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Progress Report on Parallelism in MuPAD

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(1)Progress Report on Parallelism in MuPAD Christian Heckler, Torsten Metzner, Paul Zimmermann. To cite this version: Christian Heckler, Torsten Metzner, Paul Zimmermann. Progress Report on Parallelism in MuPAD. [Research Report] RR-3154, INRIA. 1997. �inria-00073535�. HAL Id: inria-00073535 https://hal.inria.fr/inria-00073535 Submitted on 24 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est 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..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Progress Report on Parallelism in MuPAD Christian Heckler Torsten Metzner Paul Zimmermann. N ˚ 3154 Avril 1997 ` THEME 2. ISSN 0249-6399. apport de recherche.

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(6)       . Unit´e de recherche INRIA Lorraine Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY (France) 615 rue de Jardin Botanique, BP 101, 54600 VILLERS L ES T´el´ephone : (33) 83 59 30 30 – T´el´ecopie : (33) 83 27 83 19 Antenne de Metz, technopˆole de Metz 2000, 4 rue Marconi, 55070 METZ T´el´ephone : (33) 87 20 35 00 – T´el´ecopie : (33) 87 76 39 77. "!.

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(9) ¨d dªŸŒ©H— ypcfsk t bdVl j t h S t `dinVhDink t VlH|8`dlni t `do8kBV^ Vl°t ¼}Vi j t iš`ahDbaVH|BVD^ h t `di™ypj t |8Vl<V³>VW#y2baVDlUb j}l8V^ W­jz_>in`dq>k8V |8kb¢jz_8erjzefV k Y˜ y{cgk hVl}|8Vk2³i-mrypVl|2VBy{j j:bdbEV^ bd`alšW#VVDib nVD^ i¹j:ijzhink8VDbU|8VBlncg_`aW6y8b¢j:_>i¹jzin`dc:_™° ºp’Cú·ú• µ  – hEjzbahDk8bMc t W6Vb-²Uy{j t j:bdbV^ bd`dlnW6Vf²UW­jzh S `d_8V j„T W«V^ W6cf` t Vypj t ivj:eV^ Vf²HhEjzbahDk8bˆ|2`alši t `ao2k Vf^ ² ¼}V_>i t VŒ¼ S j t bdVl ½V t W#`dinVf² — k s Y˜.

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(14) ƒ BR S VŒy{jzy{V t `dlc t erjz_8`V|%j:l =cfbdbdc:›BlE°  Vhin`dcg_P|8Vlšh t `dopVli S Vhk tšt V_>ilni¹jzinVc:Bi S Vl S j t VD|>¬ W6 VW#c t m®‡gV t lš`acgt _Ocz — S k s ˜² jz_8|Ol S c:›Bt l j t Vj:bBS V³8j:W6y8t bdV›B`di S i S t V s ¼} y t cgincfi-m>ypV#‡gV t S lš`acg_™° S VDc:h›in`dcf`diˆ_®›BX`dbd|8bUVoplšVYh `d`dW6opVy8lbdVi W#VV_>b¢j:išV_8|™eg° k{j:egVBhcf_2lni k8hišlYcz i V|8`dlni `do8k8inVD|­‡gV lš`dcf_™²8jz_8|#eg`a‡gVl}lncfW6V `d_>inl  } 8­ ‘DE  -_i S `dl™lšVhin`dcg_™²:›}V  t lni<|8Vlnh t `do{VBi S V}lnm>_>i¹j:³cz — k s Y˜ y{j t j:bdbdVbglni¹jzinVW6V_>inl =c t l S j t V|>¬-W#VDW#c t m W±j:h S t `d_8Vl²Ei S V_ S c:›%i S VlnV}lnivj:inVDW#V_>inlj t V}`aW6y8bdVWV_>inVD|™²Ejz_8|ef`d‡gVlncgW#VHin`dW#`d_8egl8Mc t ypcfbdm>_8cgW#`¢j:b 0jzhinc ` Ejzin`dcf_™° . 

(15)      . R }› c r`d_8|2lc:}y{j t j:bdbdVbHhcf_8lši t k8hinlj t V j‡fjz`db jzo8bdVcf_®l S j t V|>¬-W#VW6c t m6hcfW6y8k8išV t lžy{j t jzbabdVb

(16) "!g¬ bdc>cfy8ltj:_8|%y{j t jzbabdVbo8bdcrh>lE° s j t j:bdbdVb#!g¬-bdcrcgy8lj t VV³8j:hišbam%bd`fVlnVDqrk2V_>in`¢j:b}bdcrcgy8lE²HVD³rhDVy8ii S V gVm>›ˆc |%$`dlˆlšk8o8lšin`dink8išV|orm'&("!(*)+)*,*)pž - º{“ ` - “:º  lši¹j t i ’º V_8% | . ”8“:”  =µ  “. /21>”8’Eµ uz² 43 0 6ª 5M7 Ÿ 5ïª *µ 8‘» - º{“ RBS V . 0“ /91>”8’µ fVDmr›}c t |V_{jzo8bdVli S Vk8lšV t išc|8VhDb j t Vj:k8³>`dbd` j t m‡fj t `¢j:o8bdVl²f› S `dh S j t VBbdcrhj:b8išcVEj:h S y t c>hVlnlšc t ° s j t j:bdbdVbUo8bdc>h >lj t V|8:V {_8V| jzl}=cfbdbdc:›BlEž . ”8“:Ž*µ ;</28 . 0“ /21>”8’Eµ uz² 43 6ª 5M7 °° Ÿ 5>= 6ª 5M7 Ÿ 5@? *µ 8‘» . ”2“ › S V t Vi S Vlni¹jzinVW6V_>inl6ª 5M7Ÿ 5 = inc6t :ª 5M Ÿ 5A?±j t VV³>VS hDk8inS V|`d_#y{Sj t jzDbabdµ VBpb-° Ž *µ ;<_8/2V8 c: i S Vlšt Vlnivj:inVDW#V_>inlBW±jm `dinlšVb hcf_8lš`dlniBc:lšVq>k8V_>in`¢j:b<`d_8lni k8hiš`dcf_8l²8`d_#› `dh hEj:lšVi V hcf_8lši k8hi`dlB_8VV|8VD|™ž µ BpŽ*µ ;/98 6ª 5M7 °° Ÿ 5>= µ*8‘» 6ª 5M7Ÿ µ5 B ? RBS VDlnV±i S8t V= V#hDcf_8lši t k2hinl C. ”8“:Ž*µ ;/98 ² µ BpŽ*µ ;/98 j:_8|%y{j t jzbabdVb - º{“ ¬-bdcrcgy j t V6i S V6cf_8bdm®y{j t jzbabdVb hcg_8lni t k8hDinl j‡fj:`db¢j:o8bdV`d_%i S V#W#`dh t c:¬-y{j t j:bdbdVb ïl S j t V| ¬0W6VW#c t m ‡fV t ln`dcf_%cz — k s Y˜° RBS Vm j t V ÒMÔÖ:D Õ:ËUÑÌ:Û%×nÎMÑ×ÒMÕ:ޚ×nÑҚÖ:è>Î=ËUØMÒMÌ:Ú:ÍÏÞnÎUØMΙ×nÖ:ÔÔÒMá:×äÏךÔÎ{äÏÎMØMÑÌ:×HÔvë¹Ú:ÔDÎ=ÍÏךäÏÔÒ{á:ØMäÏÑB׈Ø=ÍÏךÖÎ=ØpØMÌ:Ü:×ÔÒMÎMÔ× äÏäÏ×nÚ:ä:×nÜ:ÖÒMØ=Ñ6ÍÏÔEä>ÒMÔëÛBךÒ=Î8ÎMÍÏÑDÑÖÖ>Ô}í>ÐUÎ=× Ì:E×nÚ:ÒM×ךÖEÜ:ØMÍÏÔÔÒMäÔÛBäÏäÏךÔä8Þ-Ø=Ì:ÔDÍÏÖ:Î=Ó×Îè ÔÒM×Ü:ÒMÑEޚ×nÎMÎ=ךÕÍÏÖÔ $. ). F¢æ5:GF¢Ù.

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(33) X y t `dc>efh`d_ŒVlšX*ln)c )*t l)›}=€fV X t Vy cfc>o2hi¹Vj:ln`dlš_8c VD| l ›B`di S i S VŒy t cfS išcfi-m>y{Vy{j t j:bdbdVb<‡fV tnt t ln`dcf_c: — k s YS ˜ =c t  `dbdt `dhcf_®\ t j:y S `dhlW±t j:h S `d_8Vl²8› S V t VEj:li S V iš— `aW6s`d_8efl `a_i VlnVqU°Uhcgbdk8W#_®hDc Vlnypcf_2|Oinc±i Vlnivj:_8|{j |%lšVq>k8V_>in`¢j:b‡fV ln`dcf_™²Hj‡fj:`db¢j:o2baVcg_ jzbab k R Y˜Miny±ln`dinVDlE° S t S t t t t‡gV lš`acg›}_ ct k8hcf_8W6lW#WVk2_>h inS llšhEbac:j:›}_NV opt V­i S W±j:_j:|8i V±S Vj:oplncfVq>k8k8iVDi _rišV`¢j:lnVbcgin_8`dW#Vf²U`d_8Veglšy{lEVD° h`¢4<j:` bdbdm±lnišbd=mrc ²t i V± cf‚_2)Vv¬0) y ¢ic>S hVŒVlšhDlncfc lni y{=jc t jzi baSbdVV b bdc>h ­cgy{t V t j:iš`dcft _8l W­jmt V³>RBy8b¢S j:`d_±i S jzi °  Vhcg_8|8bdmr²2i S VS lnypVVS |>¬-k8y|8c>Vl_8cfiBe t c:S ›ba`d_8t Vj t bdm6›B`ai S t i S V _>k8WopV czy c>hVlšlnc lE° `dlW±jmo{V|8k8Višc­jhEj:h Vy V_2cfW#VD_{j8²2ln`d_8hVi Vy{j j:bdbdVb™lšqrkpj `d_8e =k2_8hint `dcg_%y{V t =c t W#lW6c t VŒhcg_8hk tšt Vt _>iW6Vt W#c t m®j:t hhDVlnlšVlE° 1 V‡fV t i S VbdVlnl²ˆ›BS `di S €fX6y t c>hVDlnlnc t l² j 0jzhinc cz t jzo{cgk8iˆ‚hj:_t opVl¹j‡gV|Mc lnq>k{j t `d_8e|8VDe VDV ' ))*)ypcfbdm>_8cgW#`¢j:bdlE²D› V_8hDVYcg_8VBhEjz_V³>y{VhDi jlš`aW6`db j lnypVV|8k8y =c ypcfbdm>_8cfW6`¢j:b>0jzhinc ` Ejzin`dcf_™°  } 8­ ‘D   4<`defk t VY€V³>y8b¢j:`d_8li S VhDcf_8hVDy8inl c:™W#`dh t c:¬-y{j t j:bdbdVbd`dlnWj:_8|6W­j:h t c:¬-y{j t j:bdbdVbd`dlnW`d_ — k s ˜°rwjzh S hbdk8lšinV tt `dlBW­j:|2VBc:lnV‡gV t jzb™y t c>hVlnlšc t lY›}c t >`a_2eŒcg_­jl S j t VD|­W6VW#t c t 6m t ïW#`dh t c:¬-y{j t j:bdbdVbd`alš6W ° RBS V hbdk8lšinV lhEj:_±hcgW#Wk8_8`dhEjzinV}o>m­j >`d_8|#cz W#VDlnl¹jzefVv¬-y{jzlnln`d_89e ïW­jzh cz¬0y{j jzbabdVbd`dln6W D° {VDba|. :5:Ö<;õ:ð¹ø.

(34) & Micro-Parallelism P1. P2. ..... Micro-Parallelism P1. Pk. Shared Memory. ..... Pl. Macroparallelism. Pm. P1. P2. ..... Pn. Shared Memory. Shared Memory. Micro-Parallelism. Micro-Parallelism. 4<`defk 4. P2. ..... Shared Memory. Network. P1. P2. t V€8ž — `dh t c:¬-y{j t j:bdbdVbd`alšW j:_8|±W­jzh t cz¬0ypj t jzbdbaVDba`dlšW®°. 

(35)    4 " . ˜ k8t Vinc i S V 0j:S hDi#t i S j:i#›}V S j— ‡fV®s hDcf_8hVD_rS i t t jzinV|cgk t V t Uc t inl±cf_ i S S V!¢l S j t V|>¬-W#VW6c t m y{j t jzbabdVb ‡g— V lšs `acg_ jz_8|t cgi V j:lnypVhišlc: k t Y˜ i V VB`dl<_8c|8`dlni `do8k8inVD|3ï`a_i V}lnV_2lnVBc:pW#Vlšl¹j:egVv¬-y{j:lšln`d_8e k Yt ˜‡gVS lš`dcf_®j‡fj:`db¢j:o8bdVBlšc0jt ° t — s t S› `d4h S cgjW4t Vi _>V}k8‡rW`dVo{›BV typVcf|`d_> i™t cfczW 8W­Œj:h išc c:¬- y{°#j wj:bdjzbdh VS bd`dlnhW bdk8lniškV t Y`d_˜ inkhcgt _%_8ln`d`dllniš=lHc c:t  W#j Vln| ViHc:Yc:pj6`a_2_>|8k8VWypVo{_8V |8t Vcz_> iHt hk8bdk8_8lš_8in`dV _8e l — k s Y˜ fV t _8Vbdlj:_8|W6`dh t cz¬0y{j t jzbabdVbd`dlnW`dl

(36) ¢c t hj:_#opV k8lnVD|#`d_8ln`d|8VBVD‡fV t m#hbdk8lšinV t ° RBS V — k s Y˜ b¢j:_2efk{jzeft VBy S t c:‡r`d|8tnVDt lBinci S V t k8lnV t t jlnypVh`¢jzbUhcf_8hDVt y8iišcŒy t cfe t jzW `d_6i SRB`dS l}|8`dlni t `dt o8k8inVD|#V_>‡>` t cf_2W#V_>iE° ½ c:t ›ˆV‡gV i VYhDk V_>ik8lšV `d_>inV 0j:hDVY`dl}_8cfiB‡gV my2baVj:l¹t jz_>iBinck8lnVft ° V`d_rišV -j:hVhcg_8ln`dlnišlBc:H_8Vi ‡fj `¢RBj:o8S bdVljzlB›ˆVDbab™jzlBq>k8tVk8Vlj:_8|±y8`dypVlBopVi-›ˆVDV_®t |8` UV SV_>i hbdk8lnišV l° t S t V 38© S 5&*§f£ ¢8§ =t ¨d©ªˆj VhDcfW#t W6cf_ incYjzbdbfhbdk8lšint V lE²glnci VmhEjz_ŒopVˆhDcf_8lš`a|2V V|j:l<l j V|W#VDW#c t m < opVi-›}VV_i V|8` UV V_>i hbdk8lnišV l ¢`-° Vf°8egbacgo{j:bp‡fj `¢j:o8bdVl ž ; MºpŽ ”  š” ; MºpŽ ”  š” UŽ -_®i S V  t lni =c t W i S VŒ_2Vi‡fj t `¢j:o8bdV (#`dl t VEj:|®cgk8iE°}½ V t 'V (`dlhDcf_8lš`a|2V t V|incopVŒj‡fj t `¢jzo8bdV_{jzW#V lšc t `di`dlS _8cfiV‡fj:bdk{jz5 inV|™° RBS V t Vlšk8bdiŒczi S t V t VEj:| cfypV t jzin`dcf_%`dli S V_OVD‡fj:bdk{5 j:išV|™°-_%i S V6lnVhDcf_8| =c W t i VS ‡fj:bdk8VBcz `dl jzlnlš`a5eg_8V|6in'c S ({°Ht ½ V Vjzerj:`d_ S (`dl}_8cfi}S V‡fj:bdk{jzinVt|±j:_8| V‡fj:bdk{jzinVlBj:lBk2lnk{jzb0° ˜ k `a_2ei VVD‡fj:bdk{j:iš`dcf_±c: ²8_8ccgi V i¹jz2l ±hEjz_®h j:_8egVi V_8Vi ‡fj `¢jzo8bdV (p°} lnlš`def_8`d_8e#j‡fj:bdk8V išc#j_8ViB‡fj t `¢j:o8bdV`dlBi S V t Vú=c t VŒj:_®j:išcfW6`ahBcgy{V t j:iš`dcf_™°     . F¢æ5:GF¢Ù.

(37) ‰. cluster 3. cluster 2. writequeue("a",1,5). writequeue("a",1,7). . . . . .. queue named "a" of cluster 1. cluster 1. 4<`defk. :5:Ö<;õ:ð¹ø. t Vƒ{ž k8Vk8VDl.

(38) ') wt j:h S hbdk8lšinV t hEj:_ S j‡fVŒjz_rm6_>k8Wo{V t c:q>k8Vk8Vl²Uj:_8| jhDbak2lninV t hj:_®› t `aišV`d_®j:_>m#q>k8Vk2V3ïlnVV {egk Vƒ ž ¸Œ“0/M’µ B Hµ Hµ n” 4/ G1>”   µ ½ V t V (+)81,t `alBi S VV³>y t Vt lnlš`dcf_®›S S `dh St ›B`dbdbUopVY› t `dininVD_®inci S Vqrk2RBVk8S V7t (cz‘i S Vhbdk8lnišV t >° ~hbdk8lnišV hEj:_ cf_8bdm VEjz|Oi V' lniVbdVW#VD_ric:`dinlq>k8Vk8Vf° V V±V³>`dlninlj±o8bdc>h>`a_2e j:_8|Oj _8cg_>¬-o8bdcrh >`d_8e t VEjz|„jzhin`dcg_­cg_®jq>k8Vk2Vfž “:µ>”{

(39) » B  µ Hµ n” “:µ>”{

(40) » B  µ Hµ n”  Mº™•  ½ V t V (`dlHi S VB_{j:W6Vc:{i S t VBq>k8t Vk8VB— i S V}s hbdk8lnišV t ›B`dbdb t VEj:| t cfW °M— {i S VBq>k8VDk8VB`dl VDS W#y8i-mi S V=k2_8hin`dcgS _ hEjzbd4b ! ,*("$ .1 ,81,Q.}›B`dbab VDink _ j k ˜cfofuvVhDiBc: i-m>ypV˜  18 5° -_±i VlšVhcf_2|®hEj:lšVi V =k2_8hRBin`dS cg_­t hj:bd4b !,+("$ .1,.1,Q.B›B`dbdbpo8bdcrh 6k8_>in`dbUi S V t V`dlj‡fj:bdk8VYt j‡fjz`db jzo8bdVBt `d_#i S Vq>t k8VSk8Vf° V VV³>`dlninl±cg_8V®lnypVh`¢jzbq>k8Vk8V_{j:W6V |  "!8+®`d_NV‡fV m«hbdk8lšinV ° B=išV i Vlnm>lšinVW `al `d_8`din`¢jzba` t VD|%V‡fV t mOhbdk8lšS inV t h S Vh >l`aišlq>k8Vk8V±›B`di S t i S V±_{jzW#S V !8+6ty{V t `dcr|2`ahj:bdbdm › S VD_P`di S j:l _8c›}c inc|8S cp9° MYi V#t q>k8VDk8S V±hS cf_>i¹jz`d_8t lŒj:S _ V³>y VDlnln`dcg_™²i `dlŒcg_8V#`dl VEjz|Pj:_2|%V‡fj:bdk{jzinV|™°S S _ V³>htVDy8in`dcf_±`dl}i V hbdk8lnišV S f²>› `dh _2V‡fV h Vh >l`din l !8+Bq>k8Vk2Vf²8o8k2iˆhcgW#Wk8_2`ahj:inVDl ›B`di i V k8lšV jz_8|±S efVišlivj:2l >lB`d_#i `dlB› jm>° F `ai q>k8VDk8Vl}`aiH`dlypcflšln`do8bdVBinclšV_8|±j:_V³>y t Vlšln`dcf_6incVD‡fV t mŒhbdk8lšinV t ²>o8k8i`di`dl_2cfiHefk{j t jz_rišVV| i S j:i cf_8bdmcf_2VYhDbak2lninV t › tt `dinVl S `a_6i S S Vqrk2Vk8Vczjz_8cfi S t V t hbdk8lšinV t °  ci S Vhbdk8t lnišV t i S j:i t VEjz|8t lB t cfW `dinlq>k8Vt k8V6hEj:_2_8cft iopV# lšk V6S i t jzii V#‡fj:bdk8VS `dlt lnVD_ri cfWt j±lnt ypVh@` {h6hbdk8lnišV `diS › jzS _rišlŒišc VDhVt `d‡fV jz_®V³>y VDlnln`dcg_® cfW° c#i V VV³>`dlnišlYj:_2cfi V |{jzi¹jlni k8hišk V_{jzW#V|¦   ¦8©²{› `ah `dllš`aW6`db j inc q>t k8Vk2Vl t ¢lnVDV {efk t t VŒ*‚ ° 42c t i-›}t chbdk8lšinV t l }j:_8t |

(41) i S V t Vt V³>`dlnišlV³8j:hDt inbdm#cg_8Vy8`dypVYt cz_{j:W6V (i S j:i t k8k8_8_8l _8`d _8ecgW t cfhW bdk8lnhDišbaV k2lnin V t inc±

(42) hinbdck8lšhinbdV k8lniš

(43) 8V °t }j y2t `aVYypi-7V ›}( c|8k2` _8_8V t `d_8V_>e6i y8cg`dypW VlEhDž bak2lninV }inc6hbdk8lnišV

(44) ­j:_8| j6y8`dyp7V ( ¸Œ0“ /M’µ . / . µ n” 

(45) 1r”  Hµ ½ V t VBi S VtV³>y t Vlnlš`at cg_ (*)81,B`dl™› t `aišinVS _`aS±_ti S Vy2`aypt V ( c:8hbdk8lšt inV t  5° M i S `dl™ t `d_8 lši t k2hin`dcf_`al<y{V t =c t W#VD| o>m#hDbak2lninV >²>`diBh VEjzinVljy8`dy{V (› `dh k8_8l} cfW hbdk8lšinV išchbdk8lnišV ° “:µ>”{» . / . µ š” 4/ “:µ>”{» . / . µ š” 4/  Mº™•  t tYt t t i S Vy8`dyp7V (6c:ˆhbdk8lšinV t r° F«`di S i S  RBS VlnVhcgV _8|®hElnijzbd!bU,+`alB("$*i S &-V & o8,Qbdc>.h  >hE`d_8jzebab-²U‡fjj t hD` jzbak2_>lniˆinVcz i SVEj:V|2MlŒk8_8j:h_®iš`aVDcg³r_±y hEVj:lšbdln4b `d!cf,*_ ( $*&-cgW RBS V=k8_2hin`dcf_ &*)*76 DV_{jzo8bdVli S V­k2lnV t c t y t cge t jzW#W6V t inc r _2&c:,› .gi° S V6incgy{cgbacgefm%czi S V |8t `dlši t t `do8k2inVS |6lnm>lnišVW Qž t &—*)*76s D. t Višt k t _2lBi S V _rk2S Wo{V t cz™hbdk8lnS išV t — lB`d_s i S V lnm>lntišVW 38&*)*76 DBF- Vink _8l#i V#_>k8WopV cz k ˜ gV _8VDbal²`-° Vf°Ui V#lšk8W c:i V k Y˜ gV _8VDbalc:`d_8|2`a‡>`d|8k{jzb hbdk8lšinV t l t 3* "&— *)*s !6 D ft ²› S V t V Œ`dlBjt ypcfln`diš`a‡gVB`d_rišVefV t opVi-›}VV_ @jz_8| t "& *)*t 76 D#S .g² t Višk t _2lBi SS V _>k8tnWt opV cz k t Y˜ fV _8VbdlB`d_hbdk8lšinV 43"{_{j:bdbdm "&+)*76 D*)81," ,"! Vink _8lBi V`d_8|8VD³Œcz™i V hk VD_rihbdk8lšinV °  .  . .    . . .  . . . . . . . . F¢æ5:GF¢Ù.

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(83) Unit´e de recherche INRIA Lorraine, Technopˆole de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS LES Unit´e de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unit´e de recherche INRIA Rhˆone-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN 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.

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