS@H`a6CGBIg6L@BUVYhWElCK H`h6mXCRFLSxXDCFBU]/VJ%^WaYVYL@]XDTWL Q2RCFBIgL@BDVYhWERC`_XUTWL@VYB
_BD]UXlS[C6CFB[g6VYh_H`XUL@]
P - VY]g6L@][S[BDVYe3Lge6m*EGVYnVYhWEVYXU]aYC6CF^Vh3g6L[kdaYC`LfBH`h3g*\W^=^_L@B e_CG\Wh_g6]ZdH`h_g
VYXD]e_CWg6m6%jTWLaYC6CF^e_CWg6mVY]H%]ULG\WL@h_S[LCK]UXIH`XUL@JLfhnXU]XDCe3LL[kWLS[\WXULg 1CFBL@GL@BDm
]D\_SfS[L@]D]UVYFLFH`aY\WLCKXDTWL aYC6CF^VYh_g6L[kd3BIHKhWEFVYhWE 1BUCFJ XDTWL aYC`LfBle3CF\Wh_g XDCXUTWL \W^W^3L@B
e_CG\Wh_g VYh_S[aY\W]DVYGL @ C6CG^W]S@H`h*e3L hWL@]DXULgdVLGd3S[CFh6X[H`VYhWLgVYh*XDTWL e3Cgm CKH`h=CFXDT=L@B
aYCnCG^d[CFBDJ%VYhWElH- $ , 3
$(
; P aYC6CF^h=L@]DXVY]¨][H`VgXUCe_L
3
*
3 (/V¾=1CFB/L@GL@BDm aYCnCG^VYh!XUTWL
hWL@]UXdVYXD]aYC6CF^*e3CWg6m*S[CFh=]DVY]DXU]C`LfVXUTWL@BLxk=HGS[XDaYm*CGhWL aC6CG^d3CFBH%aYC6CF^61BUL@L]DLZF\=L@h_S[L
C`/Vh=]DXDBU\_S[XUVCGhW]cHK]jVYhW3EF\WBULM#`(
WCGBH%^_HKBDXUVSx\WaH`BL[kWLSx\WXDVYCFh CFBVYXUL@BIHKXDVYCFhRCKOH%aYC6CF^dXUTWLFH`aY\WLf]CKXUTWLVYh_gL[k
C`XUTWVY]jaC6CG^ H`h3gCF\=XDL@BjCGhWL@]d6X[H`GL@h%1BDCGJ(XDTWLCG\WXDL@BUJCG]DXXDC XDTWLVYhWhWL@BUJCG]DXdK1CGBDJ H
FLSxXDCFBS@H`aYaYLg )(3
$# ()9&,I
3 (6
;
jTWL!]UL@XlCK-XDTWL VYXDL@B[H`XDVYCGh GLS[XUCFBD]CK;XUTWLVYhWhWLfBDJCG]DX
aYCnCG^VYh H aYC6CF^%hWL@]UXj1CGBDJ]XDTWL )( 3
#!(*)96,*5# 3
C`/XDTWLaYC6CF^hWL@]UXOjTWLL[kWLS[\WXUVCGh*C`
3 3 3
HaYC6CF^*h=L@]DX][S@H`h=]XUTWL^3CFVYh6XD]lCKVYXU]lVYXDL@B[H`XUVCGh]U^_HFSxL!VYhH`]ISxL@h_g6VYhWEaYL[kWVS[CFEGBIH`^=TWVS@H`a
CFB[g6L@B
=LHKBDL!SxCFh_S[LfBDhWLgTWL@BULCFh=am%jVYXDTeW\WVYag6VYhWE ^_L@BÐ1LS[XaYC6CF^%hWL@]DXU]%/C eW\WVYag%]D\_SUT
H*h=L@]DXZd-L h=L@Lg XUC S[CFJ%^W\WXDL%XDTWLaYC`L@BH`h_g \W^W^3L@Be3CF\=h_g 1CFB LHFSUTaYC6CF^dXUTWL@VYB
e_CWg6VYL@]e3L@VYhWEVYJ^WaYVS[VYXDam!g6L/_h=Lg%enmXUTWL^_LfB1LSxXlaYC6CF^hWL@]UXl]UXDBD\3S[XD\WBULE1VYh6XDLfBDJLZg6VH`XDL
aYL@FL@aY]/-H`h_g XDTWLRVYhWVYXDVH`a6^WBUCFEFB[H`J 1VYhWh=L@BDJ%CF]DXaC6CG^(NjT=L@BDL[CFBDLGdWXDT=LlaYC6CF^ EFL@hWLfBIH`XUVYCFh
^WBDCGeWaYL@J S@HKhe_LBUL@]DX[H`XULg*H`]CFaYaYC`j]
" )I 3 , #6,.)( 3
$# ()9&,W &%.#&)4, V 6
3 I 3
L &/&)4,# ( 3 N R 5&)0,>(3mN
, ( 3
Y$&:>,'.yN R #' #L7:>,$()96, ( 3
$ 6#6)4, # ( 3
{N c VZZVdN R/c
;
phXDT=V]g6L/_hWVYXUVCGhdGXDTWLS[CGJ^W\WX[H`XUVCGhW]/C`¨HKhnm XCg6VY]UXDVYh_S[Xe_CG\Wh_g6]jH`BULlVYh_g6Lf^_L@h
g6L@h6X/^WBDCGeWaYL@J]/H`h_g CFh=am XDTWLjSUTWCFVS[LjCK XUTWLe3CF\=h_g EFL@h=L@BIHKXDVYCFhH`aYEFCGBDVYXDT=JJHZmH LZS[X
XDTWLfVBjVYh_gL@^_Lfh_g6L@h_SxLF
&* R; `` &*-
VYFL@h Hg6CFJH`VYh: W ]N\_ Y[NRJt `+V/L*]DLHKBISUT 1CGBH^_H`B[H`J%L@XDBUVS L[kW^WBDLf]D]DVYCGh
C`jXDTWLe3CF\=h_g6] C`yNJRH`]!H 1\Wh3S[XDVYCFhCKSN cBVN)eVZZVNJR/c fphXUTWVY]]DL@h=]DLFd/CG\WBEFC6H`a
VY]XDC _h_g XDTWL^_HKBIH`J%L@XDBUVSLxk^=BDL@]U]DVYCFh C`L[kWXDBDLfJL*FHKa\=L@]C` NhR]D\_SUTXUT_H`XXUTWL@BUL
L[kWVY]DXD] H`hN W .N c VZZdVN RVZZVN'
aGjTWLvN R ]]IHKXDVY]1m6VYhWEXDT=Le3CF\Wh_g]1CFBUJH
^WBDCZo[LS[XUVCGh*CFhXDTWLEDYXDTS[C6CFB[g6VYh_H`XULC`XDTWL VYhnXUL@BD]ULS[XUVCGhCK jVYXDT*XUTWL Tnm6^3L@BD^=aHKhWL
W]^i _i[W N+V MUQ HVQ D FM`AVF^3H`BIHKJL@XUL@BDVY@LZge6m,SN c VN e VZZVdN R/c 5a
76VYh_S[LlXUTWLSUT_H`B[HFS[XUL@BDVY]IHKXDVYCFhCKXDTWVY]^WBDCoxLS[XDVYCGh VY]-h=CFXjg6VYBDLS[XUaYm HZFH`VYaH`e=aLGdLSfH`h
]DLHKBISUT1CGBH%aYL[kWVS[CFEGBIHK^WTWVS@H`aYaYmLxkXUBDL@J%L^_CGVYhnXCKLa%jT=V]HKaaYC`j]R\W]XDT=L@hXDC
_h_g XUTWLL[kWXDBDLfJ HCKnNhRDd]DVYh_S[LXUTWLaYL[kWVS[CGEFBIHK^WTWVS@HKalCFB[g6L@BUVh=E C` FLS[XUCFBU]*VYJ^WaYVYL@]
XDTWL ]DX[H`h_g=H`BIg Q2%CFBIgL@BDVYhWEC`XDTWL@VYBA_BU]DX!S[C6CGBIg6VYh_HKXDL@]ZjT=LS[CGBDBUL@]D^3CFh_gVh=E*aYVh=LH`B
^WBDCGEFB[H`JJ%V¾hWE^WBUCFeWaYL@J8V]XUTWL@hXUC _h_g HaLxkVS[CGEFB[H`^WTWVS@HKaYamJ%VYhWVYJH`a/^_CGVYhnXCKH
^_H`B[H`J%L@XDBUVSO^_CGaYmnT=Lg6BDCGh
jTWLj]DLHKBISUTC`H`h L[kWXDBUL@JLj^3CFVYh6X/\Wh_g6LfBHJVYhWVYJVH`XUVCGhS[CFhW]UXDB[H`VYh6X¨VY]-L<; S[VYL@h6XDaYm
^_LfB1CGBDJLZg*e6m*XUTWL]DVYJ^=aLxkHKaYEFCFBUVYXDTWJ4jC`L@FL@BZdXDT=L^_H`B[H`JLfXDBDVS H`]U^_LSxXHKh_g*XUTWL
aYL[kWVSxCFEFB[H`^WT=VSfH`aWCGBIg6L@BUVYhWEC`L[kWXUBDL@J%L^3CFVYh6XD]J\W]UX-e_LL@Je3LgWgLgVYhXUTWLH`aYEFCGBDVYXDT=J
jTWVY]ljH`]g6CFhWL enm*=LH`\=XDBDVYL@BZd3jTWC^WBUL@]DLfhnXULgVYh:"8'&-XDTWLA/pDA #6#&%
3
(*7),>(
3 " 3
"
#&%%)4,
"
RH`aYEFCGBDVYXDTWJ jT=VSUTCL@BD]Zd/e_Lf]DVg6L@]CGXDTWLfB1\=h_S[XDVYCGh_H`aYVYXDVYL@]Zd H%B[H`XDVYCGh_H`a
^_H`B[H`J%L@XDBUVSO]DVYJ^=aLxk HKaEGCFBUVXUTWJ
ph B[H`XUVCGh_H`a3JCWg6LGdWA/pDA S[CFJ%^W\WXDLf]jXDTWLaYL[kWVS[CGEFBIHK^WTWVS@HKa_J%Vh=VJ \WJ C`;H ^3H`BIHKJL[
XDBUVSRB[H`XDVYCGh_H`a3^_CGaYmnT=Lg6BDCGh
mW]^N_-YAI YiJ TVNJtJViJtJV)J t `+V [M
^_CG]D]UVe=am\=h_g6L@BHGgWg6VYXDVYCFh3H`aS[CGhW]DXUBIHKVh6XU]CFh XUTWL^_H`B[H`J%L@XDL@BWd/EGVYFL@henm H]Dm6]UXDL@J
J"!lS@HKaYaLZgXUTWLW$&,>(
3K
(lCK¨XDT=L^=BDCFe=aLfJ
jTWLS[CFhW]UXDB[H`VYh6XD]lCGhXUTWL]DVYEGh CK;S[CGJ^3CFhWLfhnXU]CKOiH`h_g#H`BULhWCFXjBUL@]DXUBDVS[XDVYGLF3Vh
H%aC6CG^hWLf]DXd/CGhWaYm*XDT=LCF\WXUL@BDJ%CF]DX aC6CG^JHZm T3HZFLVYhX_hWVYXUL@aYm*J HKhnmVYXDLfBIH`XUVYCFhW]Z%$-m
" 3
J HKnVYhWEVYXD]VYh_g6L[kEFBUC`XDC`jH`B[g6]jVYhX_hWVYXmLLfhW]D\WBULXDT3H`XjVYXD]aYC`L@Bje3CF\=h_gVY] _hWVYXULFd
H`h_g e6m H`h*HK^W^WBDCG^WBDVHKXDLlXUBIHKhW]DaH`XUVYCFhLS@HKhJ H`GLlH`aYaaYC6CF^VYh_gVSxL@]j^_CG]DVYXDVYGLF
¨CH`^=^WaYmXDTWVY]lH`aYEGCFBDVYXUTWJ XDCCF\=Bj^WBDCGeWaYL@J(L!S@H`hhWCGXDVS[LXDT3H`Xje6m]DL@XUXDVYhWE
Y2I Yi WjY
i
s
W /VY]XDTWLR^WBDCZo[LS[XUVCGhCGhviCKXDTWLR^_CGam6TWLZg6BDCGh W ] hVdi y_-YJVi JUT-`Aa 4OL@h_S[LFd
e6m ]DLfXDXDVYhWE pW SN cBVN)eVZZVNJR/c %H`h_gpiUW .NJRVZZ#VN VYh LG\_H`XUVCGh ML SfH`h
S[CFJ%^W\WXULXUTWLaYL[kWVS[CFEGBIHK^WTWVS@H`aJ%VYhWVYJ!\WJ CFBjJ H kWVJ\=J d`]UL@L,"+&3OC`2SNhR#VZZVdN RH`]
H\Wh_S[XUVCGhC`-.Nc-VdNfeVZZVdNhR/c#5a
VYFL@hCF\=BCFeFo[LSxXDVYFL@]ZdZXDTWLA;pDA*H`aYEFCGBDVYXDT=JH`aYaYC`j]\W]XUClS[CFJ%^W\WXUL¨CGhWL-aYC6CF^Re_CG\Wh_g
^_LfB!S@H`aYad3aYLHFg6VYhWEXUC XUTWLBDLf]DCFaY\WXUVYCFhC`]DJ HKaYaLfBl^WBUCFeWaYL@J%]XUT_H`hVYhXDTWLSfH`]DLCKXUTWL
WCG\WBDVYL@BÐÐ CFXDfnVYh L@aYVYJVYh_H`XUVYCFh[PRa]UC_d`]UVYh_S[LA;pDAV]e_H`]ULg CFhXUTWLl]UVJ%^WaYL[k!H`aYEFCGBDVYXDT=Jd
LjT3HZFLjHJ\_SUTe3L@XDXUL@Be3L@T_HZ6VYCF\=B-jBDXZFVYh XDL@BUJ]/C`]U^_HFS[LlSxCFJ^=aLxkVYXmZXDTWLj]UVYJ^WaYL[k
VY]HS[CGhW]DX[H`h6X]DVY@L H`aYEFCGBDVYXDTWJd`H`h_g%XDT=L^3H`BIHKJL@XUBDVSlSxCFJ^=\WXIHKXDVYCFhaYLHGg6]jXDCH eWVYh_H`BUm
XDBUL@LC`/BDLSx\WBD]UVGL!S@HKaaY]jXUCXDTWLA/pDA H`aYEGCFBDVYXUTWJd`jT=CF]DL g6VYJL@hW]UVYCFh]DXUBDVS[XUamg6LS[BULH`]UL@]
jTWL@hXDT=LgL@^WXDTCK¨XDT=LSfH`aYaEFBUC`j]U]DL@LW"85&Ð(CjT=V]jXUBDL@LVY]jL[kW^WaYCFBULg%VYh H g6L@^WXUT633BD]DX
CFB[g6L@BH`h_gXDT=LBDL@]U\WaYXVY]SxCFhW]UXDBD\3S[XDLg CFhXDTWL_m6N4jLfh_S[LFdCFhWaYmXDT=LS[\WBDBUL@h6XeWB[H`h_SUT
C`jXDTWLXUBDL@L%J\W]DXe3LT=L@agVYhJL@J%CFBUmn3wlh=L S@HKh ]DT=C`(XDT_HKXVYXD]HZFL@B[H`EGLg6L@^WXUT VY]
E *1Fs/aCGE0WV=jTWL@BULV0 VY]XDTWLh6\WJe3L@BC`-S[CFhW]UXDB[H`VYh6XD]gL/_hWVYhWEXUTWLg6CFJH`VYhH`h_g
1VY]/XDTWLg6VYJL@hW]UVYCFhCK XUTWLjVYh_g6L[k]D^_HGS[LF jTWVY]-aYLHFg]-XDCH`hHZGL@BIHKEFLj]U^_HFS[LlSxCFJ^=aLxkVYXm
C`E0W1*1Fs/aCGE-0W5a
ph SxCFJ^3H`BDVY]UCFhdGXDTWLS[aHK]D]DVS@HKae3CF\Wh3gCGhJ%L@JCGBDm BDLG\WVYBDLfJL@h6XD]jCK¨XUTWL!=CF\WBUVLfB
CFXU@6VYh LfaVYJ%Vh3H`XDVYCGhV]EGVYFL@h e6mE1 0 e
5a-jT=V]e3CF\=h_g J HZme3L*VYJ^=BDC`FLZge6m
HFgWgVh=EBULg6\Wh_g=H`h6X-S[CFhW]UXDB[H`VYh6XBDLfJC`FH`a`CGBTWVY]DXUCFBDmSUTWLSU6] 4jC`L@FL@BZdZXDTWL/ CFBDJ%L@BS@HK]DL
VYJ^WaYVYL@]-CGhWLaYVYhWLH`Bj^=BDCFEGBIHKJ XUCe3L]DCGaGLg%1CGBjLHFSUT*BULg6\Wh3gWH`h_SxmSUTWLSU 1LFE_Yd=\W]DVYhWE
XDTWLhWCFhW^3H`BIHKJL@XUBDVS]UVYJ^WaYL[k HKaEGCFBUVXUTWJ H`h_g]DaYC`j]S[CFhW]UVg6L@BIHKeWaYm XUTWL*L[kWLS[\=XDVYCFhd
jTWL@BULH`]jTWVY]UXDCFBUmSUTWLSU6]lBDLZF\=VBULH`hCFB[g6L@BjC` 0 HGgWg6VYXDVYCGh_H`a3]DXUCFBIHKEFLR]D^_HGS[L1CGBL@GL@BDm
S[CFh=]DXDB[H`VYh6XEGL@hWL@B[H`XDLZg enmXUTWLHKaYEFCFBUVYXDTWJ/jTWL@BDLx1CFBULFd`XDT=LJHZkWVYJ!\WJ ]UVfLCK_XDB[HFS[X[H`eWaYL
^WBDCGeWaYL@J]jVYaaGe_Lj\=]D\_HKaaYmJ\_SUTaH`BUEFL@B1CGB/XDTWLjA/pDAH`aYEGCFBDVYXUTWJXUT_H`h 1CGB¨XUTWLl=CF\WBUVLfB
CFXU@6VYhL@aYVYJVYh_HKXDVYCFh
jTWLBUL@]D\WaYXU]^WBUCg\_S[Lge6m A/pDA HKBDLeWVYh_HKBDm]DLfaLZS[XDVYCFh XUBDL@L@]CGh XUTWL^_H`B[H`JLfXDL@B
FLSxXDCFB=dWVLFYdnSxCFh_g6VYXUVCGh_H`aLxk^=BDL@]U]DVYCFhW] C`-XDT=LlCFBDJ
) ! J t ( 3
, c 3
-(
3
ye'
jTWL@BUL$cH`h_gye H`BULL@VYXDTWL@B ]DL@aYLS[XUVYCFhXDBUL@L@]CGBFLS[XUCFBU]!CKH';hWL 1\Wh3S[XDVYCFh=]CGh =
jTWLFHFS[\WVYXmCK VY]-g6L@hWCGXDLg e6mXDTWLjLxk^=BDL@]U]DVYCFh9+3 jTWVY]-S@H`]ULjJ HZmH`BDVY]ULjjTWL@h
hWCS[CGhnXUL[kWXlVY]j]D^3LS[V _LZgVhXUTWLVYhWVYXDVH`a3^WBUCFeWaYL@JdWH`]jCFBj]DCGJLRFH`aY\WL@]RC` XDTWLfBDL jVaYa
e_LhWC^3CFVYh6XD]VYh(jVYXDT^WBUL/WkW
jTWLaYLHZGL@]CK!XDT=LXUBDL@L HKBDL*GLS[XDCGBD]*CKH';h=L1\Wh_SxXDVYCFhW]%CFh WdjH`h3gg6L@][S[BDVYe3L
XDTWLS[C6CFB[g6VYh_H`XUL@]C`XUTWL aLxkVS[CGEFB[H`^WTWVS@HKaJ%Vh=VJ\=J CK HK]j1\Wh3S[XDVYCFh=]CK W=CFB
L[k_H`J%^WaYLFd[XDTWLaYC`LfBe_CG\Wh_gCKHVYhXUTWLTWL[k_H`EGCFh_HKaFXDVYaYL/VYh_EF\=BDLM/jVaYaGe_LBDL@^=BDL@]UL@h6XDLg
H`]EUV¾2FUD I:8Jt XDT=L@hV2FUDI sJt XDTWLfh$t L@aY]ULVDGF s L@h_g6V-L@aY]DLOL@h_g6V#3jTWVY]
3 3 3
JLHKhW]/XDT_HKX-CFB-D ]DXDBUVS[XDaYmEFBULH`XUL@BXDT_HKhW8XUTWLl^3CFaYm6TWLg6BUCFhVY]-L@J%^WXm6dFH`h_g CGXDTWLfBDjVY]DLFd
V D V]]DJ HKaYaLfBXDT3H`h s=d`XDT=LjaC`L@B-e3CF\=h_g C`%HV]tWFVYh_H`aYaYm6d@CFB-D ]IH`XUVY]1m6VYhWEs x DKQ68
XDTWLaYC`L@Bje_CG\Wh_g%C`GH!VY]UD F sW
"$%!&
- '
@ */
ph"v+&Ðd6=LH`\WXUBDVYL@B]D\WEGEFL@]UXD]jXUC!\W]ULg6VYBULS[XDaYmXUTWLS[CGh_g6VYXDVYCGh_H`aWXUL@BDJ%]^=BDCWg6\_S[LZge6mA/pDA
H`]jHFS[XU\_H`ae3CF\=h_gL[kW^WBUL@]D]UVYCFhW]ZjT=LJH`VYh VgLHCK¨XUTWLaC6CG^EFL@hWLfBIH`XUVYCFh*H`aYEFCGBDVYXDT=J VY]
XDTWLfh XUCSxCFJ^=\WXDLjXUTWLS[CFh_gVXUVYCFh_HKa_XUL@BDJ%]-1CGBHKaYa_e3CF\Wh3g6]RH`h_g/RdVGCFBaC`L@BjH`h_g
\W^W^3L@Bje3CF\Wh_g%CK¨XDT=L DXDT*VYh_g6Lxk d6BUL@]D^3LS[XUVYFL@aYm624OL^WBDCG^_CG]DL@]RXCHKaEGCFBUVXUTWJ]ZdZe_CGXDT
CF^3L@BIHKXDVYhWE VYXDL@B[H`XDVYGL@aYm1BUCFJ(XUTWLCF\WXUL@BDJ%CF]DXjXUC!XUTWLVYhWhWLfBDJCG]DXVYh_g6L[k
jTWL3BD]DXH`aYEFCGBDVYXDTWJ©g6C6L@]/hWCGX¨\=]DLjS[CFh6XDLxkXU]d6H`h_g^WBDCWg6\_SxL@]aHKBDEFLfBe3CF\Wh_g L[kW^WBUL@]
]DVYCFh=]3P ]DLS[CGh_gCGhWL V]XUTWL@h^WBUCF^3CF]DLZgXDCBDLg\_S[LXDT=L]DVY@LC`e_CG\Wh_gL[kW^WBUL@]D]UVCGhW]d
H`h_g \W]UL@]XUTWLje3CF\Wh_g]-C`CF\WXUL@B-VYh_g6VS[Lf]H`]-XUTWLlS[CGhnXUL[kWX-jTWL@hSxCFJ^=\WXDVYhWERXDTWLje3CF\Wh3g6]
C`/XDTWLh=L[kWXlVYh_g6L[kXjTWL g6V L@BUL@h_S[L VYhXDTWL]DVY@L C`-BDL@]U\WaYXlXUBDL@L@]S[CGJL@]1BUCFJ XDTWLHFS[X
XDT_HKXljTWLfhhWCHFg6LZF\3H`XDLS[CGhnXUL[kWXV]R\W]DLZgdXDT=LHKaEGCFBUVXUTWJ(L[kW^WaYCGBDL@]FH`aY\WLf]lC`^_H`B[HZ
JL@XUL@B/1CGBjT=VSUT%XDTWLR]DCFaY\WXUVYCFhVY]LfJ^WXm1LlJ%LH`]U\WBDLg \W^XUCyGt C` hWCWg6Lf]jVh
XDTWLXUBDL@L@]S[CGJ^W\=XDLg jVXUTWCF\=XS[CGh6XDL[kWX(
4jC`L@FL@BZd/L%J\W]DX HFgWgTWLfBDL]DL@GL@BIHKalhWLf H`BUEF\WJ%L@h6XD]XUCXUTWL*g6VY]IS[\W]U]DVYCFh^WBDLx
]DL@h6XULgVYh "v+&DVBU]DXZdXUTWLJ\WXU\_H`a/VYh_g6Lf^_L@h3g6L@h_S[LC`lg6VY]DXDVYh_SxX!e3CF\Wh3gS[CFJ%^W\WXIHKXDVYCFh=]
H`aYaYC`j]XDC*^3L@BCFBDJ8XDTWL%jTWCFaYLaYC6CF^EFL@hWLfBIH`XUVYCFh VYh^_HKBIHKaaYL@aPRaY]DC_dXDT=L: *h=CgL@]
J HZme3LlVYEFhWCGBDLg dnHK]-XDTWLfmS[CFBUBDL@]U^_CGh_g%XDCVYXDLfBIH`XUVYCFhFLZS[XDCGB^WBUL/WkWL@]jjTWVSUTjVYaYaWhWL@GL@B
e_LRBDLHGSUTWLgg6\WBUVh=E!XUTWLlL[kWLSx\WXDVYCFhCKXDTWLRaC6CG^hWL@]DX]DL@LE"y+&D(/XDTWL]UL@J HKhnXUVS[]-C`aYC6CF^
hWL@]UXL[kWLS[\WXUVCGhL@hW]U\WBDLf] XUTWLL@h61CGBIS[LfJL@h6XC`XDTWL*S[CGh6XDL[kWXdjTWVSUT SxCFhW]UV]UXD]%C`XUTWL
e_CG\Wh_g6]CKCG\WXDL@BaYC6CF^VYh_g6VS[L@]ZVh3H`aYaYmnd6L[kW^3L@BDVYJ%L@h6XD]]UTWC` XUT_H`XHZFCGVg6VYhWEXDTWL\=]DL
C`-S[CFh6XDLxkXjaYLHGg6]jXDCH J%CFBULlXDVYJ%L[cL<; S[VYL@h6XL[kWLSx\WXDVYCFh ]DL@L]ULS[XUVCGh W(
L@XU]\=]hWCFXUL[A/pDA
3K
(* 5#&7% QI!#6 2L' U (#*HSfH`aYa/XDC*A/pDA jTWVSDT]DCGaYFL@]H`h
H`^W^=BDCF^=BDVH`XULjL[kWXDBDLfJ\WJ ^WBDCGeWaYL@J cHFSfS[CFB[g6VYhWE!XUC!XUTWLlFHKaY\WLjC`
3CK
( -jVYXUT^_H`B[H`JLfXDL@B
FLSxXDCFB 5#&% FH`BUVH`eWaYLlGLS[XUCFB I&#&6d3S[CFhW]UXDB[H`VYh6Xj]Dm6]DXUL@J!L'HKh_gS[CFh6XDLxkXE$( H`h_g%aYL@X
-/)$(+ $ :, $()9&, le_LXUTWLR1\Wh_S[XUVYCFhjT=VSUT*S[CGJ^W\WXUL@]XDTWL 3
(C`-S[CFhW]UXDB[H`VYh6XD]gL/_hWVYhWE
XDTWLe3CF\=h_g6] C`[NhRI%= VXUTXDTWLH`e3C`FLFdXDT=L:ISxCFh6XDL[kWXÐ11BUL@L H`aYEFCGBDVYXDT=JC`"v'&SfH`h e3L
jBDVYXDXUL@h*H`]lH ^_H`B[H`aYaYL@aCFh=LV._E_3sW(
jTWLXDL@BUJVYh_H`XUVYCFhC`-A/pDA #"8'&3L@hW]U\WBDL@]RXDTWL XDL@BUJVYh_H`XUVYCFhC`/XDTWVY]jH`aYEFCGBDVYXDT=J 4OC`
LfFL@BZdFXUCe_LR\W]DLZgVYh!^=BIHFSxXDVS[LlVYX-J\=]DX/e_LRJCWg6V _Lg VYh!CGBIg6LfBXUCEFVYFLj]UVYJ^WaYL@Be_CG\Wh_g
L[kW^WBDLf]D]DVYCGhW] jTWL*EGCnHKajV]%TWL@BULXUC H`XDX[H`VYh XDTWL 1CFBUJ CKLxk^=BDL@]U]DVYCFhW]^WBUCWg6\_S[Lg e6m
WCG\WBDVYL@BÐÐ CFXDfnVYhL@aYVJ%VYh_H`XUVCGh3J H kVYJH%H`h_g JVYhWVYJ H C`]UL@FLfBIH`aH';hWL 1\Wh_SxXDVYCFhW]Zd
g6L/_h=Vh=EH aYC`L@BHKh_gH`h%\W^W^3L@Ble3CF\Wh3gdWBUL@]D^3LS[XUVGL@aYmn=PlhL[k_H`J%^WaYLjC`-]D\3SDTe3CF\Wh3g6]
H`BULXUTWLe_CG\Wh_g6]RC`!VYhE_EG\WBDLMY@
jTWLBDLf]D\WaYXD]^WBUCWg6\_S[Lge6m*A/pDAS@HKh*e_L BDL@^WaHGS[Lgenm*HKh HK^W^WBUCF^WBUVHKXDLLxkXUBDL@J\=J
1L@VYXUTWL@B JVYhWVYJ!\=J CGBJ H kVYJ \WJWC`lH]UL@X!CKj1\Wh_SxXDVYCFhW] g6L/_hWVYhWEXDTWL_BD]UXS[C6CGBIg6V
h_H`XULjC`XUTWLS[CFBUBDL@]U^_CGh_g6VYhWE aLxkVS[CGEFB[H`^WTWVS@HKaWL[kWXDBUL@J\WJ #"s+&D(/P 1CFBUJ H`a6^WBUC6C`C`XUTWL
FH`aYVg6VYXmC`XDTWVY]XDBIHKhW]CFBDJH`XUVCGhjH`]-EGVGL@he6mCGaYaHKBIgdGWLHK\WXDBUVYL@BHKh_gqjVY]U]DL@X-VYh"y'&D
" 3
=3¦_~ ^_H`B[H`aYaYL@aÐA;pDA
¨~`} Y[NJUT
¦/~ /~`}ARc)R d /R c)R
¦_ D W#M~¦1/W` =9=¢>=;§¦
D WA;pDA2JVYh)V'"NcBVNfeBVZZVNJR/c &V'"NJR.VZZ#VN &#VXY[NJ TV
§¦/¢
__
¦_ D W#M~¦1/W` =9=¢>=;§¦
D WA;pDA 1JHZk V'"N c VN e VZVN R c&#V'"N R VZZVdN /&#VXY[NJ T V
§¦/¢
VEG\WBDLs=WA¨HKBIHKaaYL@a3e3CF\Wh_g%EGL@hWL@B[H`XDVYCGhH`aYEFCGBDVYXDT=J e3H`]DLZgCGh*A/pDA/
Pl]jH`hLxk=HKJ^WaYLFd6S[CGhW]DVg6L@BjXUTWLlLxk^=BDL@]U]DVYCFh*CKXDTWLaYC`L@Be3CF\=h_gCKNHVYhXDT=LT=L[k_HZ
EFCGh_H`a3XDVYaYLj1BUCFJ 3EF\WBULMFXjTWLS[CGh_g6VYXDVYCGh_H`a3XDL@BUJ UV-FUDI 8 J t XUTWL@hV-FUDIs)J t
XDTWLfht L@aY]DLED%F s L@h3g6VL@aY]DLEjL@h3g6V#^=BDCWg6\_S[LZg*e6m A;pDABULg6\_S[Lf]XUCJHZkcthV#D F s
jTWVSUTVY]jXDTWL L[kW^_LSxXDLg%CFBDJ
jTWL]DLfXCKH';hWL\Wh_S[XUVCGhW]HK^W^3LH`BUVh=EVYh*He3CF\Wh_g L[kW^WBDL@]U]DVYCFhS@HKh*e_LS[CGJ^W\6
XDLg \=]DVYhWEXUTWL1\Wh_SxXDVYCFh! "! $#%'&()*%'g6L@][S[BDVYe3LgH`e3C`FLF5jTWLjVYJ^WBUC`FL@J%L@h6XD]
CFeWX[H`VYhWLg VYh%^WBIHGS[XDVS[LH`BULlGL@BDmEGCnCWg6LfFL@h%aHKBDEGLlXDBUL@L@]]DL@GL@BIHKaT6\Wh_g6BULg6]jCK¨hWCWg6L@]/
S@H`h e3LjBDLg6\3S[Lg%XDCH`h L[kWXDBUL@J\WJ C`HjL@1\Wh_S[XUVYCFhW]Zd`jTWCF]ULjh6\WJe_LfB¨VY]/C`XDT=LjCFBIgL@B
C`-XDT=Lhn\=Je_L@BRC`S[CGhW]DXUBIH`VYh6XD] 0*CFBUL@C`FLfBdXUTWVY]]UL@XC`-\Wh_S[XUVCGhW]VY]lVYBDBULg6\Wh_g=H`h6X
VYhBIHKXDVYCFh3H`ah6\WJe3L@BD]ZdWVLFYdFXUTWL@BDL VY]jhWC 1\Wh3S[XDVYCFhVYJ^=aVYLg e6m%CFXDT=L@Bj1\=h_S[XDVYCGhW]jVYh XUTWL
]DL@XZ jTWVY]S@HKh*e_L LHK]DVYaYm ]UTWC`jh*e6mhWCFXUVYhWEXDT_HKXlLHGSUT*1\=h_S[XDVYCGh*L[kWXDB[HFS[XULg1BDCGJ XUTWL
BDL@]U\WaYXjXDBUL@LVY]lH`h*HFSxXD\_HKaL[kWXDBDLfJ\WJ 1CGBj]DCFJ%LlFHKa\=L@]jC`/XDTWL^_HKBIH`J%L@XDLfBD]
$-mS[CGaYaHK^W]DVYhWEXUTWLS[CFh_gVXUVYCFh_HKaWe_CG\Wh_g L[kW^WBDL@]U]DVYCFh=]jVh6XUCL[kWXDBUL@J H C`¨H;hWLj1\=h_SU
XDVYCFh=]d6LCGeWXIHKVYh H hWL@ 1CGBDJ(CK;H`aYEGCFBDVYXUTWJ,+-!.-/$0!1(21d]UTWC`jhVYhW3Ey=
3
- F l
=LCF\WXUaVYhWLTWLfBDLXDTWLBDL@]U\WaYXD]*HFSUTWVYL@GLg jVYXDT FHKBDVYCF\W]A;pDAce3H`]DLZgHKaEGCFBUVXUTWJ]ZN/C
]DXU\_g6m^WBIHGS[XDVS@HKa¨g6CGJ HKVh=]dGLBUL@jBDCGXDL XDTWL A/pDAH`aYEGCFBDVYXUTWJ \W]DVYhWEH J\WaYXDVY^WaYLj^WBDLZS[V
]DVYCFh%aYVe=BIH`BUm,#"54'&D( =LS[CGJ^_HKBDLg XDTWL^3L@BÐ1CFBUJ H`h3S[LC`¨XUTWVY]jVYJ^WaYL@J%L@h6XIH`XUVYC`h!jVYXDT*H
yFs ceWVYXjVYh6XDL@EGL@BlGL@BD]UVYCFh*CK=CF\=BDVYL@BDCGXD@6VYh*L@aYVYJVYh_HKXDVYCFh\W]ULgVhXUTWLA;pDA7*^=BDCo[LZS[X
H`X
i
z-S[CGaL g6L@]lVYhWL@]g6LA/H`BDVY]Z
7WH`J%^WaYLVYXDL@B[H`XDVYCGh g6CFJH`VYhW]lL@BUL^WBUCg\_S[Lg enmH^WBUCFXUCFXm6^_LCKH%^_H`B[H`aYaYL@aYVfVh=E
S[CFJ%^WVYaYL@BXjTWLS[CGBDBDLf]D^3CFh_g6VYhWE^3L@BCFBDJH`h_SxLJLHK]D\WBUL@JLfhnXU]HKBDL ]DTWC`jh*VYh*X[H`eWaYLMG
L@J%CFBDmBDLZF\=VBUL@JLfhnXU] 1CFBA;pDAce3H`]DLZgH`^W^=BDCnHGSUTWL@]H`BULEFVYFL@h1CGB!XUTWLCGBD]UXDS@H`]UL
S@H`aYaCK;LZHFSUT*L[k_H`J%^WaYLFXjVYJLRJLHK]D\WBUL@JL@h6XU]jC`-^_H`B[H`aYaYL@aA/pDA ]UXIHKh_g%1CGBjXDTWL aYCFhWEGL@]DX
]D\We=^WBDCWS[L@]U]L[kWLS[\WXUVCGhd_HKh_g%XDTWLS[\=J\WaH`XDLZgL[kWLS[\=XDVYCFh*XUVYJLjC`-H`aYa]D\=eW^WBDCWS[Lf]D]DLf]
3 3 3
5=_¦3~ L<; S[VYL@h6X^_H`B[H`aYaYL@aÐA/pDA
/~`} Y[NJUT 6¦/~ /~`}A
R
c)R d
R
c)R
¦3VD W©M~Z¦1/W`5=+=1¢>=-§¦
R W J H k1aYVY]DXÐcCK1 \Wh_S[XUVCGhW]/cA;pDA2JVYh)V'"NcVN)eVZZVNJR/c&#V
"NJRXVZVN & VXY[NJUTV
§¦(/¢
__
¦3VD W©M~Z¦1/W`5=+=1¢>=-§¦
R W JVYh 1aYVY]DXÐcCK1\Wh_S[XUVCGhW] cA/pDAL1JHZk V'"N c VN e VZXVdN R/c &#V
"N R VZVN#& VXY[NJUTV
§¦(/¢
VYEF\WBULy=WPlaYEGCFBDVYXUTWJ +-!."-"/$!1(2*1 jVYXDT%CF^WXUVJ%VY@Lge3CF\=h_g%L[kW^WBDL@]U]DVYCFh=]
=LL@BDL*h=CFXH`eWaYLXUCEFL@h=L@BIHKXDL*XUTWLaC6CG^ e3CF\Wh_g]VhXDTWLaH`]UXL[k_H`J%^WaYL\W]DVYhWE
WCG\WBDVYL@BÐÐ CFXDfnVYhLfaVYJ%Vh3H`XDVYCGhd[XDTWLjS[CGJ^W\WX[H`XUVCGhBDLG\WVYBDVYhWEJ%CFBDLXDT_HKh!XUTWLlHZFH`VYaH`eWaYL
JL@J%CFBUm]D^_HGS[LF
-H`eWaYLMFjL@J%CFBUm \=]IH`EGL$SxCFJ^3H`BDVY]UCFh e_LfXL@Lfh WCG\WBDVYL@BÐÐCGXD@6VYhdS[CFh6XDLxkXÐce3H`]DLZg
A/pDA H`h_g%^_HKBIHKaaYL@aA/pDA
lCGJ HKVh L@JCGBDmce6m6XDL@]/
gVJ 1X S[CGhW]DXUBD])0W D A/pDAFSxCFh6XDL[kWX ^_HKBIH`aYaYL@aA/pDA
y $ t=s$4 t=sFs tWsGv
8 Ms t=yFy t=yFv tWy!4
sFt MGyFs MG8ns MFvGv
$ s $ lvGt 83yWM vWvGt
jTWLBDLf]D\WaYXD]]UTWC`©XCVYJ%^_CGBDXIHKh6X^3CFVYh6XD]Z_XUTWLJ%CgL@BIHKXDLBULG\WVYBDL@J%L@h6XD]lCKA/pDA/d
L@]D^3LS[VHKaaYmO1CFBaH`BUEFL-S[CGJ^W\WX[H`XUVCGhW]d@HKh_gRXDTWLL[k_S[L@aYaYL@h6X^3L@B1CGBDJH`h_S[L/CKXUTWLHKaEGCFBUVXUTWJ
+-$."-/)!1(21 dGjTWVSUTLfFL@h]DLG\WLfhnXUVH`aYVY@LgVY]j1BUCFJv%\W^XDC 4Fv H`]DXUL@BXDT_HKhXUTWL
S[CFh6XUL[kWXe_H`]ULgFLfBD]DVYCGhBDCFJ "v+&D
P S[CFJ%^_H`BUV]UCFhRC`W^_LfB1CGBDJ HKh_S[L@]CKWXDTWL-]U\WeW^WBUCWS[L@]D]UL@]-CK=HKaYEFCFBUVYXDTWJ +-!.-/$0!1(21
*_E3%8 ]DTWC`j]XUT_H`X XDTWL@VYB!SxCFJ^=aLxkVYXmVY]!H`hL[kW^3CFhWL@h6XUVHKa-\Wh_S[XUVCGhC`jXDTWLh6\WJe3L@B
C`/^_H`B[H`JLfXDL@BU]1VLFYd6hWL@]DXUVYhWEaYL@GL@adL[k_S[L@^=XlCFBjXDT=L!S[CFJ%^W\WX[H`XDVYCGhC`-XDTWLVYhWhWLfBDJCG]DX
VYh_g6L[kjTWLfBDL XDTWLHFSxXD\_HKa¨S[CGJ^WaYL[kWVYXmS
3
7
3
#
3 ;
jTWVY]lH`aYaYC`j]-XDC^3H`BDXUVYXDVYCFh%XDTWL!SxCFJ
^W\WX[H`XDVYCGhW]j1CGBl]UJ H`aYahn\=Je_L@BU]lCK^WBDCWS[Lf]D]DCGBD]1BUCFJ XC\W^XDCL@VYEGTnX/jjVYXDTHEGC6Cg
aYCnHFg e_H`aH`h3S[LF% VYGL@hXUTWL@]UL^3L@B1CGBDJH`h_S[LBUL@]D\=aXU]dLHKBDLS[\WBUBDL@h6XDaYmCFBUnVYhWECGh$H
G\_H`h6XDVYX[H`XDVYGLJCWg6L@aC`XDTWLBD\=hWhWVYhWEXDVYJLCK+-!.-/$0!1(21*VYh CGBIg6L@BXUC^WBDLZg6VS[X
" 3
-H`eWaYL s= z¨kLZS[\WXDVYCGhXDVYJLSxCFJ^3H`BDVY]UCFhe3L@XLfL@h=CF\WBUVLfBDCFXU@6Vhd-S[CFh6XDLxkXÐce3H`]DLZg
A/pDA H`h_g%^_HKBIHKaaYL@aA/pDA
RCFJ HKVYh j]UL@BlXUVYJL]#
g6VYJ S[CFhW]UXDBU] D A/pDA ^3H`BIHKaYaLfaA/pDA
1X 0W SxCFh6XDL[kWX JHZk^_LfBl^WBUCSxL@]D] XDCGXIHKa
y $ t=sFy MFt $ tWsWM tW54 8
8 MZs 83sFu yW($Ft MFt 8 yWFy
sGt y 8nv!4WtFt 4=MF 8 MZyWtFt v $WYMt
$ s $ h FH u=MFMG86y M 86vWFy vFs=MF($WM
t=tWM t=YM
M
MZt MtGt MtGtFt
t M s y 8 v 4
.1< 4[>
2/4W>
.2¨0 44
=5!4 ,
2
4,
2
I
I I I I I I I I
VYEF\WBUL83JzkWLS[\WXUVYCFh*XUVJ%LjC`-]D\=eW^WBDCWS[Lf]D]DLf]C` +-!."-"/$!1(2*11 W $JV20 W s $
3 3 3
XDTWLCFBU6aC6HFg%CFBHEFVYGL@h$gCFJ HKVYh HKh_gXDC*e3LH`eWaYL XDC*T3H`h_g6aYL ^_H`B[H`aYaYL@a/^WBDCWS[Lf]D]DVYhWE
BDL@]UCF\WB[S[L@]CL@BULge6mJ\WaYXDVY^WBUCSxL@]D]UCFBjCFBD6]UXIH`XUVYCFhW]Z
( `)
jTWLjaYC6CF^ EFL@hWLfBIH`XUVYCFh^WBUCFeWaYL@J VY]-CFh=LjC`XDTWLRFL@m6]-XUCXDTWLREFL@h=L@BIHKXDVYCFhCK ^WBUCFEGBIH`J%]/Vh
^_H`B[H`aYaYL@aYVYH`XUVCGh!H`h3g^_HKBIHKaaYL@a=^WBDCGEFBIHKJ XDBIHKhW]CFBDJH`XUVCGh 4OC`LfFL@BZdFL[kWVY]DXUVYhWEJLfXDTWCWg6]
C`-aYC6CF^*EGL@hWL@B[H`XDVYCGhdWe_HK]DLgCFhWCG\WBDVYL@BÐÐCGXD@6VYh*^_HKVYBDjVY]DLLfaVYJ%Vh3H`XDVYCGhdF]U\ L@BR1BDCGJ
XCaYVYJVYX[H`XDVYC`hW]Vh=TWL@BDLfhnXXUCXDTWVY]HKaYEFCFBUVYXDTWJnHKh VYJ%^_CGBDXIHKh6X]U^_HFS[LS[CGJ^WaYL[kWVYXmH`h_g
H`h%VXUL@BIHKXDVYFLL[kWLSx\WXDVYCFhBDLG\WVYBDVYhWEH`aYaaYC6CF^%e_CG\Wh_g6]jXUCe_L!SxCFJ^=\WXDLg%H`XjCGh_S[LF
ph XUTWVY]j^_H`^3L@BRLVYh6FL@]UXDVYEnHKXDLg%XDT=L^WBIHGS[XDVS@HKa\W]ULC`-H aYCnCG^EGL@hWL@B[H`XDVYCGh J%L@XDT=Cg
1CFBUJL@BUaYm]D\WEGEFL@]UXDLge6m =LH`\WXUBDVYL@BVYh9"v+&D 4jVY]le_HK]DVSRVg6LHjH`]RXDC\W]UL!^3H`BIHKJL@XUBDVSRaYV¾
hWLHKB^WBUCFEFB[H`J%JVYhWE/XDC6CFaY]XDCSxCFJ^=\WXDL/aYC6CF^e3CF\Wh3g6]CGhWL-e6mCFhWLjH`h_gRL[kW^WBUL@]D]XDTWL@J H`]
S[CFh3g6VYXDVYCFh_HKaWXDLfBDJ]Zd`XDC e3LlL@FHKa\3H`XDLZgH`X-BU\Wh6XDVYJLF5jTWVY]-JL@XUTWCWg ]D\Weg6VY6Vg6L@]XDTWLRVh=V¾
XDVH`a=^WBDCGeWaYL@J Vh6XUC]UJ HKaaYL@B/CGhWL@]aLZHFg6VYhWEXDC H`hL/; S[VYL@h6XjS[CFJ%^W\WX[H`XDVYCGhdZeW\WX-^WBUCWg6\_S[L@]
aH`BDEGLHKh_g%Vh=L/; S[VYL@h6Xj]DCG\WBIS[L S[CWg6LF
=LL[kW^WaYCFBULgXDT=L!^3L@BCFBDJH`h_SxLC`H JCWg6V _Lg%H`^W^=BDCnHGSUT*jTWVSUT^WBUCWg6\_S[L@]CG^WXDV
J HKa¨e3CF\=h_gL[kW^WBDLf]D]DVYCGhW] =LH`aY]DC*]UTWC` XDTWL^3CF]U]DVYeWVYaYVXmC`l^_HKBIH`aYaYL@aYVY@VYhWEXUTWLaYC6CF^
EFL@h=L@BIHKXDVYCFh*^=BDCWS[L@]U]dXUTn\=]lBDLZg6\_S[VYhWEXDT=L!S[CFJ%^WVYaH`XDVYCGhXDVYJLFW=BDCGJ CF\WBRL[kW^_LfBDVYL@h_S[Lf]d
A/pDAe_H`]ULg H`aYEFCGBDVYXDTWJ%]_H`^=^_LHKB/XDCle3LcH`]UXDL@B/H`h_gJ%CFBUL¨BUCFeW\=]DX/XDT_HKh!=CF\WBUVLfBDCFXU@6Vh
L@aYVYJVYh_H`XUVYC`h@ph ^_H`BUXDVS[\WaHKBd`^WBUCFeWaYL@J%]/\Wh6XDBIHGS[XIHKeWaYLjjVYXDT XDTWLRaHKXDXUL@BdFg\WLjXDCVYXU]-]D^_HGS[L
BDLG\WVYBUL@JL@h6XU]d6T_HZGLe3L@L@h]DCFaYGLg%jVXUTGL@BDmEFC6CWgLxkLZS[\WXDVYCGh*XDVYJLf]
jTWL]DVYJ%^WaYVSxVXmCK-XDTWLJ%CWg6V _Lg%^_H`B[H`aYaYL@a¨HKaYEFCFBUVYXDTWJ(JH`GL@]jVYXjFL@BUm*LH`]UmXDC\=]DL
H`h_g VYJ^WaYL@J%L@h6XdZH`h_gHKaaYC`j]/XDC L[kW^WaYCFVYX/XDTWLRJ\WaYXDVY^WBDCWS[Lf]D]DVYhWES@HK^_H`e=VaYVYXDVYL@]/CK XUCg=HZm ]
^_H`B[H`aYaYL@a/CGBD6]DX[H`XUVCGhW]G=LH`BDLS[\WBUBDL@h6XDaYmCGBD6VYhWECFhXUTWL G\_HKhnXUVYXIH`XUVYFL JCWg6L@aYVYhWE
C`XDTWL^3L@BCFBDJH`h_SxL@]-C`XUTWLH`aYEFCGBDVYXDT=Jd@jT=VSUT]UTWCF\Wag TWL@aY^VYh 1\WaYaYm!L[kW^WaYCGVXUVYhWE]D\_SUT
1\Wh_SxXDVYCFh_HKaYVXUVYL@]
G- l
"YM/& PRh_S[CF\=BDXH`h_g pBDVYEFCGVh 7WS@HKhWhWVYhWE ^3CFaYm6TWLgBIH jVYXDT w aYCnCG^W] ph
L&% ;
&, 7)0,7) -3 #&, # $()3
3 #-4-
3
-
"
$#&%% )4,
" d
^3H`EFLf]lyFuWvFt=dFPljd F\Wh=LMuGuWMF
"s'&/WT3H`J]U6VcK ,JI )46,5,
3 % 3 ,>( - "
)3)
3
-P
3 $
"
#6% %.# ()96, :, # / 1
3
-&1
3
#!(
3
:>
3
$#6-:- #6#6-4-
3 -3
¨A;T ©XUTWL@]DVY]Zd jh=VGL@BD]UVYXWiLg6L!qOL@hWhWL@]Rp@dWWLfeWBD\_HKBDmMuFuGyW
"y'&3ÐCFaYaH`B[gdWA/WLHK\WXDBUVYL@BdH`h3g,_qOV]U]DL@XZ
&,X$(7:$()96, $O-7$&%
L'(
3
% , 3
&,X$(#&)0,>(3@qOL@]DLHKBISUT qjLf^_CGBDXuFy xMZvWd
i
zS[CFaYLjCFBUJ HKaL 76\6
^jiL@BDVYL@\WBUL!g6L mFCGhd5mGCFhdW=BIHKh_S[LFd3*HZm*MuGuFyW
"8 &A/ WLZH`\WXUBDVYL@B A/H`B[H`JLfXDBDVSVYh6XDL@EGL@B ^WBDCGEFBIHKJJ%Vh=EW
3 3
#
3
1
3
$# ()9&,
, 3 -4-
3 3
#!(*)96,X
3 3
#6#
dWsFs cy @s 86ys $=dWMu $ $W
" 3
"v'&A//=LH`\WXUBDVYL@BZ76L@JH`h6XDVS@HKaH`h3H`aYm6]DVY]!H`h3gJ H`XUTWL@JH`XDVS@HKa^WBUCFEGBIH`J%JVYhWEW=Pl^W^WaYV
SfH`XDVYCGh XDC^_H`B[H`aYaYL@aYVZH`XDVYCGh$HKh_gGLS[XDCGBDVYHKXDVYCFhph ¨CG]Dh_HKBIg L@XHKadLg6VYXUCFBD]Zd
#-4-
3 - #&, )$(*)3Y7: ( 3 -
"
&7)(
%)@dF^3H`EFLf]yFtGuWyFsFt=dz/aY]DLfnVYL@B7WS[VYL@h_SxL!A/\WeWaYV
]UTWL@BU] $RJl! OCFBDXUT6 4jCFaYaH`h3g@d3Mu $FuW
"'&A/W=LHK\WXDBUVLfBHKh_g X;HZje=Vc 1
3
6-/: ()9&,
3
L'$(
3 % 3 )4,1
3
8: #!(*)+&,X -/)0,21
3
#6)4
3
%.'
3
3 % -6)-&: - "
)37)
3 - C/LSUTWhWVS@HKa¨qjLf^_CGBDXd5H`e3CFB[H`XDCGVYBDLP7p@d jhWVY
A;VLfBDBDL L@X*HKBDVYL\=BDVYLFd_A/HKBDVY]dW=BIHKh_S[LFdlLZS[L@Je3L@B!MZu $Fu=
"54'& B[H`hWaY\Wh3gW
53
:>-() -3 3
)7)9&, 7)( % 3
(*)3 )Y7$#&L`
qOL[1L@BUL@h_S[L J HKh6\_H`adF=BDL@L76CK1XjH`BDL=CF\Wh3gWH`XUVCGhdGph_S`Yd RLS[LfJe_L@BMZuFu=MF
"($'& _X=CFaH`h_g%75H`J PaYC6CF^%XDBIHKhW]CFBDJH`XUVCGh!XUTWL@CGBDm HKh_gHKh HKaEGCFBUVXUTWJ
XUCJ H kVYJVY@L^_HKBIHKaaYL@aYVY]DJ / #6,X # (*)96,X &, #-4-
3 - #&, 2)(7)3Y: ( 3
L'(
3
%@d_s 38X@8nvGs 84WMGdFwS[XDCGe_L@B MuGuWMF
Unit´e de recherche INRIA Lorraine, Techn ˆopole de Nancy-Brabois, Campus scientifique, 615 rue de Jardin Botanique, BP 101, 54600 VILLERS L `ES NANCY
Unit´e de recherche INRIA Rennes, IRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rh ˆone-Alpes, 46 avenue F´elix Viallet, 38031 GRENOBLE Cedex 1
Unit´e de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex
Unit´e de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex
Editeur´
INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)
ISSN 0249-6399