On the Removal of Anti and Output Dependences
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. On the removal of anti and output dependences Pierre-Yves Calland , Alain Darte , Yves Robert , Frédéric Vivien. N 2800 Février 1996. PROGRAMME 1. ISSN 0249-6399. apport de recherche.
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(46) Sk : Sl :. S’k :. temp(f(i)) = rhs(...) f new. Sk :. lhs(f(i)) = temp(f(i)). lhs(f(i)) = rhs(...) a out rhs(...) =. a out Sl :. (a). rhs(...) = (b). *km\]n9Xha|bz?' "nOl[¶¯\;Z;k~xO\«^x;l_kJw9azpazxOw9azxOgja'^ i u9a3¡@Z]Xha&^;xOw u i ^5¡@l_azX l2Xv^;xOoh¡@Z]X_`|^1lvkrZOxJ¬ S’k : Sk :. f new. lhs(f(i)) = rhs(...) o out. Sl :. temp(f(i)) = rhs(...). Sk :. o out. lhs(...) = Sl : (a). lhs(f(i)) = temp(f(i)). lhs(...) = (b). * km\]n9X[a'b;b'"nOlh¶¯\Z;k~xO\'Z]nOl3p9nOlw9azp9azxOw9azxOgja|^ i ua3¡SZOXha^;xOw u i ^5¡SlvazXYl3X_^;xOoh¡@Z]X_`|^Kl_kmZ]xJ¬ e=qmqYl2iOajova'Xhajo2nOqml_o&^;Xha|o2n9`$`|^;Xhkjajw k©x®*km\]n9Xha6$¬ µZ;lva|l2i9^1l"o_ajq¤¡qmZOZ]pOoc^Xha'p9XhZOgjajo_o_ajwÀ^1o"l2iOa|Zl2iOazX ajw9\;ajo('l2iOa"qmajo_oZOu;OkmZ]nOogz^1o_a"kmo"^Fo_ajq¤¡^;x;lvk¦¶¯w9azpazxOw9azxOgja|qmZ]ZOp¨Z]x¨o_l2^1lvaz`&azx;l e o_k~xOgjakmlgjZ]`&ajo(¡ XhZO` e ^;xOw\;ZOajol_Z e t9kmlkmqmq·uacXhazpOq~^1gjajwu}«^x^;x;lvk w9azpazxOw9azxOgja|ajw9\;a"gjZ]`ªk©xO\$¡¯XhZ]` e ^xOw\;Z;k~xO\|l_Z e ¬ µaj{OlY ao2iOZ1 l2iOanOo_a3¡¯nOq~xOajo_oZ1¡Z]n9X*l2Xv^;xOoh¡@Z]X_`|^1lvkrZOxJ¬;/¡â al2Xv^;xOoh¡@Z]X_` ^1qmqJ;azXhl_kmgjajo Z1¡â^"w9azpazxOw9azxOgja \]X_^p9iJt9l2iOazxl3iOaZ]xOqm}gj}Ogjqrajo"l3i9^1l`|^5}'Xhaz`$^1k~x ^Xhap9n9Xha Z1 w9azpazxOw9azxOgja|gj}Ogjqrajo g Z]xOqm} `|^1w9acn9p kml2i ajw9\;ajoq~^;uajqmajw "i Z]X p9n9XhaZOnOl2p9nOlw9azp9azxOw9azxOgja$gj}]gjqmajo g Z]xOqm}«`$^1w9acn9p¨kml2i ajw9\;ajoq~^;uajqrajw i ¬ JºO¾ 5º
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(109) *¬ "km;azx&^=\]Xv^;p9i j g9%(':) i t3 aYgjZ]xOovl2X_nOgjlY^;x&^1w`&kmo_ovk©uOqma qmZ]ZOp xOajo_ltU iOZ;o_a"w9azpazxOw9azxOgja|\]X_^p9i¨kro ¬ OZqmajl j g&%('*) i u9a¨\;km;azxJ¬eqmqYajw9\ajock~x `&nOo_lgjZOX_Xhajo2pZ]xOwl_Z²^;x;lvk*w9azp9azxOw9azxOgjajo|k~x®l2iOa|qmZ]ZOp. Ox ajo_ltU¬±Z|az^1gji¨;azXhl_aj{ % a^1o_ovZ]gjk~^1l_a^ªqrk~xOaz^;X=^;X_Xv^z} f k &Z1¡*o_kja'bz g o2^5} i ¬O£au9nOkmqmw²l2iOa"qmZ]ZOp xOajo_ltUD^1o^&o_k~xO\;qmaqmZOZ]p¨o2n9X_X[Z]n9xOw9k~xO\ % o_l2^1l_az`ªazx;l_o5¬ù±iOazXha"kmoZ]xOa"o_l2^Kl_az`&azxl e pazX;azX[l_aj{ Vt]=iOZ;o_a qma3¡Sl"i9^;xOwo_kmw9ackmo=o_k~`|pOqm} wf k g;]1i j ¬Z]X az^Kg i ajw9\;a j g +#' i ) a"Z]uOl3^1k~x ^;x^;x;lvk w9azpazxOw9azxOgja ¡¯XhZ]` e l_Z e u}k©xOovazXhl_k~xO\«^$Xha3¡SazX[azxOgjal_Z f k g;]Sl b i k~x l2iOaXhkm\]ili9^;xOw ovkrw9aFZ1¡ e ^1o(¡SZ;qmqmZ1o(' Y;Z\ ] 'j b` Z ; cZ ` %` (` e ^ wf k + g;]1i j l f k g;]l b il ` %` (` e ^ f k g ]Ai j l ±iOacqmZOZ]pxOajo_l{U kro"gjqmaz^;X[qr}^1w`ªkrovo_k~uOqra;t*^1o"l3iOazXha|^;Xha|xOajkml2iOazX Z1 xOZ]XZ]nOl2p9nOl"w9azpazxOw9azxOgjajo|k~x²kml_o w9azp9azxOw9azxOgja"\OX_^;p9i ¬;±iOkmogjZOxOo_l2X_nOgjl_kmZ]xªkrogjqmaz^;Xhqm}«pZ;qm}ùxOZO`&k~^1qk~x$l2iOao_kja=Z1¡ t;^;xOw'l3iOaXhajo2nOqml*¡@Z;qmqmZ1o k~`|`&ajw9k~^1l_ajqm};¬. . ¾Jz¾ ç5»
(110) !
(111) 5H+>6? =1 "8JV%'(4 7 5 &+24 " /+,(;+2 ; ? !" _+V _)*+ ¯x$l2iOkmoVOajgjl_kmZ]x¨ au9Xhkma }¨o2°1ajl_gji&o_Z]`ªaiOazn9Xhkmo_l_kmgjol_Z xOwZ]nOlY=iOkrgji$;azXhl_kmgjajo Z1¡ l2iOa=w9azp9azxOw9azxOgja\OX_^;p9i. j 9g %('*) i Z1¡¢^ªqrZOZ]pxOajo_lo3iOZ]nOqmw u9aFl2X_^;xOo[¡SZ]Xv`&ajwo_Z|l2i9^Kll2iOazXhaXhaz`$^1k~xOoZ]xOqm} p9n9XhaFgj}]gjqmajo"k~x ¬9£a \;km;a|lh Z iOazn9Xhkmo_l_kmgjozt*uZ;l2i 9nOkml_a«x9^Kl2n9X_^1q@¬â±iOa& Xho_lFZ]xOa'`&km\]ilcua';azX[} aj{9p9azxOovkra'k~x®l2iOa' Z]X[o_lgz^Ko_a;t u9nOl gjZ]nOqmw u9a=Z1¡âk~xl_azXhajo_l¡@nZ]X*o2`$^1qmqw9azp9azxOw9azxOgja&\]Xv^;p9iOoz¬]±iOao_ajgjZOxOw'Z]xOa"^1qm^5}]o Xha ùnOk~Xhajo=^p9Zqr}9xOZ]`&k~^1q l_k~`&a;¬9@l X_n9xOo k~x¨l_k~`&n a )g g % wl ) i i t5iOazXha kmol3iOaxOn9`&u9azXZ1¡âl2Xv^;xOoh¡@Z]X_`&ajw&;azXhlvkrgjajo5tùiOazxOgja^ Z]Xho_l gz^1o_auZ]n9xOw )g % g % l ) i i ¬ >/10 !8 GOAJ%D O " FhJI8 @5M=
(112) D 8 MH V8CG GF 5CK
(113) D 8 H S9A8 J)5hK ^5}«ua;tl2iOa`ªZ;o_lx9^1l3n9X_^1qâiOazn9Xhkmo_lvkrg&kmo=l_Z'u9nOkmqmw l3iOaci}ùpazXh\]X_^;p9i j 9g %(' i ZK¡·l2iOaFgj}]gjqmajo=Z1¡ ¬ kro w9a3 xOajwH^1o^¨gjZqrqmajgjl_kmZ]x®Z1¡Yo2n9uOovajl_o " % tViOazXha|az^Kg i " kro"l3iOacovajl"Z1¡l2iOa&;azX[l_kmgjajo"Z1¡(^;xajqmaz`&azxl2^;Xh} gj}Ogjqra D Z1¡ ¬SOaja*km\]n9Xha|b5"¡@Z]X l3iOai;}9p9azXh\OX_^;p9i ZK¡âZ]n9X l2^Xh\;ajl aj{ù^`|pOqma;¬9 n9Xhl3iOazX_`&Z]Xha=az^1gji¨;azXhl_aj{ k~x " kmoc`|^X_°1ajw
(114) , ! !
(115) 3%' k¤¡ D kmocu9X[Z]°1azxiOazx kroFl2X_^;xOo[¡SZ]Xv`&ajw tk@¬ña;¬ kmoc`|^;Xv°1ajw
(116) , 7! !
(117) 3%R k¤¡l2iOa k~x¶¯gjZ]`&k~xO\ajw9\;aFZ1¡ $k~x D kmo^x ^;xl_kJZ]X Z]nOl3p9nOlw9azpazxOw9azxOgja;t ^;xOw l2iOaFZ]nOlh¶¯gjZ]`ªk©xO\¨ajw9\;a&^ Z1 Z]X^;xl_k w9azp9azxOw9azxOgja;¬ 6. . . . . :. . . . 9. . . à à Û . 9.
(118) S4 S1. S2 S3. *km\]n9Xha&bzS' }ùpazXh\]X_^p9i&Z1¡ l2iOa=l2^;Xh\;ajl*aj{9^;`|pOqma;¬O±iOal3i9Xhajaajqmaz`&azxl2^;Xh}|gj}Ogjqmajo^;Xhao2iOZ1xªkml2i|w9k % azXhazxl ^;X_X[ZóD¡@Z]X_`|^Kl_oz¬ "xOgja kmo(u9nOkmqmlztV a^;p9pOqm}^F\]Xhajajw9}|o_l3X_^1l_aj\;}|^;xOwl3X_^;xOoh¡@Z]X_`l2iOaazXhl_aj{ iOkmgjiu9ajqmZ]xO\;ol_Zt9^;xOw kmo
(119) 3 1 !C!
(120) ,%' ¡@Z]Xjt1l2iOa=`|^1{Ok©`$^1qx]n9`&uazX*Z1¡Jo2n9uOo_ajl_o " &¬;£aw9ajqrajlva^1qmqgj}Ogjqmajol3i9^1l* azXha\;Z;k~xO\"l2i9X[Z]nO\]i ^;xOw|¡@Z]X¢iOkmg i ^1ou9Xhaz^°z^;uOqma;¬O£a"Xhajw9Zl2iOa=Z]pazX_^1l_kmZ]x¨n9x;lvkrqJl2iOazX[aXhaz`$^1k~xOo(xOZFgj}]gjqma"k~x¨l2iOa\OX_^;p9i kml2i
(121) , 7! !
(122) 3%R azXhl_kmgjajo ¬ ±iOa|wX_^5u9^1gj°Z1¡ l2iOkmociOazn9Xhkmo_lvkrgkmoFkrlvociOkm\]i®gjZ;o_l"k~xl3iOa| Z]Xho_l=gz^1o_a;¬eqml2iOZ]nO\Oi²l2iOkmokmo|n9xOqmk©°Kajqr} l_Z i9^;p9pazxJtâl3iOa«xOn9`&u9azXFZ1¡gj}]gjqmajo&gz^;x u9a¨aj{ùpZ]xOazxl_k~^1q k©x®l2iOa¨o_kja )g % Il ) i Z1¡l2iOa|\]X_^p9iJtâ^;xOw l2iOa gjZ]xOo_l3X_nOgjl_kmZ]x¨Z1¡ `&km\]i;ll2iOazXha3¡@Z]Xhai9^5;a^|p9XhZOiOk©uOkml_km;aFgjZ;o_lz¬ Jº º
(123) Q/j(Q/ R R Qçºù¿ j¾H Jº²j Vº]$º R ~º £a o2iOZ1iOazXhal3iOal3X_^;xOoh¡@Z]X_`|^Kl_kmZ]x"Z1¡Jl2iOa l2^;Xh\ajlâaj{9^;`|pOqmaZ1¡MOajgjl_kmZ]x|9¬pnOo_k~xO\Fl2iOkmoiOazn9Xhkmo_l_kmg;¬*kr\On9Xhab( o2iOZ1o=l2iOa|i;}9p9azX[\]X_^;p9igjZ]X_X[ajo2p9Z]xOw9k~xO\l_Z'l3iOacw9azpazxOw9azxOgja'\OX_^;p9iZK¡*km\]n9Xhabz9¬V±iOa&l2^;uOqma'u9ajqmZ1 g *k¤¶ \]n9Xha|b5 i o2iOZ1o ¡@Z]X az^1gji¨;azXhl_aj{iOZ1 `|^;x}'ajqmaz`&azxl2^;Xh}¨gj}Ogjqrajo"k~xOgjq~nOw9a&kml^1o=^
(124) 3 7!C!
(125) ,%' ;azXhl_aj{J¬
(126). 9. . . . :. . S4 S1. S4 S2. S1. S2 S3. S’3. S3. (b). (a). *km\]n9Xha|b(?' }ùpazXh\]X_^;p9i¨ZK¡âl2iOa=l2^;Xh\;ajlaj{ù^`|pOqma^ i u9a3¡@Z]Xha"l2X_^xOoh¡SZOX_`|^1l_kmZ]x$^;xOw«u i ^5¡SlvazX*l2X_^xOoh¡SZOX_`|^1l_kmZ]x Z1¡YxOZ]w9a e z¬
(127). e W eMn e e b b *kr\On9Xha'bz?'ùµ=n9`&u9azXZ1¡âgjk~XhgznOkml_o"iOkmgjik~xOgjq~nOw9a|^&;azXhl_aj{^1o
(128) 3 7!C!
(129) ,%' ¬ ëÙ@Îhï_Я2 Ü ÷Ù/Ú1ÎЯï Ü@Î÷|î2ë¦ÎÙ ïЯïÓ@î3Î ÎÙ@VÐ¯Ý Ó@yJÎï_÷5ðñOë¦Ü@Îë¦Ý÷3ë¦ÏzÙ@z;Ó@ЯÎÙYÑ_ð¦î3Ñ_Ó/Û1Îh÷1Ýö2ίÙ*Û|ÎÛzÓ@Ó/Ñ Ó Ð¯1ïvïvÙ@ÛóÛóÎhÙ ï_ò2~ЯÑvÜë¦Ð¯Ù ÷ Ý yJë¦÷1Ó Îh!Ó/z;÷óÎ7ÎÜhÐ ïvÓ/Úó2~÷óÑ_Ù@ÎÐΠЯÓ/ÎâyJ÷óÎ Î*Î*3÷óÝ ë¦Ù@ïvÓ@ï î2Ù 3Îë¦ïÝ Ó@Ù/Ñ ïvÚ(ð9z1zzÛjÙ@ЯÎ7ÎhÚóÓï Ý zKë¦ÎÓ¢Ð*ð¦
(130) Ñï229Ó@yJÎÙ/ÐÚ(÷5z1ë¦ÑvÜ@Ù@Û|÷"ÎhÓ@ï_Ü7Ù ÛzÑvÛzåÓ@yJïvï_ëñÛóå"Ù éÓ@ïÑ"Ù/ò5ëñÛóÎð¦övÎ7ð¦Ó@Î ÎÓ/÷óÄ Î Üh åjܯð¦Ä Î î3ÎЯÎÓ@ð¦Î Ù@Î Ä Ä . . . . . . . . . .
(131) . . . . . . æçßàâæ~è.
(132) e=gjgjZ]Xhw9k~xO\ l_Z l2iOkmo'l2^uOqra;t l3iOa iOazn9Xhkmo_l_kmg Xho_l¨l2X_^xOoh¡SZOX_`&o'xOZOw9a e 1¬Y±iOai}ùpazXh\]X_^p9iÀZ1¡l2iOa²xOaj \]X_^p9i&kmoo3iOZó=x&k~x|*km\]n9Xha&b(9¬1e=oYl3iOai}ùpazXh\]X_^p9i|o_lvkrqmqVgjZ]x;l3^1k~xOo
(133) 3 7!C!
(134) ,%' xOZOw9ajozt9^;xOw'^Ko eMn kmo l2iOaZ]xOqm}
(135) , ! !
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