HAL Id: inria-00071924
https://hal.inria.fr/inria-00071924
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-
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non,
The Petri net synthesis problem for automatic graphs
Eric Badouel, Philippe Darondeau
To cite this version:
Eric Badouel, Philippe Darondeau. The Petri net synthesis problem for automatic graphs. [Research Report] RR-4661, INRIA. 2002. �inria-00071924�
ISRN INRIA/RR--4661--FR+ENG
a p p o r t
d e r e c h e r c h e
THÈME 1
The Petri net synthesis problem for automatic graphs
Eric Badouel — Philippe Darondeau
N° 4661
Décembre 2002
12436587:9<;,=?>@BA6CEDGF$H,36A63JIIK@0L09<24=?M;@B9<>ON
PRQTSU:V:WYX[Z\4]^V4_a`cb(Vde]^fc]gdhSU:V4]
iGj^kalgVdemon
Z!_apqp<kaj^drqVj^VtsuQqVjusuQqVv<wnBxaxTWX[ye\tsVUzqjhV{a|a|B{+X}{a~)p_aVt]
(,4 #qa a `cd^kU_#d^saju_pqQ]Y_aj^Vp<k]h]zqfEvcvqdhVaju_pqQ]d^Q_ju_#d^kav_,pqj^Vt]V4vBdh_#dhkvzof
vqdhV_a`cd^kU_#dh_TKdhQqVfvTs`TrqVj^rq]_a]_p_j^d^s`q_j:s_]VG_vTrmo\vqV4j^`qV4]QT_]]^Qqk#vv_
]^VU:vT_B kajh!d^Q_#d,d^QqV4f+_aj^V _v$Vbod^Vv]kavkarcVdhVjhU:vq]d^sGpq`T]^QTrck#v)aju_pqQT]4 ¡eVjhVa¢£V£pqjhkap<k]^V
_pqjhkosV4rc`Tj^V(d^QT_d:rcVtsrcV4]¤jhkaU¥d^QTV0_a`cd^kU_#d^s¦pTj^Vt]V4vdu_#dhkvk_§aju_pqQQTVd^QTVj:d:U_¢fzV
jhV4_aVtr$]kU:kajhpqQqs_fYzof+dhQqVRj^Vt_asuQT_azqVR]dh_#dhV£jh_apqQ)ka<]^kaU:V£Tvqd^VRiVd^jhcvqVd4ad^Qd^ju_v]d^kavT]
v0zqlgV4s¨dhkvEdhQ0_azV4]ka,VtrcaVt] ]!_a`cd^kU:_d^sYaju_pqQT]sk#©aV4j U:kj^V$skavTs`qj^jhVvBd]fc]d^V4U:]d^QT_av
pT`T]Qrck#vaju_pTQT]dhQq]:pqj^kcsV4rc`qjhV(U:pqjhk#©aV4])k#©aV4j4G_avTrvTrcV4V4rª]^`qzT]^`qU:V4]4GdhQqVvqVd¦]^fBvBdhQqV4]^]
pTj^kcsVtrc`qjhVY]d^`rcVtrV4_jhV4j£¤kjpq`T]^QTrck#vaju_pqQ]
«E¬B?®h¯)° #±,a _a`cd^kU_#d^sYaju_pqQ]ciVdhj^²vTVdh]4qjhVkvT]c]^fovdhQqV4]^]
³K´<µ¶4·q¸¹²·¦º¨»¨¼¨½t¸¢¾<¿uÀ¨Á¢Â¢ÃtĸŲ¿uÆÇgÈÉÀ¨Á¢ÂËÊÇÌ¢¿uâÀ¨Á¢ÇgÍÎ,ÏÀ¨ÍÐ4ÑJÇgÒÔÓBÕÁ¢Â¢Ö×ÆÇØÑÕÒgÆ+Ù
ÚÛ¤ÜÛ
¶4ÝK¸#Ò¿uÆÏ¢Á¢ÞâÇG¹?Ç¿uÁ¢Í×Ö×ÇgÁB¸#ßT»¨à¨½uáâ
Ü
Çg¢¢ÇgÞÅ?ÇgâÇØãËÊÃ#¿uÈÉÀ¨Â¢Ã¢Ç¿uÁÎ,Ö×ÈÉÖ×ÞÔ¿tÕäÈÙ
.%0 %)'¦ *%+:'
V4]jh_apqQqVt]_`cdhkaU_#dhB`qVt]!]kvBdrcV4]eaju_pqQTV4] Tvq]!k`EvqTvq]erckvqvq\4]!pT_j`qvqV$pqjh\
]^VvBdu_#d^kav(jh_d^kavqvqV4VaÉ_d^Vr_`qd^kaU_d^V4]KTvq]$`qvTVYk`(rcV4`cb¦zT_vrcV4]4csV4] jh_apqQqV4]£skvdhV4vqvqV4vd
V4v pT_j^d^s`qVj+Vt]ajhV4]4Vdmc\vqV4j^`qV4]+_0U:kavBd^jh\rq_v]+`qv d^ju_¢©¢_a¤kvTrq_d^V`Tj`]Y¤kj^U:V4vd
`TvqVVbod^V4vT]^kvpTj^kpqj^V!rqV4] jh_apqQqV4] rqVd^ju_vT]^d^kavT] r_`qd^kaU_d^V4]Krc\dhVjhUvq]gdhV4]+pTVa !k`T] pqjhk!
p<k]^kav]sc`TvqVpqjhkos\4rc`Tj^VB`qqrc\4srqVa"YpT_ajdhj rcVR_Ypqj^\t]V4vBdh_#dhkv_a`cd^kU:_d^#B`qVr`Tvjh_apqQqV£Tvq
k`vcTvTJ<]$pV4`cd&%dhj^Vjh\4_a]\':]kUkj^pTQq]^U:VpTj^St]pT_jV+aju_pqQTV+rqV4]U_j(B`T_V4] _as4sV4]h]^zqV4]
r)`qvªj^\t]Vt_`rcV¦iVdhj^£Tvq6 _¢©aVtsrcV4]$d^ju_vT]^dhkvT]V4vzqlgV4s¨dhkvª_¢©aVts:Vt]$\d^#`TVddhV4]rcV4]_jus]$rq`
jh_apqQqV +*£kaU:U:V(V4]aju_pTQqV4]_`cdhkaU_#dhB`qVt]ska`q©ojhVvBd)pq`T]rcV]^fc]gdhSU:V4]skvTs`Tj^jhVvBdh],B`qV(vqV
V$¤kvBdeVt]ejh_apqQqVt]!rcV$dhjh_avT]^dhkvT]ercVt]e_a`cd^kU:_d^Vt]-pqVa²sVdd^Vpqjhkos\4rc`Tj^V)_aU:\kj^VTVd+rcV$É_ad
jhV4ska`q©ojhVa¢_pTj^kcs\trc`qjhV£pTj^kpkB]\4V _avdh\jhV4`qjhVU:VvBdpk`qj _Y]^fBvBdhQqS4]^VrcVRj^\t]Vt_`cb)rcVRiVdhj^.YpT_ajdhj
rqV4]Raju_pTQqV4]RrcV+dhjh_avT]d^kav] r)_a`cd^kU:_d^Vt]rc\dhVjhUvq]gdhV4]&pqVa
/ ° t ® 10 aju_pTQqV4]R_`qd^kaU_d^#`TV4]4Bjh\4]^V4_a`cb(rcViVdhj^6qjh\kvT]c]^fovdhQqS4]^V
( )
'-K$
PRQTV)pqjhkazqVU kaKvqVd]^fBvBdhQqV4]^]YskvT]^]dh]v rcV4srqvqQqVdhQqVjd^QTVjhVVbo]dh]Y]kUViVd^jhvqVd]h_#d
]¤fBvqa©aV4v]p<V4s<s_#dhkvT]4 *Øv8d^QTVV4©aVvBd+k£_p<k]^dh©VrcVts]kav]^`TsuQ_iVdhj^vqVd]^Qqk`qrzV
skavT]d^jh`Tsd^V4r£kad^QqV4j^]V(¤VVtrczT_asu]^Qqka`TrªQqkap<V¤`qfzV0pqjhk#©orqV4r©ovqvTrcs_d^kavT]Qof d^QqV
©Vv]^p<V4ss4_#d^kav]s4_vqvqkadz<V:j^Vt_4V4r0zof8_¦iVdhj^vqVd4 *Øv§¤j^kvBdkKd^Qq]YaVvTVju_]hsuQqV4UV?d^QqV
_akajhdhQqU:s]^ka`cdhkvT]$pqjhk#©orqV4r]^kEÉ_jvd^QqVd^V4jh_d^`qjhV(_pTpqf dhkU)`suQj^Vt]gdhj^s¨dhV4rs_a]h]^V4]$ka
]^p<V4ss4_#d^kav] +*Øv_Ejh]d]dh_aaVa,dhQqV]^fovBd^QqVt]]$pqjhkazTV4U¥Q_a]zV4VvªrcVt_dd^Q_a]h]`qU:vq8dhQT_#d
]^p<V4ss4_#d^kav]Y_aj^VTvqd^V:_z<VVtrjh_apqQT]a©aV4vzofEVbcpqsd$]^Vdu]+k£]dh_d^V4]_avTr8_z<VV4rd^ju_vT]^
dhkvT]4 _azV4V4r jh_apqQT])_aj^V¦]h_r d^k§zVj^Vt_4V4r zof§Tvqd^V(iVd^jh£vqVdh]4 _azV4V4rv#lgVts¨dh©Vf§kv
dhjh_avT]^dhkvT]RdhQqVf _jhV:]^kaU:kajhpqQqsd^k0dhQqVj)jhV4_suQT_zTVU_jhBvqEjh_apqQT]4 mo`TsuQ]^pVtss_d^kavT]
Q_¢©aVe_`qvq#B`qVeU:kcrcV²_avTrdhQqVf¦s4_vqvqkad£z<VY]dh_d^VtrdhQqka`qd_skUpTVd^Veovqk#V4rqaVeka²d^QqV+]dh_#dhV4]
kad^QT]RU:kcrcV6cQqsuQ(]R_]d^jhkavqj^V B`qjhVU:V4vd]^VrckaU U:VdRvpqju_asd^s_a<]^fc]gdhVU]RrcV4]^v $,EkajhV
k#©Vjt#kavqf$vqdhVd^ju_v]d^kav:]fc]d^V4U:]K_jhVRskavT]^rcVjhV4rQqVjhV4_],d^QqVU_aj^ovq+aju_pqQT]kavqdhV!iVdhj^
vTVdh]£U_¢f V<zVYvqTvqd^Va GPRQTV4]^V!U:du_#dhkvT]KQ_¢©aV!z<VV4v(dhV4r¦p_j^d^_fv]`qz]V `TVvBd ]d^`TrqVt]
- vªd^QqVkavTVQT_avTr²K_ rcVts]kavQT_a]z<VVvzqjhka`qQdd^k§d^QqV0pqjhkazTV4U k]^fovdhQqV4]^vqvqVdh]¤jhkaU
vcTvTdhVjh_apqQT]a©aV4vzofEaju_pqQaju_U:U_ju]d^QT_d]d^kE]h_¢fEskavBdhVbod 6¤j^V4Vjh_apqQT].¤kjpq`]QTrqk#v
jh_apqQT]0/21346 !PRQq]!]É_aj!¤jhkaU ]`5sV4vBd4?]vTsV)Va T d^QqVskaU:pqVdhV)j^rkav0d^QTV)p<k]^dh©V`_j^d^Vj
kaRd^QqVpq_avqV(])_EdgfBpTs4_KVbq_U:pqV¦ke_EvTkavskvBd^Vbod J¤jhVV¦jh_apqQ dhQjhVa`T_aj)]gdhj^`s¨d^`Tj^VdhQT_#d
U_¢f8z<V¦jhV4_aVtrzof§_vqdhV¦iVdhj^GvTVd4 - vqVU:QBd$j^Vt_a]^kavT_azqf8rq]hs_ajhr§d^QT]Vbq_U:pqVzBf§U
p<k]^vqz<ka`TvTrcV4r _]fovTsuQqjhkavT]^U kav i,Vd^jh vqVdu]zq`cdV©Vv `qvTrcV4j+d^QT_dpqj^k#©o]kd^QTV:skavBd^Vbod 6¤j^V4V
jh_apqQT]_jhV(É_aj¤j^kU VbcQT_`]gdhvq§d^QTV0s_]^]kaeU_jhBvq aju_pqQ])kaeiVd^jhvqVdh]4 - vd^QqV0kdhQqVj
Q_vTr²?_¦rcVts]kavEQ_a]!z<VV4v8zqjhka`qQBdd^kdhQqV)pqjhkazTV4U&kaK]fovBd^QqVt]vqvqVdh]e¤j^kU _(rc]ÉlgkavBdpT_aj
ka²j^V4a`q_jG_vTa`T_aaV4]4kvqVeVbopTj^Vt]^]^vTj^V `TjhV4rz<VQT_¢©oka`qju]K_avTrd^QqVkdhQqVj£VbcpqjhV4]h]vq+¤kajhzqrqrcV4v
z<VQ_¢©Bka`Tjh]edhQqka`cd+U_ovq_avBf0jhV¤V4j^V4vTsV)d^k0]gdu_#d^Vt]618794Ô i,Vd^jhvqVd+j^Vt_4V4]]`TsuQ ]p<V4s<s_
dhkvT] ,du]R_avqa`_aVYskvdu_vT] _²j^V `TjhV4rz<VQ_¢©Bka`Tjh] _vTrvqk¤kajhzqrqrcVvzV4QT_¢©ok`qjt PRQT]R_]^k
])É_j$¤jhkaUz<Vvq§]^`5¦sVvBdtG]vTsVU:k]d]dh_#dhVÔzT_a]^V4r pqjhkap<Vj^d^V4])kdhQqV]^fc]gdhVU`qvTrqVjrqV4]^v
s4_vqvTkdzV+¤kj^U`q_#d^Vtr_]R_avqa`_aVvs`T]kav]
*Øv d^Qq]pT_p<Vjt! VUpTj^k#©Vk#©aV4j¦pqjhV©oka`T]£kj^zof d^vq¤`qjdhQqVjdhQqVdg kjhV4]d^jhsd^kavT]
U:V4vdhkvqvqVtr(_z<k#©aV ,PRQTVYU_avj^Vt]`qd]R_av(V!:?V4sd^©aV]^ka`cd^kav(d^kd^QqV+vqVd]^fovBd^QqVt]] pqj^kzqVU ¤kaj
_a`cd^kU_#d^saju_pqQT]4 `qd^kaU_d^s)jh_apqQT]Y_jhVVvqV4jh_afvcTvTdhVaju_pqQT]QqkB]V]Vd+ka£]dh_#dhV4]+_vr
_z<VV4rd^ju_vT]^d^kavj^V4_d^kavT]_jhVRj^V4pqjhV4]^VvBd^Vtr)zofTvTdhVe_`cdhkaU_#du_+_vTr)d^ju_vT]hrc`TsVju] .¤kjdg£k1Jdu_p<V
_a`cd^kU_#dh_;/¨ PRQqV+s_a]h] k_a`cd^kU:_d^sejh_apqQT]RskvBdh_vT] ]gdhj^s¨dhf¦d^QqVs_a]h]£k,skavBd^Vbod6¤j^V4Veaju_pTQT]
QTVvTsV£dhQqVvqV4]^fBvBdhQqV4]^] _akajhdhQqUO¤kaj _a`cd^kU:_d^s aju_pqQT],U:pqj^k#©V4],k#©aV4jdhQqVR¤kajhU:Vj _akj^d^QTU
,8kajhVk#©aV4j4c_¦]d^ju_d^¤kajh _ajhr_arq_apcdh_d^kav0kdhQqV)vqV4Ë_akajhd^QqU _k#!] dhk¦Vbod^V4vTr0vqVdY]^fBvBdhQqV4]^]
¤jhkaU _a`cd^kU:_d^saju_pTQT] dhk< 2=&>?0'#¨ `cdhkaU_#dhs]p<V4s<s_#dhkvT]£j^Vt]V4U)zqV!U)`suQ
_a`cd^kU_#d^s¦jh_apqQT]4Gzq`cd:dhQqVfvqV4V4rvqkdrcVTvqV0_`qvTB`qV0_z<VVtraju_pTQ @*ØvªÉ_s¨dt£_`qd^kaU_d^s
]^p<V4ss4_#d^kav]!U_¢fQ_¢©aV)_av8_jhzqdhjh_aj^fvo`qU)z<VjYk U:kcrcV]eju_vTavq:¤j^kU Vjhkdhk¦vqTvqdgfa ]_
jhV4]^`qd4qkvqVU_¢f¦vqk#O]Vt_jusuQ_a`cd^kU_#d^s_af¤kje_:iVd^jhvqVded^Q0U_jhBvqjh_apqQvE_v0vcTvqd^V
É_aU:fka$_azV4V4r jh_apqQT]4zq`cdd^QqV4j^V8]_pqjhsVEd^kp_¢f]^vsV_a`cd^kU:_d^s8]^pVtss_d^kavT]_jhV
_rcU:d^d^VtrcfrqA5¦s`Tdd^k j^d^V +B V0Vt_¢©aV¦kpV4v¤kaj¤`qj^d^QTVj:]gdh`Trcf d^QTVB`qVt]gdhkvkepTj^kcrc`TsvT
_a`cd^kU_#d^s+]p<V4s<s_#dhkvT]Rzofd^ju_v]_#dhkv¦¤jhkaUU:kajhVskvB©VvqVvBd!]^pVtss_d^kav_avqa`T_aaVt]
PRQqVj^V4U_vqvT]Vts¨dhkvT]kdhQqV(p_p<Vj_aj^V(kajh_vT4V4r _a]$¤kak#!]4 8moV4sd^kavT]:{E_vr~8zqjhvT
dhQqVYvqV4sV4]h]^_aj^f:zT_suBj^k`qvTr:ka_`cdhkaU_#dhsY]d^jh`Ts¨dh`qj^Vt]£_avTr¦iVd^jh<vTVd!]fovBd^QTV4]^]4 GmoV4sd^kavT] n_avTr
pTj^kpkB]V$_]k`cdhkv0d^kdhQqVi,Vd^jhvqVd+]^fovBd^QqVt]]!pqjhkazTV4U ¤kj_`qd^kaU_d^saju_pqQ] emoVts¨dhkvx¦_vr
7$]QTk#_$¤`q?Vbq_U:pqV+_vTr¦_sk`qvBd^V4j ÔVbq_U:pqVa moV4sd^kav<3)VbBdhVvTrT]GdhQqV+j^Vt]`qdh]K¤jhkaU_`qd^kaU_d^s
jh_apqQT]d^k_a`cd^kU:_d^s£]^p<V4ss4_#d^kav] PRQqVRpT_p<Vj ],¤kak#£Vtr$zof)_+]QTkaj^d _apqp<VvTrcb)kv]^VU:6vTV4_j
]^`qzT]^Vdu]Rk_vr?
Ë 0%$-£ $" %$'¦,) *%+- .%
ju_pTQT]+dhQ_v "" =! _]kEs_aV4r 0u, VjhVV4©ak©V4r8zof§mc\vq
V4j^`qV4]Y_a]+_]U:kokdhQVbBdhVvT]^kav k* _¢foVf0aju_pqQ]ka£_`qd^kaU_d^s)j^k`qpT]4 *ØvkjhrcV4jYd^kj^Vt_asuQ_av
V&5¦sVvBdrqV4s]^kv)k?s_a]h]s_apTj^kzqV4U]kv)j^k`qpT]4¢_a`cd^kU_#d^s ]gdhj^`Tsd^`qjhV4],ka * _¢foV4faju_pqQT] VjhV
rqVTvqVtrE_a]!¤kak#!]zofGpT]d^V4v & 61W4| 4 _¦j^V4a`q_j_vq`T_Vkavd^QTV]^Vdekaj^k`qp0aV4vqVju_#dhkaju]
]^`qpqpTV4]d^QqV£j^V4pqjhV4]^VvBdh_d^©aV4]<¤kj©Vj^d^sVt] .¤6 Va j^k`qpVVU:V4vdu]/²d^Qqjhka`TaQ+d^QTVKV©#_a`T_d^kav$kac£kjhrq]
vd^QqVj^k`qppT_jh],ka<V `T©#_VvBdGjhVpqjhV4]^VvBdu_#d^©aVt]_aj^V_ssV4pcd^VtrzBf_YTvqd^V!_`qd^kaU_d^kavd^Q:dg k
du_p<V4] .Js_aV4rd^QTV ! =8 9!/<_avTrd^QqVjhQBde_as¨dhkv0kV4_asuQaV4vqVju_#dhkajR]!rcVvqV4rE]^U:_aj^f¦zof_
vqdhV)_`qd^kaU_d^kav0d^QEdg k:dh_apVt]6.Js_V4r0_ !8'"8& /¨ #qkjepqjhV4s]^kavT]4cd^QTV$j^Vt_arcV4j]ej^V¤Vjhj^Vtr
dhk 1Wt| 46
mo\4vqVjha`TV4]]£rcVTvTdhkv¦k_`qd^kaU_d^s!aju_pTQT] 1×{~ 4?j^Vdh_avT]G¤j^kU dhQq]G¤ju_U:V kajh_]GU`TsuQ(_a]
p<k]h]^zqVd^k0_asskaU:kcrq_#dhV_a]^k(vqkv§QTkaU:kaVvqV4ka`T]eaju_pqQT]Y]^`TsuQ§_]edhQqVrcVdhVjhU:vq]d^s)pq`]QTrqk#v
jh_apqQT]4 moVdd^vqe¤kos`T]kv$jhV4_asuQ_zqV aju_pqQ]²dhQ$kavTVGj^kokad4#_vTr+d^Q_ercVd^VjhU:vT]d^sG_z<Vvq!ka
VtrcaVt]¢QqV£V4]h]V4vBd^_f_arT_pcdu]dhQqV£_azk#©V£rcVTvqd^kav$zof]`TzT]gdhdh`cd^vq+]V B`qVvTsV4]kq_z<V],kav$jhkokd^Vtr
p_#d^Q]vdhQqV0aju_pqQ¤kaj:]V `TVvTsV4]kj^k`qpVvqV4jh_d^kaju]4 PRQqVVB`T_aV4j_avTrdhQqVU)`qd^pqV4jh]
.ÉkavqV8¤kaj(Vt_asuQ ]^fBUzkev d^QTV_pqQT_azVdk_z<V]/¦_jhVrcVvqV4r _](_z<k#©aVEzofTvqd^Vdg£k1Jdu_p<V
_a`cd^kU_#dh_TQTk#£V4©aVjedhQ]kU:V]^vq<s_vBdrqA:?VjhVvTsV4]4 *Øv 1Wt| 46<dg k(_d^V4j^v_#d^©aV$dgfBp<V4]k dg k!
du_p<V_`cdhkaU_#du_ VjhV+skav]rcVjhV4rcvT_U:VfdhQqV %$#%0#% R_`cdhkaU_#du_$dhQT_#d!jhV4_r¦dhQqVjRdh_apVt]v
_adhVjhvT_#dhkv4_avTrd^QqV2 %$#%0#% ²_a`cd^kU_#dh_T mofovTsuQqjhkavTka`T]<ju_#d^kav__vq`T_V4]¤kajhU _!zkokaV4_av
_aVzqju_q#V4vBdh_vqrcVtsrq_zTdgfk<d^QqVRjh]dGkajurcVj,dhQqVkj^fk²]fovTsuQqjhkavTka`T],_`cdhkaU_#dhsRj^k`qpT]4#zq`cd
_]fovTsuQTj^kvqka`T]$j^V4a`q_j_avqa`T_aaVt]$rck vqkd¤kajhU _8z<kBkVt_vª_aVzqju_q _adhQqka`qQd^QTVf_jhV !
#" & 0 "!&#=#T *Ôd+]+ kaj^d^Q8vTkd^vqd^Q_#d'& # (!)$ %$#%0#% _`qd^kaU_dh_q²QTVjhV
kvqVdu_p<V(s_avqvqkdzV¦jhV4_r_QqVt_ar§¤jhkaU d^QqV¦kad^QqV4j)zof Ukj^V:dhQT_v TboVtr+* VddhVju]_jhV(_s¨d^`_f
]^`5¦sVvBd£¤kajR]`qpTpqfovq_]fovTsuQTj^kvqka`T]K_`cdhkaU_#dhseajhka`qpT]KdhQ_a`cd^kU:_d^se]d^jh`Ts¨dh`qjhV4] 1Wt| 46 PRQqV
dg k!6dh_apV_`cdhkaU_#du_$`T]^V4r(v+1{a~ 4?d^krqVTvqV+_`qd^kaU_d^s]d^jh`Ts¨dh`qjhV4]£kaju_pTQT]£_aj^VezT_a]^s4_f:kad^QqV
z<ka`TvTrcV4r_]fovTsuQTj^kvqka`T],dgfop<VaBzq`cd£d^QqVzk`qvTrcVtrcvqV4]h]KskavT]d^ju_vBdG]Kd^Vtr¦_#dhVjKd^QqVjh]dGdh_apVQT_a]
z<VV4vVbcQT_`T]d^Vtr(vd^QqV$]^pVtss+s_]VYkd^QqV+U)`TdhpqVje_a`cd^kU:_dh_T *Øv0]pTdhVkdhQqV+zk`qvTrcVtrcvqVt]^]
skavT]d^ju_vBdh]4Bd^QqV+Tju]dRkjhrqVj d^QqV4kajhf¦k _`qd^kaU_d^sjh_apqQT] ]RvqkdercVtsrq_zTV 1×{| 4Ô
*ØvdhQq]K kajh?£V_arckpcd mo\4vq4Vjha`qVt]$] ©oV k_a`cd^kU:_d^sRaju_pqQ] zq`cd £Verq]hs_ajhr_azk`qvTr
VtrcvqVt]^]skvT]d^ju_vdu]¤jhkaUËQT]rcVTvqd^kavT]4 -, V¤kajhV ]dh_d^vqedhQqV]U:pqTVtr)rcVTvqd^kavT]4Vd `T],zqjhV.Tf
skaU:U:VvBd)kv d^QqV¦jhV_#dhkvT]^Qqpz<Vdg VV4v_`cdhkaU_#dhs:jh_apqQT]$_avTr d^QTV(rcVjs_a]h]k6
"" =! !GvBdhj^kcrc`TsV4rv 1W 7 4_avTr(¤`qjdhQqVj]gdh`TrcV4rv 1x4Ô `cdhkaU_#dhs+]d^jh`Ts¨dh`qj^Vt]_jhVjhV_#dhkvT_
]d^jh`Tsd^`qjhV4]:k#©Vj_v `qvTrqVjhfovq]Vd4RQqk]^V0VVU:VvBdu]¦_jhV0j^V4pqj^Vt]V4vBd^V4r_a] kajurq]k_ jhVa`T_aj
_vTa`T_aaVa / B`T_adgfª_avTrkad^QqV4jjhV_#dhkvT]:_jhVrcVtsrcV4rªzBf vqdhV0U)`qd^Jdu_p<VE_`cdhkaU_#du_8kd^QqV
]^fovTsuQqjhkavqk`T]Rdgfop<Va?_vTr8]^kdhQqV$Tju]dkajurcV4jd^QqV4kajhVt]!kaK_`cdhkaU_#dhs$]d^jh`Tsd^`qjhV4]_jhV)rcV4srT_zqVa *Øv
p_j^d^s`q_jt¢_azV4V4rjh_apqQT]U_¢fYz<V j^V4pqj^Vt]V4vBd^V4r+d^Q_`cdhkaU_#dhs£]gdhj^`s¨d^`Tj^Vt]v)QqsuQ$dhQqVR_pqQT_
z<Vd£`T]Vtrd^k$jhVpTj^Vt]V4vd ©Vj^d^sV4]K_a] kajurq]Grc:?Vju] ¤jhkaUdhQqVe_apqQ_z<VdKka?_z<V]4 ,PRQqV]^_aUVjhVU_aj^
_apqpqVt]dhkd^QqV_a]^fovTsuQqjhkavqk`T]_`cdhkaU_#dhs)]gdhj^`Tsd^`qjhV4]d^QT_d`qvTrqVjhV$dhQqV '#% =;e]d^`TrqVtr
v 1W 346 $B VrckvTkdRBvTk#QqVdhQqVj dhQqV+i,Vd^jh?vqVdjhV4_4_#dhkv¦pqjhkazqVUs4_vzVrcVtsrcV4r¦¤kaj d^QqVt]V
_jhaV4jes_a]h]^V4]!kaK_`cdhkaU_#dhs)aju_pqQ] ka`qjYpqjhV4]^VvBd!dhV4suQqvq#B`qV)kaKvqVdY]^fBvBdhQqV4]^]jhVVt]!QqVt_¢©Bfkv
dhQqV$_a]h]^`qU:pcd^kavd^QT_d!©aV4jdhsV4]R_jhV+j^V4pqj^Vt]V4vBd^V4r(zof¦_azV4]Rka,vTskaU:vTp_#d^Q] GyVTvTdhkvT]vqk#
¬ É° ¬ ¬ , 6 ° ¬ #_]fovTsuQTj^kvqka`T]?rcVd^V4j^U:vq]d^sdg k!6dh_apV _`qd^kaU_!
dhkav " ")$_a]^fovTsuQqjhkavqk`T]!_`qd^kaU_d^kav ! &#" 6"' &&"#="' # $ &
"&%('$)*+ #-, 0 = &! /.6"" ="'#%<#=>0 " &"&&1 &#% & $3245627
4
0289
2 78
#":2
7
; # )$ "#<2
7 # ! = 0% "= 03>
7!?
#% "$, "A #%
.0 @ # "$, & &$ # &!# 6A.:" # =$ 9#; ' & 0<#B2 4 %-2 8 = '
C%28
? . '! 0 #2 7
4
:2 7
8
' ' & # = & &!D) "!,
,$ & 0#24 ';& #2 7
8 ?
#%3,$ & #E28 & #2 7
4 ?
#" #%3,$
& 0 #32 7
4
%F2 7
8
';!>
7
:; "" 0 #32 4 %G2 8 IH &$ = # %$#%0#%
#_ssV4pcdh] 2A. 0"#;KJML 4N L 8OQP "#R0ST"#R6. & VU 9L A.KJL 4 ) N L 8 ) O
, 6 00 & $ #"
W£¬ X& Y,4 ¬¬ :.0 Z\[$ "& &#=# #"]RCS^"]R %!&!= ! @ 0@A. #`_
=&;! 0% &$ %$# # 6
*Øv§d^QTV:]^VB`qV4JaYJb O rcV4vqkdhV4]d^QqV:ju_#dhkvT_jhV_#d^kav¤kajhU:V4r8ka _apT_aju]Yk£ kajurq]:JML 4cN L 80OFP
"]R6SF"#Re_as4sVpqd^V4rzof:_a]^fBvsuQqj^kvqka`],_a`cd^kU_#d^kvb -#Tkaj£skavo©aV4vqVvTsVav:d^QqV`T]gdhjh_d^©aVpT_ajdu]
dhQqVYaVvTVju_#d^kjh]Jd NAe+O _vTrJefN d O k5"]RKS3"]R?zVYj^ddhVvj^Vt]p<V4sd^©aVf_a]Qd:_vTr d< PRQo`T]4B¤kaj
vT]dh_avTsV dfg d:U:Vt_vT]]Jg N dd O GPRQqjhka`qQqka`qdKd^QTVYp_p<VjthjijJk Nml3Nmn$oN1p`O rcV4vqkdhVY_aju_pTQ(v
QTsuQ_c©aV4jdhsV4]
nP
k U_¢f$z<Vj^Vt_asuQqVtr$¤jhkaUd^QqVjhkokd
n$oTP
k zBf]V B`qVvTsV4]ka?rcjhV4s¨dhV4rV4rqaV4]
v
lrq
ksSkd^Q_0]^`qjÉlgVts¨dh©VVtrcaV_z<Vvq0U_p
p&tlvu
"$]^`TsuQ§d^QT_d)rc]gdhvTsd)V4rqaV4]
d^QEskaU:U:kav0]k`qjhsVY_aj^VYU_pqp<V4rdhk:rq]d^vTs¨d!_z<V]Rvw"xix'dzy N|{|{}{N d
* $B V+Vd nF~ di n
p JnN/nZO id0_vrEzV`qvTrcVTvqVtr8kdhQqVjh]Vzof0Vbod^V4vT]^kv?£V$Vd n:~ dLi n~ d ~ L¤kjYL P "#R
rqA:?VjhVvBdR¤j^kU e .¤d^QqVV4U:pcdgf( kajur=/¨
¬ É° X^ q±z ¬ ±\J ° 5a¬ $ ¬ <h #_`qd^kaU_d^sKjh_apqQ . $ &0 Z !#
$jh_d^kavT_a²]gdhj^`Tsd^`qjhVT&i NL oNmlGN b@y N|{}{|{N b
0.!!, V
/ > #=#%|_=@ # ! &3"T 0%#; !#; =&;aJ O " =
.! L P aYJ O n$o#~ L '> # #"].!V !%$w !" nP k: n i n$o#~ L.!6&
L P aJ O
/xL oTP aYJ O # n$o i n$oK~ L o
/ lG 6# %$#%0#% #+! &"G$0 =0
aYJlG!O i'JL N L XOP aYJ O SwaJ On$oK~ Li n$oK~ L *
©/.! 0 0@
P¡ N/¢¤£
Tb-¥ +# %$# # # $ &¦"T " =0 ]aYJb-¥
O i
'JML
N L
ZOP aJ
O
SwaJ
On o ~ Ld¥5i
n o ~ L
*!
-
vTVU:aQBd¦_a]h]`qU:V 6 kT T dhQT_#daJ O ]:sk]^V4rª`qvTrcV4jpqjhVqb² B VE]Q_ vTkd:`T]VdhQq]_]
]^`qU:pcdhkv yVdhVjhUvq]gdhspq`]QTrqk#vaju_pTQT]+_jhV_pT_j^d^s`T_ajYs4_a]^Vka£_`cdhkaU_#dhsaju_pTQT]ed^Q
z<ka`TvTrcV4rdhj^V4VÔrcd^Q
dgfopqs_aVbq_aUpTVka _av8_`cdhkaU_#dhs$aju_pqQdhQ8`qvoz<ka`qvTrqV4rd^jhVV$6rodhQ
]+dhQqV(skU:pqVd^Vj^r kav d^QTVpkB]d^©aV B`T_ajdhVj$ka£dhQqV¦pq_vqV .J]V4V #, GW9/ PRQTVajhr²,QqsuQ ]
#,a`qjhV:W PRQqVj^r
kzo©Bka`]f¦]kUkj^pTQqsdhk:d^QqV$jhV4_suQT_zTV+U_jhovq:jh_apqQ0k _:kavqV$Ô]h_#¤V+iVdhj^vqVdtTU_¢f(z<V$a©aV4v
_ju_#dhkvT_²]d^jh`Ts¨dh`qj^V#&i NL o NmlGN b yN b
_a] ¤kk#!]
W aJ O i\dzRy dzR _avTr¦L o i e
{q aJ
lF!O i 'JML
N L O
L P aYJ
O
*B
~T aJb
yVO iJd
y d yO
R d y Jd d
O
R
n aJb
O
iJd
y d yO
R d 0Jd d
O R
Ë - ¨0 $" :
*Øv 1W 4£_avTr 146_z<VV4r aju_pqQ]]^kaU:kajhpqQqs$dhk0j^Vt_asuQT_azqV:]dh_#dhV:aju_pqQT]Yka£iP 6vqVdh]4d^Q_a]
U_avBfd^ju_v]d^kavT]_a]e_azV4]4?£V4j^V)]^Qqk#v0dhk]h_#d^]¤f(dg k(suQT_ajh_s¨d^V4j^]d^s_#bckU]<j^V4U:vq]hsVvBdeka
dhQqV¦jhVkvT_G_bokaU]+¤kj GVU:VvBdu_jhf§Pjh_avT]d^kavmofc]d^V4U:] 1WW"4 1Wt{94Ô ,R_a]^V4r kav d^QTV4]^V(_#bckaU]
pTj^kcsVtrc`qjhV4]rcV4srcvqY`qp<kav$dhQqVRi#P ÔvqVdrcVTvT_azqdgfkqTvTdhV£jh_apqQT],_vTr)kskavBd^VbBd J¤jhVVKaju_pqQT]
VjhVpqjhkap<k]^V4r²,jhV4]^pVts¨d^©aV4fv 1W"41{94_vr v 13 4Ô PRQqV¦k_aGkRd^QT]pT_p<Vj]$d^k8Vbod^Vvr d^QqV
rqV4s]^kv d^kª_a`cd^kU:_d^sEaju_pTQT] B VjhV4s4_!vd^Qq]]^V4sd^kav dhQqVzT_]s0pqjhvspqVt](kai#P ÔvqVd
]^fovBd^QqVt]]QqsuQd^QqVYrcV©VkapqU:V4vd£pqj^Vt]V4vBd^V4r¦vdhQq]KpT_apV4jKjhV4]dh]K`qpkv B§VYjhV¤VjKd^QqVj^Vt_arcV4j d^k
1~ 4¤kaj!_:U:kajhV$]`TzT]gdu_vBd^_²pTj^Vt]V4vdu_#dhkvka,vqVd]^fBvBdhQqV4]^]4c¤kj©¢_aj^ka`]s_a]h]Vt]RkvqVdu]Tdh_VvE_a]
_s¨dh`T_²pT_ajh_aU:Vd^V4jh] kad^QTVvqVde]^fovdhQqV4]^] ¤`qvTs¨dhkajt GyeVTvqd^kavT] ¤kk#
¬ É° +®1¬ t ;_# !$2" riJ NKNGO , =! #" > #
&#=?&!'/.Rpq_asVt]6#"Yd^ju_v]d^kavT]| "=0!Z )$ # .0 #%&#z t J S YO %<J S
Ou &U_jhovT /.! #" t u ; ]gdu_#dhVjh_apqQ /.! &
!8Z_,$ $#; Y &"'0!1c, =!=!2# &; &0 :,$ '#"#!%5.0
" &" # # #'" %()" . #"+# )$ .c.! | &%$ " 0+* P :,"J-* O/.0 J-* N% O #
" J-* O i1"J-* O+23 J-* N% O45 J% N* O ; =j^Vt_asuQT_azqV§]gdu_#d^V§aju_pqQ A.+#vqd^_4V4r ;_# !
iJ NNQN " o$O ,$)vqd^_U_jhBvq'" o 2!"'"&'# /. ' " = 0 0 &.0
" o
¬
É° Z@
- 0 0Jq
¬ °
(#f
Z &# 0ThjijJk Nml3Nmn$oN1p`O &0 #
"G$#:U:pqsdRpq_asV /.9h ;_# ! *¦iJ' **
N " N QN
"
oO
,$ #% " 0 *$0 =& T,$
h #< &!#=! &m " 6 0
t k u G&&# #; &"! " 0Y 8 =! " = =
J
n$oO i " o
J-*
O
# | &%$( ]J
n`NmnZO
%&0!' 9 ?> "#; "&"
p J nN/nZO
("
/.6#%&Q,$+$#; " # /" !0&'Z )$ Z !#+&$/"(J* O i JnfO #"/" J-* O iJnZO
00, #=& ==$ 0( !skaU:pT_avqkvU_ap A.?" 0 *-
¬ É° X^ ¬ c#c J°` ± É°c & *¦iJA'** N " NQN" oO &0 # "8! " 0
/.6h' 0 * & ]^Vp_ju_#d^VF &"'0! n #" n . 0 =#=# 0 t k u &'>?!
JnOiJnZO 0/* & vqQTzqd # &'# ,$ &0d P " &" n .
0 =# # 0+&>?!JnO J* N d O
° ° É° 0wh &= 0 = & "0@/. ;_#%& .#
# )$. > #=" &"&& /. "8& " 0! " =< -0#%& &"'0!6 &=
&$ ! #:#".! V &%$( &" n V &%$@ !#^d &8# # @( E,$ "A# n
#&& &" &$ & 0 # ' H &#/&0 0#"#6&>? =Ch
& <=0 = &" = A.)]fovBd^QqVt]Vtr8vqVd i J N " N QN b o O &&#%
&$8 # ;!&#@#=0#= #<" A! 0" # : ,$ =# ,$& > #=# /.
$, '#
#"6#= $#b
o
Ë¥4 -£u %$-£ $" %$ %$'() *%-/.$,%0
Vdch iJk Nml3Nmn$o+N1p`O zV _v$_a`cd^kU:_d^s aju_pTQk#©aVj5"\i 'dfy N|{}{|{Nd
*B¢_avTr+Vd&i N L oNmlGN
b^y N|{}{|{N b
I
z<V£_ejh_d^kavT_a]d^jh`Ts¨dh`qjhVG¤kajh: £_asuQ$U:pqsdpq_sV*¦iJ' ** N " N QN" oO kqjh_apqQ
h U_¢f¦z<VrcV4vBd^TV4rd^Qd^QqVJ ¢&4 OÔ©aVts¨d^kj
*¦i#" o J-* O=N J* N dfy O=N}{|{|{N J-* N d
O=N Jdzy N* OVN|{|{}{N Jd
N* O
.gW9/
evTrcV4j+d^Qq]jhVpqjhV4]^VvBdh_d^kav?dhQqV:U:pqsd$pq_sVt]+kh _jhVd^QqV:vBd^Vjh_apkvBdh]$k _p<kafoQqV4rcju_
skavqV+QqkB]VVvqV4jh_d^kjh]£U_¢f¦zVskaU:pq`qd^V4r¤jhkaU £PRQqVpq`Tj^p<k]^Vk,d^QqV$]^V4sd^kav]d^k¦]^Qqk# _av
V&:<Vts¨dh©V+skaU:pq`cdu_#dhkvkadhQq]eskavTVa
, fErcVvqdhkvkaGU:pqsd+pq_asV4]4©V4s¨dhkaj
i N y N|{}{|{N
N
"!
y N|{}{|{N
rcVTvqVt]_av
U:pqsdpq_sV¦kah e_avTrkvqf e_£V4vdhj^V4]
¥_jhV¦vqkav.6vqV4_d^©aVvdhVVju]$_vTrd^QqV4j^V¦Vbc]dh]
_8U:_ap# t k u ]^`TsuQ dhQT_#d$Jn o O i o
_avTr ¤kajV©VjhfV4rqaVJn`NmnZO ka h: p JnN/nXO isd¥
dhQqVv%JnO!.
¥!_avTr&JnZO i'JnfO2
¥ 4 )(
"!
¥*
+]vq0dhQqVjh_d^kavT_aG]gdhj^`s¨d^`Tj^VËkKh:d^QqV
_#d^d^V4j skavTrcd^kav(U:_¢f:z<VjhV¤kj^U`q_#d^Vtr_a]K¤kak#!] ¤kj _a¤ P- +Nm¢¤£ _avTr:¤kaj V©Vjhfp_j£ka£kjhrq]
JL 4cNL 80OP aJbB¥O
Jn o ~ L 4O.
¥,-Jn o ~ L 80O i.Jn o ~ L 4O 2 ¥ 4
)(
"!
¥*
dd^QT]:]dh_Va kvqVs4_vdu_aV_arc©#_avdu_V¦k_§sjh`Ts_£É_asd (¤kaj:_vof vqkav ÔVU:pcdgf pqjhVTb&/¤d ¥ ka
_ kajurL P aYJ O z<kd^Q n o ~ /_avTr n o ~ /¤d¥_jhVrqVTvqVtr²_vTr&Jn o ~ / N/n o ~ /¤d¥O ]_av V4rcVka
h:,_z<VVtr§d^Q-d
¥
(PRQo`T]RkavTV¦suQqkok]^V4]Yv-aYJ O dg kEj^V4pqj^Vt]V4vBdh_#dh©V4]YL _avTr(L ¤kjd^QqV
jhV4]^p<V4s¨dh©V+©aVj^d^sVt] n$oK~ /_avTr n$o~ /d
¥
qdeskaU:V4]
Jn o ~ /¤d¥O i.Jn o ~ L O iJn o ~ L O 2 ¥ 4 ( ! ¥* iJn o ~ / O 2 ¥ 4
)(
"!
¥*
*Ôd¤kk#!]Rzof_avvTrc`Tsd^kavEkvpTj^VqbcV4] /Ek kajurq]QL P aJ O d^QT_d4q L ¥ rcV4vqkdhV4]RdhQqVkcss`qj
jhVvsVsk`qvBdRka,Vd^d^V4jKd ¥ vwL$qdhQqVv
Jn$oK~ L O i o4
¥ y L ¥ SJ (
"!
¥* 2 ¥O .J{ /
]vqed^Qq]V B`T_#dhkv#_!p<k]h]zqfYvcTvTdhV ]^Vd,kqvqVt_jVB`T_dgfYkjvqVB`T_adgfskvdhjh_avBdu]²kav$©V4sd^kaj
U_¢fz<VrqVjh©V4r¤jhkaU d^QqVjh_d^kavT_a]d^jh`Ts¨dh`qjhV!kh: PRQqVYskav]gdhjh_avBdh]£_jhVd^QTV!¤kak#vqQTVjhV
JL N L ZO _vr JLQ4 N L 8 O jh_avqaVYj^Vt]p<V4sd^©aV4f¦k#©Vj +N/¢¤£ zaJb O _vTrCaJlG!O
¥ y L o ¥0S(J (
"!
¥* 2 ¥ O i .É~ /
¥ y L 40
¥0S(J
(
"!
¥* 2 ¥ O i
¥ y L 8K
¥0SJ
)(
"!
¥* 2 ¥O
.¤n;/
¥ y L ¥ S(J (
"!
¥* 2 ¥ O . 2 o . /
¥
y L ¥ S(J (
"!
¥* 2 ¥ O i
¥
y L ¥ SJ )(
"!
¥* 2 ¥O 2 4 (
"!
* .Éx /
*ÔdR]£jhV4_rcf:kaz]V4j^©V4r:dhQT_#d_vof:©aVts¨d^kj
]h_#d^]¤fBvq:_<skavT]d^ju_vBdh] rcVvqV4]R_vU:pqsd pq_asVka
h:cd^Q0skU:pT_vqkavU_p(©VvzBfV B`T_#dhkv{c PRQqV+pqjhkazTV4U vqk#]Kdhk:j^Vtrc`TsVd^QTVYvcTvTdhV+]^Vd
kaskav]gdhjh_avBdh] d^k)_avVB`q©¢_aV4vBdKzq`cd > #=G]^Vd kskvT]d^ju_vdu] -, V¤kajhV! V!a©aVRdhQqVejhV4rq`Ts¨dhkvVd
`]Rj^Vts_rcVTvqd^kavT]_avTrÉ_as¨du]
¬ É° )° tc ¬wªz ¬z;%=skU:U)`cdu_#d^©aVU_aaVA.:, #L P " R
¢ _ 0!YJML O , = & !0&'Z &#=""#JML O|£ 0 # c.!<0 0 Pj¡ N/¢¤£ = 0& !# 0" /.
!& d ¥ #IL
¬ É° Z@ c É° .0R ¬ ! i J N|~}N
}O # " &"&& /.
ju_#dhkvT_. =! m " > #= # ! & > #= =&Jk Nml3Nmn o N k N1p`O
,$ "#; = " &"&&k A. 0%#;I &"!6#"3,$ &;" &0 # " = =
aYJ O i , &# 0# 0!# # # &%&(+ 2% =! # #" e # &%!
# ! &!#
¬ É° X^ ¬ ¤®0¬ c: ¬ ! i J N}~|N
O # ^ " &"&&6A.
vqV4_aj6. $ &" !& " ~! R3, =! " P > # " &"&& A." # R
0 " " &"&& A. 0 =0
GP # G ~# q Y > # 6 #=# /. 8# " &"&&6/.
60<$]^VU:6vTV4_j6" &"&& A.$