Graph Rewrite Systems for Program Optimization

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Graph Rewrite Systems for Program Optimization

Uwe Aßmann

To cite this version:

Uwe Aßmann. Graph Rewrite Systems for Program Optimization. [Research Report] RR-2955,

IN-RIA. 1996. �inria-00073743�

ISSN 0249-6399

a p p o r t

d e r e c h e r c h e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Graph Rewrite Systems For Program Optimization

Uwe Aßmann

N˚ 2955

August 1996

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Intermediate code

Transformation

Frontend

Backend

....

Analysis 1

Analysis n

IC+information

Intermediate code’

Specification

Specification

Specification

generates

generates

generates

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n

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 œš #SšO­! ά„· # S †¹sšO¥Ÿ¬

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Ä A3i³LRJRQSJKT,LKZVOX,X,M;QSJRVaT@Bj UJKT > F V _ @3X@SŽO@ _ VOL…†WIsQ’T,JKMPIXSˆ}z PASRÇL~[(48$i _ Ma`b@BQ’T,XVOI¼@Bj]NO@3TM9JKT,XLRVa[@BL¸ˆ => @ņWIsQ’T,JKMPIX€,‚ƒ@B„…z†Jˆ‡ˆAT@,>Sz…‡`TL1[‰I i _ MzŽ;JRj]@"XMPW _ Q’@UVOIsj!T,V _ NO@ST+IM;j]@BX …†M _ @BjNO@BXSˆ ®°@"X,V€Z5T > VaTŠzYfKL—JRX  †¹s¶B ,l–B¹ž7TM^€:‚ƒ@B„…z$‹z 'üVOIj{‡ˆAT@,>Sz…‡!$‹z '¿ > JRLK@€:‚ƒ@†„…z $‹z'üVOIjŒ‡ˆAT@,>Sz…‡!$‹z 'YV _ @š1.SšP¶(–B¹ž¸ˆ f.LRL]VOja`bVOQ’@BIT I]M7j]@BXþMO…V I]M7j]@¾V _ @QSVOLRLK@BjHJKT,X¹³–S R¤£™-*·O¥½œB–Sž¸ˆ =>

@½…¸WIQ’T,JKMPI° †¹a l–ª¤P¥b–(–4TIŽ[IMO…þVIM;j]@½JRXÃT

>

@¾I;WA5[@

_

MO…þJRIQ’MPA„JRI]N

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_ LKZ«j]@’•I]@‚T > @ƒ…¸WIQ’T,JKMPI¨·O›7ž? l–ª¤P¥b–(–!TM [@vT > @oIWsA'[@ _

MO…UMPW]TNOMPJRIN¼JRIQSJRj]@BIT@Bj]NO@BXH·O›7ž?–ª¤O¥–*– $O 'Œ0 ˑe,z“f
LÂ˔€,‚ƒ@B„…z$‹z'•0’OjË㈠f¯IM;j]@ƒUJKT

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Ä …ÃI]@BQ’@BX,X,V _ ZO”~@ƒJRIj]@¼T > @ƒQ’MPA3iMPI]@BIT,XMO…Vg(~‘“N _ Vai > F~0 $IŒ—JMLt—J OQPASR — ',”@;T@BIj‚T > @…¸WIQ’T,JKMPIsXOQPA˜R+VOIjZz PA˜RgTM'X@ST,XMO…JKT@BA‚XS”VOIsj3WX@ †¹a l–ª¤P¥b–(–~VOIj9·O›7ž? l–ª¤P¥b–(– _ @BXT _ JRQ’T@Bj TM„X,i@BQSJΕQ¾@Bj]NO@LRVa[@BLRXSˆ

egMPIlT _ V _ ZTM¾MOT > @ _ Vaii _ MPVOQ > @BXJRI!T > @gLRJKT@ _ VaT,W _ @Ã+@gjM½IMOT j@BVOL"JKT > V _ [sJKT _ V _ Z5A5WsLKT,JΑ“N _ Vai > XSˆ  @BQSVOWX@ L2l $Iq I qV(48 ',”O[@STÕ~@S@BI5Tª+M¾I]M7j]@BXYX,@SŽO@ _ VOL;@Bj]NO@BX V _ @+VOLRLKMz~@Bj¿” > M€+@SŽO@ _ ”PT > @BJ _ LRVa[@BLRX A!WXT¿[@+jJRXT,JRIQ’TSˆ ‰]M _ T > @'isW _ iMPX@BXgMO…MOisT,JRA„JK‡BVaT,JKMPIŒJKT"JRX.I]MOT"I@BQ’@BX,X,V _ ZoTMƒVOLRLKM€X@SŽO@ _ VOLY@Bj]NO@BX.MO… T > @'X,VOA„@LRVa[@BL¿[@STª+@S@BI TÕ~M„IM;j]@BXx@Bj]NO@BX _ @Si _ @BX@BIlTÃi _ @BjJRQSVaT@BXÃMPIoT > @BJ _ XMPW _ Q’@VOIsjvT,V _ NO@STUI]M7j]@½MO[]`b@BQ’T,XS”sVOIjoV!i _ @BjJRQSVaT@@BJKT > @ _ > MPLRjsXUM _ j]M;@BX"I]MOT > MPLRjˆ UMz~@SŽO@ _ ”¿JKT½JRX"ŽO@ _ Z9JRA3iM _ T,VOITTM > VBŽO@jJ@ _ @BIT _ @BLRVaT,JKMPIX.[@BQSVOWsX@+@'gVOIT"TM _ @Si _ @BX@BIT5jJ@ _ @BIT 7JRIjXMO…UJRI]…†M _ A„VaT,JKMPIˆc¾I¼T > @ƒMOT > @ _ > VOIj” VOLKT > MPW]N > VaTT _ JK[sWT@BX'…†M _ I]M;j@BXA„V€Z°[@ JRIT _ M7jWQ’@Bj”¿T > JRX¾JRXLK@S…»T¾MPWT¾…»M _ T > @3X*V O@MO…+X,JRA„isLRJRQSJKTªZOˆ Ä …~j]@BX*JKNPI]@Bj¼QSV _ @S…¸WLRLKZ$†MPILKZJRIT@SNO@ _ ŽaVOLRW]@BXS”üMPILKZ A‚VaT,Q > JRI]N5JRI _ WLK@BX!',”7I]M;j@UVaTT _ JK[sW]T@BX+j]M5I]MOT~Q > VOI]NO@.T > @ _ @BX,WLKT,X MO[sT,VOJRI]@BjvJRI3T > JRX~isVai@ _ ”l[@BQSVOWX,@UT > @SZ3QSVOI [@X@S@BIVOXÃ@;T@BIsX,JKMPIHMO… T > @I]M7j]@LRVa[@BL¿X@ST (43½ˆ d7@SŽO@ _ VOLMOi@ _ VaT,JKMPIX+MPIh(~‘“N _ Vai > XgV _ @¾XT,VOIsjV _ j‚@;T@BIX*JKMPIXgMO…üX@STgMOi@ _ VaT,JKMPIXgMzŽO@ _ I]M;j@BX+VOIsj„@Bj]NO@BXBˆf (~‘“N _ Vai > F
0
F ‰ 5oF  JRXUQSVOLRLK@Bj«›7¹ ¸·O¹MO…F ‰ VOIj|F 

”JK…#F 0 $I{—
5ZI{—:JMLt—
5}Lt—,JMOQPA˜R —
%5NOQPASR — ',ˆ

d7JRA‚JRLRV _ LKZ3j]@’•³I@T > @=(~‘“N _ Vai > ‘ªJRIT@ _ X@BQ’T,JKMPI«VOIj3T > @Y(~‘“N _ Vai > ‘ªJRIQSLRWX*JKMPI¾ˆ => @ O  .–S¥–S¹³¶(–ŠF 0
F ‰ F  MO…F ‰ VOIj F  JRX F10 $Ir0 I —
I — ,JMLÂË3^JOQPASR —
Ë3 ',ˆ => @X,@STUMO… VOLRL#(~‘“N _ Vai > XUM€ŽO@ _ (¨JRX.QSVOLRLK@Bj  w ˆ => @X@STMO…VOLRL (~‘“N _ Vai > XÃM€ŽO@ _ V!• 7@BjHX@STMO… I]M7j]@BX I ” w 3 ‘0 $ w Ë3=J*5ŠJ ',”…†M _ A„XÃV5•³IsJKT@LRVaTT,JRQ’@Oˆ f-(~‘“N _ Vai > ‘ªA3M _ i >

JRX,A’> mF-[ Q’MPIX*JRXT,XMO…gTÕ~M…†WIsQ’T,JKMPIX > 3 QI — [IkVOIj7> 8 QL — [ L

 > JRQ > …¸WLΕ³LTÕ~MƒQ’MPIjJKT,JKMPIX" /†{su È/YyzÈ;wBÈ;y]{]x$†Ì"

O7fhI —  OQPA˜R $/>B3$O ''0'OQPASR — $O ',”

 z=fhL — !z:PA˜R$/>B8;$‹z ''#0
z:PASR — $‹z' $?"ü|!$†ÍþÈS"ü|PÈ|þyzȝwBÈ7y7{x$†ÌE" z\f Lt— y€:‚ƒ@†„…z 8 $/> 8 $‹z''0 > 3 $‹€,‚ƒ@B„…z 8 $‹z'',”h‡ˆAT@,>Sz…‡ 8 $/> 8 $‹z''0 >B3 $‡A!@,>Sz ‡ˆ8  $‹z'' Ä …[MOT > > 3 VOIjy> 8 V _ @vJRIa`b@BQ’T,JKŽO@O”#>«JRX'QSVOLRLK@Bj¨ †¹….B–,¶€ž¡ »¢a–’ˆ Ä …Ã[MOT > > 3 VOIjy> 8 V _ @vX,W _ `¦@BQ’T,JKŽO@O”#>¼JRX!QSVOLRLK@Bj œ’›;¥/.B–,¶Bž¡ †¢£–Sˆ Ä …#>JRX¾JRIa`b@BQ’T,JKŽO@„VOIjŒX,W _ `¦@BQ’T,JKŽO@3JKTJRX¾VOI´ œ’·O­·O¥ª§™ œŸ­5ˆ{> 3! :> 8‹V _ @!MO…»T@BI¼@;T@BIj@BjŒTMvX,@ST,X¾MO… I]M7j]@BX½VOIj@Bj]NO@BXSˆ Ä …QSLK@BV _ … _ MPA Q’MPIlT@7TS”> 3! :> 8‹V _ @VOLRXMoQSVOLRLK@Bj >sˆ®°@!MO…†T@BIŒ _ JKT@Œ>SFJRIXT@BVOjŒMO…> $‹F' TM„X*VBŽO@[ _ VOQ O@ST,XSˆ   y€{$y€ÈË‚y†$»x€È9y /†È;w Ä IvM _ j]@ _ TMj]@’•³I@.N _ Vai > _ @S _ JKT@ _ WsLK@BXÃVOIjƒTM3VOLRLKM€—X,MPA3@¾A3M _ @.@7i _ @BX,X,JKŽO@BI@BX,XņM _ T > @BJ _ LK@S…»Tb‘ > VOIjoX,JRj]@BXS” +@¾• _ X,T > VBŽO@TM‚@;T@BIsjoT > @j]@’•IJKT,JKMPIoMO… V}(~‘?>!@BA:CEDˆ vÈ! #"%$»x$†Ì" 
(#";‘“N _ Vai >%$

0 $IKJMLNJaOQPA˜R:JMOmz >SA ‡Mz&('5 œ½š^(Y¸¤P¥¦šŸ§s™ šO›P¤O­3–S¹žª– 5S¬š #’›7¹s¶Bž¡ ¸·O¹H¹³–Õ¤PšOžª– ˆ

LŽ[) ©þ™ ¸¶’™«žª– 眂©þ™7–Sž†™]–S¥‚šP¹²– €¤–•z°· #„ž†™7–„¤O¥bšŸ§™  œ„¹³–ª¤lšOžª– +*¦¹³–Õ¤PšOžª–  $‹z'=0 ž¡¥Ÿ›]–,'+

$ ™;š£œvšŒ¥b– ΚOžª–  ( ¡¤O¥bšŸ§™ $-.,/ © †ž†™ŒšY¹–ª¤PšPžª– – €¤l–’œ'œ’ž¡¥( ç§P§7– 0* $-.,/ 0 $I|JML

e,zNfgLÂËOmz >SA ‡Mz& $‹z'401‡b@†ƒEzBjBJMOQPA˜R1'+ ;ü™7–

œB–Sžg· #'š(#"¡¤O¥bšŸ§™lœ' Ñœ'¶(šK– 32 w4

+

f ¤O¥bšŸ§™º¥–S©¥Ÿ »žª–9¥(›-R–{@g0 $57698 l9:'Q’MPIX,JRXT,X!MO….VK– #Ÿž?ª™;šO¹ °œŸ ,l–;5-f<2 w 4 ”V´¥(›-R–H †¹¢zšP¥( ¸šO¹ž 8f= w *b »¹sžª–S¥ #’šP¶Ÿ–¾¤O¥bšŸ§™'5”VOIjoVo¥Ÿ K¤a™ž?ª™;šO¹ „œŸ ,l–: f= w $¸‰ JKNPW _ @ -',ˆ $U@SNPVaT@Bj9@Bj]NO@BXV _ @¾MPILKZƒVOLRLKM€+@Bj JRI‚LK@S…»Tb‘ > VOIsj‚X,JRj@BX T > @SZ!V _ @"WX@Bj3TMT@BXT+T > @UVa[sX,@BIQ’@UMO…¿X,W[N _ Vai > XBˆ Ä I3T > @NPJKŽO@BI‚•sNPW _ @BXMO…N _ Vai > _ @S _ JKT@ _ WsLK@BX +@UVOII]MOT,VaT@.VI]M7j]@Ã[Z!JKT,XLRVa[@BL¸”;VOIj3JK…I]@BQ’@BX,X,V _ Z5[;Z!VOI„JRj]@BIT,JΕ³QSVaT,JKMPIƒIWA5[@ _ ”PJRIM _ j]@ _ TM'JRj@BIlT,JK…†Z T > @¾X,VOA„@.IM;j]@BX~MPIoLK@S…†Tb‘VOIsj _ JKN > Tb‘ > VOIsjƒX*JRj]@BXSˆ 8VOQ > _ WLK@¾VOLRXM > VOXgV!IVOA3@ $ > @ _ @?>ÊŸ‘*p',ˆ ®°@¾VOLRXM3WX,@"T > @ …†MPLRLKM€UJRIN„T@ _ A‚X"

A

B

C

A

B

D

A

B

C

D

::=

RS-1

a

b

a

c

a

b

c

‰ JKNPW _ @ ¾f"[M€ŽO@% (¿@S…†Tb‘5VOIj _ JKN > Tb‘ > VOIj²X*JRj]@Oˆ
.MzUI$LK@S…†T" _ WLK@9j]@BLK@ST,JKMPI‹JKT@BA„XBˆ
.M€"I²A„JRjsjLK@% _ WLK@ JRIŽ£V _ JRVOIlT  ˆ
.M€UI _ JKN > T" _ WLK@VOjjJKT,JKMPIJKT@BA„X

rule invariant K

left hand side L

right hand side R

redex invariant gK

redex gL

transformed redex gR

g

g

g

‰ JKNPW _ @'nEh@BLRVaT,JRIN„MO… V _ WLK@VOIj9V _ @Bj]@vUJKT > T > @¾M7QSQSW _,_ @BIQ’@Y> I /  $@('j‘0'I  $ _ WLK@T@BXT"I]M;j@BX!' L  /  $@ '`‘0'L $ _ WLK@T@BXTU@Bj]NO@BX!' ( /  3 $@('‘0'OQPASR $I  ' $ _ WLK@T@BXT"I]M;j@LRVa[@BLRX ' (  /  8 $@('‘0
z:PA˜R $L ' $ _ WLK@T@BXTU@Bj]NO@LRVa[@BLRX!' (  /  $@('‘0
(  /  3 5h(  /  8 $ _ WLK@T@BXT"LRVa[@BLRX ' I  $@('j‘0'I  $ _ WLK@JRIŽ£V _ JRVOIlTUIM;j]@BX ' L  $@('‘0'L $ _ WLK@JRIŽ£V _ JRVOIlT@BjNO@BX!' (  3 $@('j‘0'OQPA˜R$I ' $ _ WLK@JRIŽ£V _ JRVOIlTUIM;j]@LRVa[@BLRX!' (  8 $@('j‘0
z PASR  $L  ' $ _ WLK@JRIŽ£V _ JRVOIlT@BjNO@'LRVa[@BLRX!' (  $@('j‘0
(  3 5h(  8 $ _ WLK@JRIŽ£V _ JRVOIlTULRVa[@BLRX!' I($@ '‘0
I I  $ _ WLK@VOjjsJKT,JKMPI9I]M7j]@BX!' L  $@('j‘0iL L $ _ WLK@VOjjsJKT,JKMPIH@Bj]NO@BX!' (  3 $@('‘0'OQPASR  $I  I  ' $ _ WLK@VOjjsJKT,JKMPI9I]M7j]@¾LRVa[@BLRX!' ( 8 $@('‘0
z:PASR $L L ' $ _ WLK@VOjjsJKT,JKMPIH@Bj]NO@LRVa[@BLRX!' (  $@('‘0
(  3 $@('5o(  8 $@(' $ _ WLK@VOjjsJKT,JKMPI9LRVa[@BLRX!' I  d $@('‘0'I I  $ _ WLK@j]@BLK@ST,JKMPII]M7j]@BX!' L  d $@ '`‘0'L  L  $ _ WLK@j]@BLK@ST,JKMPI9@Bj]NO@BX!' ( d 3 $@('j‘0'OQPASR $I  I  ' $ _ WLK@j]@BLK@ST,JKMPII]M7j]@LRVa[@BLRX!' (  d 8 $@('j‘0
z:PA˜R $L L ' $ _ WLK@j]@BLK@ST,JKMPI9@Bj]NO@LRVa[@BLRX!' ( d $@('j‘0
( d 3 $@ 'E5K( d 8 $@(' $ _ WLK@j]@BLK@ST,JKMPILRVa[@BLRX!'  )/†È{%-/?$†|P{x$†ÌE" f _ WsLK@t@=0 $5 6 8 l :'þJRX+VaiisLRJRQSVa[sLK@.TM5V{(~‘“N _ Vai > F!”T > @™7·£œŸžY¤P¥¦š’§™l”JK…T > @"…»MPLRLKMzUJRI]NQ’MPIjJKT,JKMPIX > MPLRj ! x " °{x€|†%$?"$# ! w Yu%#³yz{%ºx€È;wSx " ‰ JRIj°VOI«JRIa`¦@BQ’T,JKŽO@o(~‘“N _ Vai > ‘ªA3M _ i > JRX,A~>À5 -.,/ [F”üQSVOLRLK@Bj‹·€¶*¶B›7¥  ¥b–B¹s¶(–Sˆ#> $5 -.,/

'JRXQSVOLRLK@Bj ¥b– l–,˜P”sJKT,XI]M7j]@BX¾¥– l–*˜o¹s· l–’œ(”sJKT,X@Bj]NO@BX¥– l–*˜9– €¤–ŸœŸˆ

 @BQSVOWX@ 5 -.,/ JRXWX@Bj¿” I]@SNPVaT@Bjƒ@Bj]NO@BX+… _ MPA 5´V _ @½I]@SNPLK@BQ’T@Bj”VOIj‚+@.VOLRXM!Va[[ _

::=

A

B

A

B

::=

A:1

A:2

A:1

A:2

A

B

C

::=

A

B

D

GRS-1

GRS-2

GRS-3

a

a

b

NOT c

d

NOT c

d

e

a

f

a

g

‰ JKNPW _ @'qE
Ç5%VX,JRA„isLK@¾N _ Vai > _ @S _ JKT@XZ7XT@BA œŸ› ª¤P¥¦š’§™'MO…F!”]J¸ˆ@OˆsJK…m>B3$I 'JRXÃT > @X@STÃMO… _ @Bj]@oI]M7j]@BX+… _

MPA > 5”I]MOTVOLRLLRJRI ;JRI]N@BjNO@BXÃMO…þIM;j]@BXgJRI

>B3 $I 'ÃI]@S@BjTM‚[@!JRIh> 5Uˆ => @N _ Vai > F > 5¨JRX½QSVOLRLK@BjÁ¥b–’œŸž~¤P¥¦š’§™lˆ 8jNO@BX"… _ MPAŽF > 5JRITMZ> 5 A„V€Z‚@]JRXT $ՙ ,1l–B¹«– €¤–Ÿœ* PšO¹¤㠆¹¤o– €¤l–’œ',ˆ ! x  " "üÈ #³{]x€È;Í$È;Í%#³È9xzȝwSx $‹z}0 $O ‰ JMO  J P ')f L  JB¹³–Õ¤PšOžª– ($‹z ' 0 ž¡¥Ÿ›– ' 0 $/> 3 $O ‰ 'MJ> 3 $O  'MJ P 'Œf6LV—” J¸ˆ@oJK…UVOI6P‘“@Bj]NO@oJRI<LÁJRX!I]@SNPVaT@Bj”~[@STÕ~@S@BI´T > @vQ’M _*_

@BXiMPIjJRI]NI]M7j]@BX!JRIy> 3 $I  '¾IM7P‘“@Bj]NO@oA„V€Z

@7JRX,TSˆ ! x " yzÈ/†{]x†$†ÌE"´|PÌ “ /†ÈlxzÈS"üȝw€wx€È;wSx az}0 $O ‰ JMO  J P'Œf6L ($@('ˆ $/> 3 $O ‰ 'MJ> 3 $O  'MJ P ' f L — J¸ˆ@OˆþI]MOT'VOLRL @Bj]NO@BX > JRQ > T > @VaiisLRJRQSVaT,JKMPIMO…%@5+MPWLRjHVOjjHV _ @VOL _ @BVOj]ZoJRI|Fˆ Ä …@.JRXVaiisLRJRQSVa[³LK@O”!@Uj@ _ JKŽO@BX… _ MPA F‹T > @(~‘“N _ Vai > ¯[Z'i@ _ …»M _ A„JRI]N"T > @g…»MPLRLKMzUJRI]NXT@Si³XJRI!M _ j]@ _ $ O †¥b–*¶Bž l–B¥( †¢zšPž¡ †·O¹oF [  ' !

{ " yzȝÍþÈºÈÍ #sȍÍYÈ /†Èx$†ÌE" f.LRL _ @Bj]@3@Bj]NO@BXÃA‚VaT,Q > @Bjƒ[;Z _ WLK@½j]@BLK@ST,JKMPIv@Bj]NO@BXÃV _ @½j]@BLK@ST@BjL — ‘0'L — >B8;$L  d $@('' ! {  " %$†ÍYÍþÈ "ÁÈ;Í%#³ÈÍþÈ/†Èx$†Ì" f"LRL _ @Bj]@'@BjNO@BX V _ @j]@BLK@ST@Bj„ > JRQ > V _ @UJRIQSJRj]@BITTMI]M7j]@BXA„VaT,Q > @Bj[;Z _ WLK@ j]@BLK@ST,JKMPI9IM;j]@BXxL — ‘0iL — e $O ‰ JMO  J P'fKL — ËO ‰  O  f`> 3 $I d $@(''Mj ! { " "üÌÍYȍÍþÈ/†Èx$†ÌE" f"LRL _ @Bj@‹I]M7j]@BXoA„VaT,Q > @Bj[;Z _ WLK@°j]@BLK@ST,JKMPI I]M;j@BXoV _ @°j]@BLK@ST@Bj6I{— ‘0 I{— >B3 $I d $@ '' ! { " "üÌÍYȍ{ÍþÍ$»x†$†ÌE" fºX,@STþMO…³I]@SÁI]M;j@BXYQ’M _,_

@BXiMPIjJRI]N"TM I  $@('JRX VOjj@BjTM^FÇ I — ‘0'I — 5 e OjË\f

I $@('MJMOQPA˜R — $O '0'OQPASR $ 'Mj

! { " È;Í%#³È{sÍYÍ#$»x$†ÌE" f X,@ST„MO…I]@S @BjNO@BX‚Q’M _,_ @BXiMPIjJRI]N°TM<L  $@(''JRXƒVOjj]@BjÁTM F!”ÃJK…½T > @SZÁV _ @I]MOT VOL _

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A:R1

A:R2

A:1

A:2

A:3

A:4

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occurrence g

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VOI #;‘“@Bj]NO@ $†T*r(',ˆ Ç5Ÿ‘ÕCƒVOjjsX"VOI
‘“@Bj]NO@![@STÕ~@S@BITÕ~M ¿‘ªI]M7j]@BX$¸VPn('

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]5Ÿ‘ÕC‚JRIV iMPX,X,JK[³LK@ > MPXT½N _ Vai > ˆ =U> @ _ WLK@!A„VaT,Q > @BX¾VoI]MPI7‘ªJRIjWsQ’@BjX,W[N _ Vai > Q’MPIT,VOJRIJRI]NoT > @ _ @Bj@I]M7j]@BX¿!VOIj ¿ˆ Ä I°T > JRXisVai@ _ ~@‚WX@3VoXi@BQSJΕ³QN _ Vai > _ @S _ JKT@‚Vaii _ MPVOQ > ˆ $"MPI]@3MO…gT > @3Vaii _ MPVOQ > @BX _ @SiM _ T@Bj¼JRI°T > @ LRJKT@ _ VaT,W _ @¾X@S@BA„XTM5A„@S@STT > @.@;@BQSWT,JKMPI‚@! ƒQSJK@BIQ’Z„Q _ JKT@ _ JRV5 > JRQ > V _ @ _ @B\;WJ _ @Bj‚…†M _ i _ MON _ VOA MOisT,JRA„JK‡BVaT,JKMPIˆ => JRXYQ’MPIQ’@ _ IXþ@BA'[@BjjsJRI]NÃMO… _ WLK@BXB”£NPLRWsJRI]N.Q’MPIsjJKT,JKMPIXS”OVOIj > JRjj]@BI@Bj]NO@BXSˆ ®°@ _ @BXT _ JRQ’TþT > @ _ WLK@+VaiisLRJRQSVaT,JKMPI Q’MPIjsJKT,JKMPIXgTM'T > MPX@"T > VaTgQSVOIƒ[@UT@BX,T@BjoJRI‚Q’MPIX,T,VOIlTgT,JRA3@. > @BI‚V!X@ST~MO…YI]M;j@BXMO…¿T > @ > MPX,T~N _ Vai > JRXgWIj@ _ JRIŽO@BXT,JKNPVaT,JKMPIˆ => WX +@.WsX@' †¹¢zšP¥( ¸šO¹ž _ WLK@"@BA5[@BjsjJRI]N   ‰YE¾mOn“”]J¸ˆ@Oˆ+@.jM'I]MOTgVOLRLKM€ > JRjsj]@BI3@Bj]NO@BX+TM5[@ _ @BjsJ _ @BQ’T@Bj”7XisLRJKTM _ Q’MPVOLK@BX*Q’@Bjˆh@Bj@;@BX~VOLK+V€Z;X _ @S…†@ _ TMV½• ;@Bj3I;WA5[@ _ MO…IM;j]@BXVOIj@Bj]NO@BXSˆ Å M _ @"LRJK[@ _ VOL @BA5[@BjjJRI]NoA„@BQ > VOIJRX*A„X½+MPWLRj¼…†M _ AVOjjsJKT,JKMPIVOL _ WsLK@3VaiisLRJRQSVaT,JKMPI±Q’MPIjJKT,JKMPIXSˆvf.LKT > MPW]N > T > JRX _ @BX@BA5[sLK@BX T > @'VOLKNO@S[ _ VOJRQ5Vaii _ MPVOQ >  8 

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A:1

::=

A:1

A:2

B

A:1

A:1

::=

A:1

A:2

A:3

A:2

TGRS-1

NTGRS-1

terminator

any

non-terminator

any

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A

A

A

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termination-subgraph

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A

A

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termination-subgraph

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A

A

A

B

B

termination-subgraph

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A

A

A

B

B

termination-subgraph

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B

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B

B

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