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Optimal Time and Minimum Space-Time Product for Reversing a Certain Class of Programs

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(1)Optimal Time and Minimum Space-Time Product for Reversing a Certain Class of Programs José Grimm, Loïc Pottier, Nicole Rostaing-Schmidt. To cite this version: José Grimm, Loïc Pottier, Nicole Rostaing-Schmidt. Optimal Time and Minimum Space-Time Product for Reversing a Certain Class of Programs. RR-2794, INRIA. 1996. �inria-00073896�. HAL Id: inria-00073896 https://hal.inria.fr/inria-00073896 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. Optimal time and minimum space-time product for reversing a certain class of programs Jose´ Grimm, Loı¨c Pottier, Nicole Rostaing-Schmidt. N˚ 2794 Fe´vrier 1996 THE`ME 2. ISSN 0249-6399. apport de recherche.

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(253) wbl¨u;onpVp ˆ

(254) f : Ìtzmnced)xktž_”uNtPs)p ceˆbwStžd)xtž_bd xd)s)p lb‰¨ced)c{d) ? 

(255) Áxžwb{bzp wStzmnlbd Akpr£ tž_bdko|mnlbdvur

(256) xzySxztžd)c urlb‰Axžpro|œd$m|t`ª`m|tž_2mu 3œurs)p {SmcedUtž_bp65‰ ?

(257) APexžwb{bzp wStzmnlbd A9¬ ^`_bd

(258) AP xwb{bzp wStzmnlbdËmnx uNonxžpÐs)p ceˆp xžd)‰ pr£u¨onp6p ˆ tž_”uNt;mnxÌd9ÀSd)s)wStžd)‰ dUm|tž_bd)Ìwblœtzm|oPu zd)xžmn‰bw”uNoEmnx`onpvªÍd)tž_”urlËu  uNonwbd p k‘ %tzmnced)xv¬ ).    !

(259) "  "       #

(260) A

(261) P" . . mnq wbzd+:   Nu o|oEq žurˆb_Apr£tž_bdPmnlSm|tzm§uNo& ”p tžžurlˈbzp q žurc±¬ . . ')(  ( ‚. $W.&%m!v-/!"0 .<f0 .,%" * — 7€+ T_UVEWWXY, —.- ˜vš € ךb™N×/-˜v™MšS˜ Ò+”Ó. ( 0  3f02.. . ƒwbJˆbzp q žurc ª¥urx°®”zxztJ‰Sm|¢d)zd)lœtzm§uNtžd)‰Ì{œyewbxzmnlbqtž_bdxzturlb‰”urz‰;zdUœd)zxždcep6‰bd

(262) pr£tž_bdoT_U

(263) VEW10 ‡p xztuNmnlbqS‘’  , u1ExzySxztžd)c±¬ T_U

(264) VEWWXYkˆd)£¤p zcex

(265) urwStžp c=uNtzmnsP‰Sm|¢d)zd)lœtzm§uNtzmnp l2{œy;ˆbzp q žurc. R WXYm/. ôqprs ut.  .

(266) ‘vŽ. . $   

(267) H# 

(268)  # !2 

(269) . tžžurlbxž£@p zc=uNtzmnp lbxv¬Ì†Htžx$mnlbˆbwStmnx u± ”p tžžurl ˆbzp q žurc urlb‰ u o|mnxztpr£Í urm§ur{Sond)xv¬Ì†¡tPzdUtžwbzlbx u2lbdUª ”p tžžurl ˆbzp q žurc £¤p  s)p ceˆbwStzmnlbq tž_bd ‰bd)m| uNtzm|œd)xv¬C†­l tž_bd zdUœd)zxžd±cep6‰bd ¼°dvurs_ q d)lbd)žuNtžd)‰Ìxžwb{bzp wStzmnlbd

(270) mnxJs)p ceˆp xžd)‰Ápr£t¡ªÍpˆ”urtžx’¬^`_bd

(271) ®”zxztJˆ”urtÍs)p ceˆbwStžd)xÍurlb‰;xu)œd)x¯mnl onp6svuNorurm§ur{Sond)xktž_bd  uNonwbd)xktž_”uNt$urzd%cep6‰Sm|®”d)‰«ª_bd)l2tž_bd%p mnqrmnl”uNo°ˆbzp q žurc mnxkd9Àbd)s)wStžd)‰¿¬ ^`_bd xžd)s)p lb‰ ˆ”urtes)p ceˆbwStžd)xetž_bd ‰bd)m| uNtzm|œd)x%mnlÐzdUœd)zxdÌcep6‰bd wbxzmnlbq2tž_bdÌxu)œd)‰  uNonwbd)x ª_bd)zdÁtž_bdUyCurzd=lbd)d)‰bd)‰¿¬  p lbxžd)}Vwbd)l©tzo|yœ¼+ª_bd)lÎu  urm§ur{Sond=mnxcepV‰Sm|®”d)‰ÐuNtdvurs_Îxztžd)ˆCpr£ uÌonp6p ˆ¿¼&uNo|oJpr£¯m|tžx$ uNonwbd)xurzd=xuvœd)‰¨mnlCurlCurzžu)y ª_bp xždexzNm M)d=mnx$d)}Vw”uNo°tžp tž_bdelVwbc{d)pr£ m|tžd)žuNtzmnp lbxpr£tž_bdPonpVp ˆ¿¬    # !

(272) "  "        #

(273) A

(274) P" A

(275) P"  #

(276)   . . mnq wbzd“ qRƒ‰SySxžxžj)d$xztžžuNtžd)qryœ¬ Rƒl±p wb`d9ÀburceˆSond ¼bwbxzmnlbq TSUVEWWXYbªÍd$_”u)œdPp {StuNmnlbd)‰ tž_bd$ˆbzp q žurc ª_bp xd$svuNo|oEq žurˆb_ mnxƒˆbzd)xžd)l©tžd)‰2mnl« mnq wbzd “S¬^`_bd     ˆbzp q žurc svuNo|onxtž_bd

(277) " ±xžwb{bzp wStzmnlbd ¼ª_bd)zd tž_bdːruNonwbd)xÌpr£ urzdËs)p ceˆbwStžd)‰

(278) f : ¨tzmnced)x urlb‰ xu)œd)‰¾mnl url urzžuvy¾pr£PxzmNM)d f

(279) : S¬ ^`_bd xwb{bzp wStzmnlbd$

(280) " "¨svuNo|onx%tž_bd

(281) P"¨xžwb{bzp wStzmnlbd ª_bd)zd mnx%s)p ceˆbwStžd)‰ ‘ Atzmnced)x’¼ urlb‰ uNo|o

(282) pr£km|tžxemnlœtžd)zced)‰Sm§uNtžd2ruNonwbd)xÌurzdAxuvœd)‰ mnl¾url¾urzžuvy pr£kxžNm M)d ‘ S¬ ^`_bd    xwb{bzp wStzmnlbd$mnxlbprt‰Sm|¢d)zd)lœtzm§uNtžd)‰«{d)svurwbxždkm|t`mnx

(283) s)p lbxzturlœtv¬¯^`_bd

(284)  " " Á 0p tžžurlËs)p6‰bdkmn x xžtžzwbsUtžwbzd)‰Ëurxxž_bp’ªl±mnlA mnq wbzdŒ6¬ ')( ( A  - ˜  К ( -˜v™MšS˜ Ò +S@Ò — ½Cd _”uvœd urˆbˆSo|mnd)‰ tž_bd   Ÿ¡xztžžuNtžd)qryÐtžpËtž_bd c=uNmnlÐonpVp ˆ {©y s_”urlbqrmnlbqC{©y _”urlb‰Ðtž_bd ˆbzd9¦ 6mnp wbxPurwStžp c=uNtzmnsvuNo|o|y¨q d)lbd)žuNtžd)‰Cˆbzp q žurc±¬ ½Cdezd)ˆSo§urs)d)‰¨tž_b.d

(285)   " Axwb{bzp wStzmnlbd={œy¨u xwb{bzp wStzmnlbd%tž_”uNts)p ceˆbwStžd)x$tž_bd

(286)    ‰bd)m|ruNtzm|œd)xPmnl¨zdUœd)zxždecepV‰bdewbxžmnlbq tž_bdep ˆStzmnc=uNo   Ÿ¡xztžžuNtžd)qryœ¬$†­l2tž_bH d P %ˆbzp q žurc±¼¿tž_bd

(287) " "±xžwb{bzp wStzmnlbd mnxtž_bd%xurced%urxƒtž_bd%p mnqrmɦ l”uNo

(288)  " "=ª`m|tž_bp wSt

(289) tž_bd%

(290) f : exztžd)ˆ±onp6p ˆ¿¬

(291) " "Ás)p lœtuNmnlbxƒp lSo|y tž_bdP‰bd)m|ruNtzm|œdˆ”urtpr£

(292) " "0¬ ^`_bdÌlbd)d)‰bd)‰ zd)s)p ceˆbwStuNtzmnp l urlb‰ xžtžp žurq dÁpr£`tž_bd mnl©tžd)zced)‰Sm§uNtžd ruNonwbd)x urzd ‰bp lbd{œyÌtž_bd

(293)  P   " AP "

(294)   exžwb{bzp wStzmnlbd$ª_Smns_AmnceˆSond)ced)l©tžx

(295) tž_bd   Ÿ¡xztžžuNtžd)qryœ¬ ½Cd tž_bd)lurˆbˆSo|mnd)‰ tž_bd¨p ˆStzmnc=uNoPxztžžuNtžd)qry tžp {prtž_ onpVp ˆbx±cep6‰Sm|£@yVmnlbq tž_bd œd)sUtžp   tž_bdec=uNmnlConpVp ˆ¨mnl tž_bd

(296) " "±xwb{bzp wStzmnlbd;urlb‰ tž_bdemnlblbd)$onp6p ˆ mnl

(297) Pb¬e†­l«tž_bd

(298) AP" xwb{bzp wStzmnlbd ¼,tž_bdÁtžd)xžt p lÎtž_bd;zd)xžmn‰bw”uNoͪ¥urx%zd)ˆSo§urs)d)‰Ð{©yÐuAonpVp ˆÐpr£P‘ 2xztžd)ˆbxv¼+tžp Vlbp’ª +H tž_bdlVwbc{d)kpr£Jd9ÀSd)s)wStžd)‰2m|tžd)žuNtzmnp lbxPurxƒmnlËtž_bd s)p lbxzmn‰bd)zd)‰ cep6‰bdUopr£Jˆbzp q žurcex’¬ .   .

(299) 

(300)   "!#

(301) $%! & '()**

(302) +. ‘E:.   AP "  P         P  

(303) " #

(304) /P  

(305)  #

(306)

(307) "/

(308) AP"/P   

(309) /

(310) AP". . ' P.     "  AP   9P      . 

(311)  P 

(312) PP  "

(313) /

(314) AP"   

(315) P  

(316) /P " 

(317)  #

(318)   . . ' P. mnq wbzdŒ J‚VtžzwbsUtžwbzdpr£tž_bd. P  

(319)  P   . . AP.

(320)  " " Áxwb{bzp wStzmnlbd ¬ 

(321)   "  !

(322) " "  #

(323)  

(324) !

(325) P" #  P   "

(326) " "!

(327) "  #     # 

(328) #!A

(329) P. mnq wbzdF9 qRƒˆStzmnc=uNo&xztžžuNtžd)qry±p l tž_bdPc=uNmnlAonpVp ˆ¿¬ ^`_Smnxl©wbc{d)pr£¥xztžd)ˆbx$mnxxžw98=sUmnd)lœtPtžpA®”lb‰ tž_bd=s)p zzd)sUtxžpronwStzmnp l¨tžp tž_bd=o|mnlbdvurPxzySxztžd)c±¬ ^`_bd   Ÿ¡xztžžuNtžd)qry ªÍurxƒsvuNo|ond)‰AtžpÁs)p ceˆbwStždtž_bd

(330) P=‰bd)m| uNtzm|œd)xv¼wbxzmnlbqÁu=_”urlb‰Sªm|tztžd)l cep6‰Sm|®”svuNtzmnp l pr£

(331) tž_bd P ˆbzp q žurc±¬Î^`_bd P ¨svuNo|o¥q žurˆb_ mnxeˆbzd)xd)lœtžd)‰ mnl mnq wbzdAS¬ ^`_bd P  ˆbzp q žurc s)p ced)xe£¤zp c u«cep6‰Sm|®”svuNtzmnp l pr£

(332) tž_bd P ±ˆbzp q žurc±¬Î^`_bd

(333) " " xwb{bzp wStzmnlbdÁmnxtž_bdÁxurced urx

(334)  " "Ëd9Àbs)d)ˆSttž_”uNt m|t‰bp6d)x lbprtsvuNo|o

(335) AP"«urlb‰Ðxu)œdÁtž_bd s)p zzd)xˆp lb‰Smnlbq mnl©tžd)zced)‰Sm§uNtžd zd)xžwSo|tžx ? urxËxž_bpvªl mnl mnq wbzdÎ

(336) Œ A9¬Š^`_bd¨s)p ceˆbwStuNtzmnp l(pr£

(337) P"Cmnx;ˆd)£@p zced)‰ {œy tž_b d

(338)  P   " AP "

(339) APCxžwb{bzp wStzmnlbd ¼

(340) svuNo|ond)‰ {©y

(341)  " "  0¬ ^`_bd)l

(342) AP" =s)p lœtuNmnlbxp lSo|yÌtž_bd$‰bd)m| uNtzm|œdˆ”urtpr£

(343) P0¬ ”zp curl±mnceˆSond)ced)l©tuNtzmnp l2ˆprmnlœtƒpr£6mndUªP¼Stž_Smnx

(344) c=urlSmnˆbwSo§uNtzmnp l2pr£mnlblbd)onpVp ˆbx`mnx

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