Compilation - TAUmaon/teaching/2016-2017/compilation/compilatio… · Control-Flow Graphs • A...

CompilationLecture7

IR+OptimizationsNoamRinetzky

BasicCompilerPhases

Sourceprogram(string)

lexicalanalysis

syntaxanalysis

semanticanalysis

Codegeneration

Assembler/Linker

Tokens

Abstractsyntaxtree

Assembly

FramemanagerControlFlowGraph

IROptimization

Optimizationpoints

sourcecode

Frontend IR Code

generatortargetcode

Userprofileprogramchangealgorithm

Compilerintraprocedural IRInterprocedural IRIRoptimizations

Compilerregisterallocationinstructionselection

peepholetransformations

IROptimization

• Makingcodebetter

IROptimization

• Makingcode“better”

“Optimized”evaluation

arrayaccess

abase index

w=0 w=0

w=1w=0

_t0= cgen( a+b[5*c])Phase2:- useweightstodecideonorderoftranslation

_t0Heaviersub-tree

Heaviersub-tree

_t0 = _t1 * _t0

_t0 = _t1[_t0]

_t0 = _t1 + _t0

_t0_t1

_t1_t0 = c

_t1 = 5

_t1 = b

_t1 = a7

Butwhatabout…

a:=1+2;y:=a+b;x:=a+b+8;z:=b+a;

a:=a+1;w:=a+b;

OverviewofIRoptimization

• FormalismsandTerminology– Control-flowgraphs– Basicblocks

• Localoptimizations– Speedingupsmallpiecesofaprocedure

• Globaloptimizations– Speedingupprocedureasawhole

• Thedataflowframework– Definingandimplementingawideclassofoptimizations

ProgramAnalysis

• Inordertooptimizeaprogram,thecompilerhastobeabletoreasonaboutthepropertiesofthatprogram

• Ananalysisiscalledsound ifitneverassertsanincorrectfactaboutaprogram

• Alltheanalyseswewilldiscussinthisclassaresound– (Why?)

Soundnessint x;int y;

if (y < 5)x = 137;

elsex = 42;

Print(x);

“Atthispointintheprogram,x holdssome

integervalue”

Soundnessint x;int y;

if (y < 5)x = 137;

elsex = 42;

Print(x);

“Atthispointintheprogram,x iseither137

or42”

(Un)Soundnessint x;int y;

if (y < 5)x = 137;

elsex = 42;

Print(x);

“Atthispointintheprogram,x is137”

Soundness&Precisionint x;int y;

if (y < 5)x = 137;

elsex = 42;

Print(x);

“Atthispointintheprogram,x iseither137,

42,or271”

Semantics-preservingoptimizations

• Anoptimizationissemantics-preserving ifitdoesnotalterthesemanticsoftheoriginalprogram

• Examples:– Eliminatingunnecessarytemporaryvariables– Computingvaluesthatareknownstaticallyatcompile-time

insteadofruntime– Evaluatingconstantexpressionsoutsideofaloopinsteadof

inside• Non-examples:

– Replacingbubblesortwithquicksort (why?)– Theoptimizationswewillconsiderinthisclassareall

semantics-preserving

AformalismforIRoptimization

• Everyphaseofthecompilerusessomenewabstraction:– Scanningusesregularexpressions– ParsingusesCFGs– Semanticanalysisusesproofsystemsandsymboltables

– IRgenerationusesASTs• Inoptimization,weneedaformalismthatcapturesthestructureofaprograminawayamenabletooptimization

VisualizingIRmain:

_tmp0 = Call _ReadInteger;a = _tmp0;_tmp1 = Call _ReadInteger;b = _tmp1;

_L0:_tmp2 = 0;_tmp3 = b == _tmp2;_tmp4 = 0;_tmp5 = _tmp3 == _tmp4;IfZ _tmp5 Goto _L1;c = a;a = b;_tmp6 = c % a;b = _tmp6;Goto _L0;

_L1:Push a;Call _PrintInt;

VisualizingIRmain:

_tmp2 = 0;_tmp3 = b == _tmp2;_tmp4 = 0;_tmp5 = _tmp3 == _tmp4;IfZ _tmp5 Goto _L1;

c = a;a = b;_tmp6 = c % a;b = _tmp6;Goto _L0;

Push a;Call _PrintInt;

end 19

Basicblocks

• Abasicblock isasequenceofIRinstructionswhere– Thereisexactlyonespotwherecontrolentersthesequence,whichmustbeatthestartofthesequence

– Thereisexactlyonespotwherecontrolleavesthesequence,whichmustbeattheendofthesequence

• Informally,asequenceofinstructionsthatalwaysexecuteasagroup

Control-FlowGraphs

• Acontrol-flowgraph(CFG)isagraphofthebasicblocksinafunction

• ThetermCFGisoverloaded– fromhereonout,we'llmean“control-flowgraph”andnot“contextfreegrammar”

• Eachedgefromonebasicblocktoanotherindicatesthatcontrolcanflowfromtheendofthefirstblocktothestartofthesecondblock

• Thereisadedicatednodeforthestartandendofafunction

Typesofoptimizations

• Anoptimizationislocal ifitworksonjustasinglebasicblock

• Anoptimizationisglobal ifitworksonanentirecontrol-flowgraph

• Anoptimizationisinterprocedural ifitworksacrossthecontrol-flowgraphsofmultiplefunctions– Wewon'ttalkaboutthisinthiscourse

Basicblocksexerciseint main()

int x;int y;int z;

y = 137;if (x == 0)

z = y;else

x = y;

START:_t0 = 137;y = _t0;IfZ x Goto _L0;t1 = y;z = _t1;Goto END:

_L0:_t2 = y;x = _t2;

Dividethecodeintobasicblocks23

Control-flowgraphexerciseSTART:

_t0 = 137;y = _t0;IfZ x Goto _L0;t1 = y;z = _t1;Goto END:

_L0:_t2 = y;x = _t2;

Drawthecontrol-flowgraph

int main() int x;int y;int z;

y = 137;if (x == 0)

z = y;else

x = y;

Control-flowgraphexercise

_t0 = 137;y = _t0;IfZ x Goto _L0;

_t1 = y;z = _t1;

_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

_t0 = 137;y = _t0;IfZ x Goto _L0;

_t1 = y;z = _t1;

_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

_t0 = 137;y = _t0;IfZ x Goto _L0;

_t1 = y;z = _t1;

_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

y = 137;IfZ x Goto _L0;

_t1 = y;z = _t1;

_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

_t1 = y;z = _t1;

_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

z = y;_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

z = y;_t2 = y;x = _t2;

y = 137;if (x == 0)

z = y;else

x = y;

Localoptimizations

z = y; x = y;

y = 137;if (x == 0)

z = y;else

x = y;

Globaloptimizations

z = y; x = y;

y = 137;if (x == 0)

z = y;else

x = y;

Globaloptimizations

z = y; x = y;

y = 137;if (x == 0)

z = y;else

x = y;

Globaloptimizations

z = 137; x = 137;

y = 137;if (x == 0)

z = y;else

x = y;

LocalOptimizations

Optimizationpath

IR Control-FlowGraph

CFGbuilder

ProgramAnalysis

AnnotatedCFG

OptimizingTransformation

TargetCode

CodeGeneration

(+optimizations)

donewithIR

optimizations

IRoptimizations

ExampleObject x;int a;int b;int c;

x = new Object;a = 4;c = a + b;x.fn(a + b);

_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = _tmp2;x = _tmp1;_tmp3 = 4;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = a + b;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

Forbrevity:Simplified IRforprocedurereturns

ClassObjectmethodfn(int);

Explainingtheprogram

Size ofObject

ObjectClass

Forsimplicity,ignorePoppingreturnvalue,

parametersetc.

PointstoObjectC

Startoffn

CommonSubexpressionElimination

• Ifwehavetwovariableassignmentsv1=aopb…v2=aopb

• andthevaluesofv1,a,andbhavenotchangedbetweentheassignments,rewritethecodeasv1=aopb…v2=v1

• Eliminatesuselessrecalculation• Pavesthewayforlateroptimizations

CommonSubexpressionElimination

• Ifwehavetwovariableassignmentsv1=aopb[or:v1=a]…v2=aopb[or:v2=a]

• andthevaluesofv1,a,andbhavenotchangedbetweentheassignments,rewritethecodeasv1=aopb[or:v1=a]…v2=v1

• Eliminatesuselessrecalculation• Pavesthewayforlateroptimizations

Commonsubexpressionelimination_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = _tmp2;x = _tmp1;_tmp3 = 4;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = a + b;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

Object x;int a;int b;int c;

Commonsubexpressionelimination_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = _tmp2;x = _tmp1;_tmp3 = 4;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = _tmp4;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

Commonsubexpressionelimination_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = _tmp2;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = _tmp4;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

Commonsubexpressionelimination_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = _tmp2;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = c;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

CopyPropagation

• Ifwehaveavariableassignmentv1=v2thenaslongasv1andv2arenotreassigned,wecanrewriteexpressionsoftheforma=…v1…asa=…v2…providedthatsucharewriteislegal

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = _tmp2;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = c;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = c;_tmp6 = *(x);_tmp7 = *(_tmp6);Push _tmp5;Push x;Call _tmp7;

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = a + b;c = _tmp4;_tmp5 = c;_tmp6 = *(_tmp1);_tmp7 = *(_tmp6);Push _tmp5;Push _tmp1;Call _tmp7;

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = _tmp3 + b;c = _tmp4;_tmp5 = c;_tmp6 = *(_tmp1);_tmp7 = *(_tmp6);Push _tmp5;Push _tmp1;Call _tmp7;

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = _tmp3 + b;c = _tmp4;_tmp5 = c;_tmp6 = *(_tmp1);_tmp7 = *(_tmp6);Push c;Push _tmp1;Call _tmp7;

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = _tmp3 + b;c = _tmp4;_tmp5 = c;_tmp6 = ObjectC;_tmp7 = *(_tmp6);Push c;Push _tmp1;Call _tmp7;

IsthistransformationOK?Whatdoweneedtoknow?

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = _tmp3 + b;c = _tmp4;_tmp5 = c;_tmp6 = ObjectC;_tmp7 = *(_tmp6);Push c;Push _tmp1;Call _tmp7;

CopyPropagation_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp3;_tmp4 = _tmp3 + b;c = _tmp4;_tmp5 = c;_tmp6 = ObjectC;_tmp7 = *(ObjectC);Push c;Push _tmp1;Call _tmp7;

DeadCodeElimination

• Anassignmenttoavariableviscalleddeadifthevalueofthatassignmentisneverreadanywhere

• DeadcodeeliminationremovesdeadassignmentsfromIR

• Determiningwhetheranassignmentisdeaddependsonwhatvariableisbeingassignedtoandwhenit'sbeingassigned

DeadCodeElimination

_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp0;_tmp4 = _tmp0 + b;c = _tmp4;_tmp5 = c;_tmp6 = ObjectC;_tmp7 = *(ObjectC);Push c;Push _tmp1;Call _tmp7;

DeadCodeElimination_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp0;_tmp4 = _tmp0 + b;c = _tmp4;_tmp5 = c;_tmp6 = ObjectC;_tmp7 = *(ObjectC);Push c;Push _tmp1;Call _tmp7;

DeadCodeEliminationObject x;int a;int b;int c;

_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;_tmp2 = ObjectC;*(_tmp1) = ObjectC;x = _tmp1;_tmp3 = _tmp0;a = _tmp0;_tmp4 = _tmp0 + b;c = _tmp4;_tmp5 = c;_tmp6 = ObjectC;_tmp7 = *(ObjectC);Push c;Push _tmp1;Call _tmp7;

valuesneverread

DeadCodeEliminationObject x;int a;int b;int c;

_tmp0 = 4;Push _tmp0;_tmp1 = Call _Alloc;

*(_tmp1) = ObjectC;

_tmp4 = _tmp0 + b;c = _tmp4;

_tmp7 = *(ObjectC);Push c;Push _tmp1;Call _tmp7;

Applyinglocaloptimizations

• Thedifferentoptimizationswe'veseensofaralltakecareofjustasmallpieceoftheoptimization

• Commonsubexpressioneliminationeliminatesunnecessarystatements

• Copypropagationhelpsidentifydeadcode• Deadcodeeliminationremovesstatementsthatarenolongerneeded

• Togetmaximumeffect,wemayhavetoapplytheseoptimizationsnumeroustimes

Applyinglocaloptimizationsexample

b = a * a;c = a * a;d = b + c;e = b + b;

Whichoptimizationshouldweapplyhere?

b = a * a;c = b;d = b + c;e = b + b;

Commonsub-expressionelimination

b = a * a;c = b;d = b + c;e = b + b;

b = a * a;c = b;d = b + b;e = b + b;

Copypropagation

b = a * a;c = b;d = b + b;e = b + b;

b = a * a;c = b;d = b + b;e = d;

Commonsub-expressionelimination(again)

Othertypesoflocaloptimizations

• ArithmeticSimplification– Replace“hard”operationswitheasierones– e.g.rewritex = 4 * a; asx = a << 2;

• ConstantFolding– Evaluateexpressionsatcompile-timeiftheyhaveaconstantvalue.

– e.g.rewritex = 4 * 5; asx = 20;

Optimizationsandanalyses

• Mostoptimizationsareonlypossiblegivensomeanalysisoftheprogram'sbehavior

• Inordertoimplementanoptimization,wewilltalkaboutthecorrespondingprogramanalyses

Availableexpressions

• Bothcommonsubexpressioneliminationandcopypropagationdependonananalysisoftheavailableexpressionsinaprogram

• Anexpressioniscalledavailable ifsomevariableintheprogramholdsthevalueofthatexpression

• Incommonsubexpressionelimination,wereplaceanavailableexpressionbythevariableholdingitsvalue

• Incopypropagation,wereplacetheuseofavariablebytheavailableexpressionitholds

Findingavailableexpressions

• Initially,noexpressionsareavailable• Wheneverweexecuteastatementa=bop c:– Anyexpressionholdinga isinvalidated– Theexpressiona=bop cbecomesavailable

• Idea:Iterateacrossthebasicblock,beginningwiththeemptysetofexpressionsandupdatingavailableexpressionsateachvariable

Availableexpressionsexample

a = b + 2;

b = x;

d = a + b;

e = a + b;

d = x;

f = a + b; b = x, d = x, e = a + b

b = x, d = a + b, e = a + b

b = x, d = a + b

a = b + 2

b = x, d = x, e = a + b, f = a + b

a = b + 2;

b = x;

d = a + b;

e = d;

d = b;

f = e; b = x, d = x, e = a + b

b = x, d = a + b, e = a + b

b = x, d = a + b

a = b + 2

b = x, d = x, e = a + b, f = a + b

a = b + 2;

b = x;

d = a + b;

e = a + b;

d = x;

f = a + b; b = x, d = x, e = a + b

b = x, d = a + b, e = a + b

b = x, d = a + b

a = b + 2

b = x, d = x, e = a + b, f = a + b

a = b + 2;

b = 1;

d = a + b;

e = a + b;

d = b;

f = a + b; b = 1, d = b, e = a + b

b = 1, d = a + b, e = a + b

b = 1, d = a + b

a = b + 2

a = b, c = b, d = b, e = a + b, f = a + b

a = b + 2;

b = 1;

d = a + b;

e = a + b;

d = b;

f = a + b; b = 1, d = b, e = a + b

b = 1, d = a + b, e = a + b

b = 1, d = a + b

a = b + 2

a = b, c = b, d = b, e = a + b, f = a + b

a = b;

c = b;

d = a + b;

e = a + b;

d = b;

f = a + b; a = b, c = b, d = b, e = a + b

a = b, c = b, d = a + b, e = a + b

a = b, c = b, d = a + b

a = b, c = b

a = b, c = b, d = b, e = a + b, f = a + b

a = b;

c = b;

d = a + b;

e = a + b;

d = b;

f = a + b; a = b, c = b, d = b, e = a + b

a = b, c = b, d = a + b, e = a + b

a = b, c = b, d = a + b

a = b, c = b

a = b, c = b, d = b, e = a + b, f = a + b

a = b;

b = 1;

d = a + b;

e = d;

d = a;

f = e; a = b, c = b, d = b, e = a + b

a = b, c = b, d = a + b, e = a + b

a = b, c = b, d = a + b

a = b, b = b

a = b, c = b, d = b, e = a + b, f = a + b

Livevariables

• Theanalysiscorrespondingtodeadcodeeliminationiscalledlivenessanalysis

• Avariableislive atapointinaprogramiflaterintheprogramitsvaluewillbereadbeforeitiswrittentoagain

• Deadcodeeliminationworksbycomputinglivenessforeachvariable,theneliminatingassignmentstodeadvariables

Computinglivevariables• Toknowifavariablewillbeusedatsomepoint,weiterateacrossthestatementsinabasicblockinreverseorder

• Initially,somesmallsetofvaluesareknowntobelive(whichonesdependsontheparticularprogram)

• Whenweseethestatementa=bopc:– Justbeforethestatement,aisnotalive,sinceitsvalueisabouttobeoverwritten

– Justbeforethestatement,bothbandcarealive,sincewe'reabouttoreadtheirvalues

– (whatifwehavea=a+b?) 93

Livenessanalysisa = b;

c = a;

d = a + b;

e = d;

d = a;

f = e; b, d, e

a, b, e

a, b, d

b, d - given

Whichstatementsaredead?

DeadCodeEliminationa = b;

c = a;

d = a + b;

e = d;

d = a;

f = e; b, d, e

a, b, e

a, b, d

DeadCodeEliminationa = b;

d = a + b;

e = d;

d = a; b, d, e

a, b, e

a, b, d

b, d 96

LivenessanalysisIIa = b;

d = a + b;

e = d;

d = a; b, d

a, b, d

LivenessanalysisIIa = b;

d = a + b;

e = d;

d = a; b, d

a, b, d

Which statements are dead?

Deadcodeeliminationa = b;

d = a + b;

e = d;

d = a; b, d

a, b, d

Which statements are dead?

d = a + b;

d = a; b, d

a, b, d

LivenessanalysisIIIa = b;

d = a + b;

d = a; b, d

d = a + b;

d = a; b, d

d = a;

Ifwefurtherapplycopypropagationthisstatementcanbeeliminatedtoo

Acombinedalgorithm

• Startwithinitiallivevariablesatendofblock

• Traversestatementsfromendtobeginning• Foreachstatement

– Ifassignstodeadvariables– eliminateit– Otherwise,computelivevariablesbeforestatementandcontinueinreverse

Acombinedalgorithma = b;

c = a;

d = a + b;

e = d;

d = a;

f = e;

c = a;

d = a + b;

e = d;

d = a;

f = e; b, d

c = a;

d = a + b;

e = d;

d = a;

f = e; b, d

c = a;

d = a + b;

e = d;

d = a;

b, d 109

c = a;

d = a + b;

e = d;

d = a;

Acombinedalgorithm

a = b;

c = a;

d = a + b;

e = d;

d = a;

Acombinedalgorithm

a = b;

c = a;

d = a + b;

d = a;

c = a;

d = a + b;

d = a;

c = a;

d = a;

c = a;

d = a;

High-levelgoals

• Generalizeanalysismechanism– Reusecommoningredientsformanyanalyses– Reuseproofsofcorrectness

• GeneralizefrombasicblockstoentireCFGs– Gofromlocaloptimizationstoglobaloptimizations

ProgramAnalysis

• Reasonsaboutthebehavior ofaprogram• Ananalysisissound ifitonlyassertsancorrectfactsaboutaprogram

• Ananalysisisprecise ifitassertsallcorrectfacts(ofinterests)

• Soundanalysisallowsforsemantic-preservingoptimizations– “Moreprecise”analysesare“moreuseful”:mayenablemoreoptimizations 120

Examples

• Availableexpressions,allows:ØCommonsub-expressionseliminationØCopypropagation

• Constantpropagation,allows:ØConstantfolding

• Liveness analysisØDead-codeeliminationØRegisterallocation

Localvs.globaloptimizations

• Anoptimizationislocal ifitworksonjustasinglebasicblock

• Anoptimizationisglobal ifitworksonanentirecontrol-flowgraphofaprocedure

• Anoptimizationisinterprocedural ifitworksacrossthecontrol-flowgraphsofmultipleprocedure– Wewon'ttalkaboutthisinthiscourse

Formalizinglocalanalyses

a = b + c

OutputValueVout

InputValueVin

Vout = fa=b+c(Vin)

TransferFunction

AvailableExpressions

a = b + c

OutputValueVout

InputValueVin

Vout =(Vin \ e|econtainsa)4 a=b+c

Expressionsoftheformsa=…andx=…a…

LiveVariables

a = b + c

OutputValueVout

InputValueVin

Vin = (Vout \ a) 4 b,c

LiveVariables

a = b + c

OutputValueVout

InputValueVin

Vin = (Vout \ a) 4 b,c

Informationforalocalanalysis

• Whatdirectionarewegoing?– Sometimesforward(availableexpressions)– Sometimesbackward(livenessanalysis)

• Howdoweupdateinformationafterprocessingastatement?– Whatarethenewsemantics?– Whatinformationdoweknowinitially?

Formalizinglocalanalyses

• Defineananalysisofabasicblockasaquadruple(D,V,F,I)where– D isadirection(forwardsorbackwards)– V isasetofvaluestheprogramcanhaveatanypoint

– F isafamilyoftransferfunctionsdefiningthemeaningofanyexpressionasafunctionf:Vt V

– I istheinitialinformationatthetop(orbottom)ofabasicblock

AvailableExpressions

• Direction: Forward• Values: Setsofexpressionsassignedtovariables• Transferfunctions: GivenasetofvariableassignmentsVandstatementa=b+c:– RemovefromVanyexpressioncontainingaasasubexpression

– AddtoVtheexpressiona=b+c– Formally:Vout =(Vin \ e|econtainsa)4 a=b+c

• Initialvalue: Emptysetofexpressions

LivenessAnalysis

• Direction: Backward• Values: Setsofvariables• Transferfunctions: GivenasetofvariableassignmentsV

andstatementa=b+c:• RemoveafromV(anypreviousvalueofaisnowdead.)• AddbandctoV(anypreviousvalueofborcisnowlive.)• Formally:Vin =(Vout \ a)4 b,c• Initialvalue: Dependsonsemanticsoflanguage

– E.g.,functionargumentsandreturnvalues(pushes)– Resultoflocalanalysisofotherblocksaspartofaglobalanalysis 130

Runninglocalanalyses

• Givenananalysis(D,V,F,I)forabasicblock• AssumethatD is“forward;”analogousforthereversecase

• Initially,setOUT[entry]toI• Foreachstatements,inorder:

– SetIN[s]toOUT[prev],whereprev isthepreviousstatement

– SetOUT[s]tofs(IN[s]),wherefs isthetransferfunctionforstatements

Kill/Gen

GlobalOptimizations

High-levelgoals

• Generalizeanalysismechanism– Reusecommoningredientsformanyanalyses– Reuseproofsofcorrectness

• GeneralizefrombasicblockstoentireCFGs– Gofromlocaloptimizationstoglobaloptimizations

Globalanalysis

• Aglobalanalysisisananalysisthatworksonacontrol-flowgraphasawhole

• Substantiallymorepowerfulthanalocalanalysis– (Why?)

• Substantiallymorecomplicatedthanalocalanalysis– (Why?)

Localvs.globalanalysis• Manyoftheoptimizationsfromlocalanalysiscanstill

beappliedglobally– Commonsub-expressionelimination– Copypropagation– Deadcodeelimination

• Certainoptimizationsarepossibleinglobalanalysisthataren'tpossiblelocally:– e.g.codemotion:Movingcodefromonebasicblockinto

anothertoavoidcomputingvaluesunnecessarily• Exampleglobaloptimizations:

– Globalconstantpropagation– Partialredundancyelimination

Loopinvariantcodemotionexample

while (t < 120) z = z + x - y;

w = x – y;while (t < 120) z = z + w;

valueofexpressionx– yisnotchangedbyloopbody

Whyglobalanalysisishard

• Needtobeabletohandlemultiplepredecessors/successorsforabasicblock

• Needtobeabletohandlemultiplepathsthroughthecontrol-flowgraph,andmayneedtoiteratemultipletimestocomputethefinalvalue(buttheanalysisstillneedstoterminate!)

• Needtobeabletoassigneachbasicblockareasonabledefaultvalueforbeforewe'veanalyzedit

Globaldeadcodeelimination

• Localdeadcodeeliminationneededtoknowwhatvariableswereliveonexitfromabasicblock

• Thisinformationcanonlybecomputedaspartofaglobalanalysis

• HowdowemodifyourlivenessanalysistohandleaCFG?

CFGswithoutloops

140Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

b = c + d;e = c + d;Entry

CFGswithoutloops

141Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

a, b, c, d

a, b, c, d a, b, c, d

a, b, c, db, c, d

a, b, c, d

a, c, d

Whichvariablesmaybeliveonsomeexecutionpath?

CFGswithoutloops

142Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

a, b, c, d

a, b, c, d a, b, c, d

a, b, c, db, c, d

a, b, c, d

a, c, d

CFGswithoutloops

143Exit

x = a + b;y = c + d;

a = b + c;

b = c + d;Entry

CFGswithoutloops

144Exit

x = a + b;y = c + d;

a = b + c;

b = c + d;Entry

Majorchanges– part1

• Inalocalanalysis,eachstatementhasexactlyonepredecessor

• Inaglobalanalysis,eachstatementmayhavemultiplepredecessors

• Aglobalanalysismusthavesomemeansofcombininginformationfromallpredecessorsofabasicblock

CFGswithoutloops

146Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

a, b, c, d

a, b, c, d a, b, c, d

a, b, c, db, c, d

b, c, d

c, d Needtocombinecurrently-computedvaluewithnewvalue

Needtocombinecurrently-computedvaluewithnewvalue

CFGswithoutloops

147Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

a, b, c, d

a, b, c, d a, b, c, d

a, b, c, db, c, d

a, b, c, d

CFGswithoutloops

148Exit

x = a + b;y = c + d;

y = a + b;x = c + d;a = b + c;

a, b, c, d

a, b, c, d a, b, c, d

a, b, c, db, c, d

a, b, c, d

a, c, d

• Inalocalanalysis,thereisonlyonepossiblepaththroughabasicblock

• Inaglobalanalysis,theremaybemanypathsthroughaCFG

• Mayneedtorecomputevaluesmultipletimesasmoreinformationbecomesavailable

• Needtobecarefulwhendoingthisnottoloopinfinitely!– (Moreonthatlater)

• Canorderofcomputationaffectresult?149

CFGswithloops• Uptothispoint,we'veconsideredloop-freeCFGs,whichhaveonlyfinitelymanypossiblepaths

• Whenweaddloopsintothepicture,thisisnolongertrue

• NotallpossibleloopsinaCFGcanberealizedintheactualprogram

IfZ x goto Top

x = 1;

x = 0;

x = 2;

CFGswithloops• Uptothispoint,we'veconsideredloop-freeCFGs,whichhaveonlyfinitelymanypossiblepaths

• Whenweaddloopsintothepicture,thisisnolongertrue

• NotallpossibleloopsinaCFGcanberealizedintheactualprogram

• Soundapproximation:AssumethateverypossiblepaththroughtheCFGcorrespondstoavalidexecution– Includesallrealizablepaths,butsomeadditionalpathsaswell

– Maymakeouranalysislessprecise(butstillsound)– Makestheanalysisfeasible;we'llseehowlater

CFGswithloops

152Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

b = c + d;c = c + d;IfZ ...

• Inalocalanalysis,thereisalwaysawelldefined“first”statementtobeginprocessing

• Inaglobalanalysiswithloops,everybasicblockmightdependoneveryotherbasicblock

• Tofixthis,weneedtoassigninitialvaluestoalloftheblocksintheCFG

CFGswithloops- initialization

154Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

b = c + d;c = c + d;Entry

CFGswithloops- iteration

155Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

156Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, b, c

157Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, b, c

158Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, b, c

159Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, b, c

160Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, b, c

161Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

162Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

163Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

a, c, d

a, b, c

164Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

a, c, d

a, b, c

165Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

a, c, d

a, b, c

166Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

a, c, d

a, b, c

167Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, b, c

a, c, d

a, b, c

168Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, c, d

a, b, c

a, c, d

a, b, c

169Exit

a = a + b;d = b + c;

c = a + b;a = b + c;d = a + c;

a, bb, c

a, c, d

a, b, c

a, c, d

a, b, c

Summaryofdifferences

• Needtobeabletohandlemultiplepredecessors/successorsforabasicblock

• Needtobeabletohandlemultiplepathsthroughthecontrol-flowgraph,andmayneedtoiteratemultipletimestocomputethefinalvalue– Buttheanalysisstillneedstoterminate!

• Needtobeabletoassigneachbasicblockareasonabledefaultvalueforbeforewe'veanalyzedit

Globallivenessanalysis• Initially,setIN[s]=foreachstatements• SetIN[exit]tothesetofvariablesknowntobeliveonexit(language-specificknowledge)

• Repeatuntilnochangesoccur:– Foreachstatements oftheforma=b+c,inanyorderyou'dlike:• SetOUT[s]tosetunionofIN[p]foreachsuccessorp ofs• SetIN[s]to(OUT[s]– a)4 b,c.

• Yetanotherfixed-pointiteration!

Globallivenessanalysis

IN[s2] IN[s3]

OUT[s]=IN[s2]4 IN[s3]

IN[s]=(UT[s] – a)4 b,c

Whydoesthiswork?• Toshowcorrectness,weneedtoshowthat

– Thealgorithmeventuallyterminates,and– Whenitterminates,ithasasoundanswer

• Terminationargument:– Onceavariableisdiscoveredtobeliveduringsomepointofthe

analysis,italwaysstayslive– Onlyfinitelymanyvariablesandfinitelymanyplaceswherea

variablecanbecomelive• Soundnessargument(sketch):

– Eachindividualrule,appliedtosomeset,correctlyupdateslivenessinthatset

– Whencomputingtheunionofthesetoflivevariables,avariableisonlyliveifitwasliveonsomepathleavingthestatement

AbstractInterpretation

• Theoreticalfoundationsofprogramanalysis

• Cousot andCousot 1977

• Abstractmeaningofprograms– Executedatcompiletime

Anotherviewoflocaloptimization

• Inlocaloptimization,wewanttoreasonaboutsomepropertyoftheruntimebehavioroftheprogram

• Couldweruntheprogramandjustwatchwhathappens?

• Idea:Redefinethesemanticsofourprogramminglanguagetogiveusinformationaboutouranalysis

Propertiesoflocalanalysis

• Theonlywaytofindoutwhataprogramwillactuallydoistorunit

• Problems:– Theprogrammightnotterminate– Theprogrammighthavesomebehaviorwedidn'tseewhenweranitonaparticularinput

• However,thisisnotaprobleminsideabasicblock– Basicblockscontainnoloops– Thereisonlyonepaththroughthebasicblock

Assigningnewsemantics

• Example:AvailableExpressions• Redefinethestatementa=b+ctomean“anowholdsthevalueofb+c,andanyvariableholdingthevalueaisnowinvalid”

• Runtheprogramassumingthesenewsemantics

• Treattheoptimizerasaninterpreterforthesenewsemantics

Theorytotherescue

• Buildingupallofthemachinerytodesignthisanalysiswastricky

• Thekeyideas,however,aremostlyindependentoftheanalysis:– Weneedtobeabletocomputefunctionsdescribingthebehaviorofeachstatement

– Weneedtobeabletomergeseveralsubcomputationstogether

– Weneedaninitialvalueforallofthebasicblocks• Thereisabeautifulformalismthatcapturesmanyoftheseproperties

Joinsemilattices• Ajoinsemilatticeisaorderingdefinedonasetof

elements• Anytwoelementshavesomejointhatisthesmallest

elementlargerthanbothelements• Thereisauniquebottomelement,whichissmaller

thanallotherelements• Intuitively:

– Thejoinoftwoelementsrepresentscombininginformationfromtwoelementsbyanoverapproximation

• Thebottomelementrepresents“noinformationyet”or“theleastconservativepossibleanswer”

Joinsemilatticeforliveness

a, b a, c b, c

a, b, c

Bottomelement

Whatisthejoinofbandc?

a, b a, c b, c

a, b, c

Whatisthejoinofbandc?

a, b a, c b, c

a, b, c

Whatisthejoinofbanda,c?

a, b a, c b, c

a, b, c

Whatisthejoinofbanda,c?

a, b a, c b, c

a, b, c

Whatisthejoinofaanda,b?

a, b a, c b, c

a, b, c

Whatisthejoinofaanda,b?

a, b a, c b, c

a, b, c

Formaldefinitions

• Ajoinsemilatticeisapair(V,7),where• Visadomainofelements• 7 isajoinoperatorthatis

– commutative:x7 y=y7 x– associative:(x7 y)7 z=x7 (y7 z)– idempotent:x7 x=x

• Ifx7 y=z,wesaythatzisthejoinor(leastupperbound)ofxandy

• Everyjoinsemilatticehasabottomelementdenotedz suchthatz 7 x=xforallx

Joinsemilatticesandordering

a, b a, c b, c

a, b, cGreater

Joinsemilatticesandordering

a, b a, c b, c

a, b, cLeastprecise

Mostprecise

Joinsemilatticesandorderings

• Everyjoinsemilattice(V,7)inducesanorderingrelationshipb overitselements

• Definexb yiffx7 y=y• Needtoprove

– Reflexivity:xb x– Antisymmetry:Ifxb yandyb x,thenx=y– Transitivity:Ifxb yandyb z,thenxb z

Anexamplejoinsemilattice

• Thesetofnaturalnumbersandthemax function• Idempotent

– maxa,a=a• Commutative

– maxa,b=maxb,a• Associative

– maxa,maxb,c=maxmaxa,b,c• Bottomelementis0:

– max0,a=a• Whatistheorderingovertheseelements?

Ajoinsemilatticeforliveness

• Setsoflivevariablesandthesetunionoperation• Idempotent:

– x4 x=x• Commutative:

– x4 y=y4 x• Associative:

– (x4 y)4 z=x4 (y4 z)• Bottomelement:

– Theemptyset:Ø4 x=x• Whatistheorderingovertheseelements?

Semilatticesandprogramanalysis

• Semilatticesnaturallysolvemanyoftheproblemsweencounteringlobalanalysis

• Howdowecombineinformationfrommultiplebasicblocks?

• Whatvaluedowegivetobasicblockswehaven'tseenyet?

• Howdoweknowthatthealgorithmalwaysterminates?

• Howdowecombineinformationfrommultiplebasicblocks?– Takethejoinofallinformationfromthoseblocks

• Whatvaluedowegivetobasicblockswehaven'tseenyet?– Usethebottomelement

• Howdoweknowthatthealgorithmalwaysterminates?– Actually,westilldon't!Moreonthatlater

• Howdowecombineinformationfrommultiplebasicblocks?– Takethejoinofallinformationfromthoseblocks

• Whatvaluedowegivetobasicblockswehaven'tseenyet?– Usethebottomelement

• Howdoweknowthatthealgorithmalwaysterminates?– Actually,westilldon't!Moreonthatlater

Ageneralframework

• Aglobalanalysisisatuple(D,V,7,F,I),where– D isadirection(forwardorbackward)

• Theordertovisitstatementswithinabasicblock,nottheorderinwhichtovisitthebasicblocks

– V isasetofvalues– 7 isajoinoperatoroverthosevalues– F isasetoftransferfunctionsf:Vt V– I isaninitialvalue

• Theonlydifferencefromlocalanalysisistheintroductionofthejoinoperator

Runningglobalanalyses

• Assumethat(D,V,7,F,I)isaforwardanalysis• SetOUT[s]=z forallstatementss• SetOUT[entry]=I• Repeatuntilnovalueschange:

– Foreachstatements withpredecessorsp1,p2,…,pn:• SetIN[s]=OUT[p1]7 OUT[p2]7 …7 OUT[pn]• SetOUT[s]=fs (IN[s])

• Theorderofthisiterationdoesnotmatter– Thisissometimescalledchaoticiteration

Forcomparison• SetOUT[s]=z forall

statementss• SetOUT[entry]=I

• Repeatuntilnovalueschange:– Foreachstatements

withpredecessorsp1,p2,…,pn:• SetIN[s]=OUT[p1]7OUT[p2]7 …7 OUT[pn]

• SetOUT[s]=fs (IN[s])

• SetIN[s]= forallstatementss

• SetOUT[exit]=thesetofvariablesknowntobeliveonexit

• Repeatuntilnovalueschange:– Foreachstatements ofthe

forma=b+c:• SetOUT[s]=setunionofIN[x]foreachsuccessorx ofs

• SetIN[s]=(OUT[s]-a) 4 b,c

Thedataflowframework

• Thisformofanalysisiscalledthedataflowframework

• Canbeusedtoeasilyproveananalysisissound

• Withcertainrestrictions,canbeusedtoprovethatananalysiseventuallyterminates– Again,moreonthatlater

Globalconstantpropagation

• Constantpropagationisanoptimizationthatreplaceseachvariablethatisknowntobeaconstantvaluewiththatconstant

• Anelegantexampleofthedataflowframework

exit x = 4;

z = x;

w = x;

y = x; z = y;

x = 6;entry

exit x = 4;

z = x;

w = x;

y = x; z = y;

x = 6;entry

exit x = 4;

z = x;

w = 6;

y = 6; z = y;

x = 6;entry

Constantpropagationanalysis

• Inordertodoaconstantpropagation,weneedtotrackwhatvaluesmightbeassignedtoavariableateachprogrampoint

• Everyvariablewilleither– Neverhaveavalueassignedtoit,– Haveasingleconstantvalueassignedtoit,– Havetwoormoreconstantvaluesassignedtoit,or– Haveaknownnon-constantvalue.– OuranalysiswillpropagatethisinformationthroughoutaCFGtoidentifylocationswhereavalueisconstant

Propertiesofconstantpropagation

• Fornow,considerjustsomesinglevariablex• Ateachpointintheprogram,weknowoneofthree

thingsaboutthevalueofx:– x isdefinitelynotaconstant,sinceit'sbeenassignedtwo

valuesorassignedavaluethatweknowisn'taconstant– x isdefinitelyaconstantandhasvaluek– Wehaveneverseenavalueforx

• Notethatthefirstandlastofthesearenot thesame!– Thefirstonemeansthattheremaybeawayforx tohave

multiplevalues– Thelastonemeansthatx neverhadavalueatall

Definingajoinoperator• ThejoinofanytwodifferentconstantsisNot-a-Constant

– (Ifthevariablemighthavetwodifferentvaluesonentrytoastatement,itcannotbeaconstant)

• ThejoinofNotaConstantandanyothervalueisNot-a-Constant– (Ifonsomepaththevalueisknownnottobeaconstant,thenon

entrytoastatementitsvaluecan'tpossiblybeaconstant)• ThejoinofUndefined andanyothervalueisthatothervalue

– (Ifx hasnovalueonsomepathanddoeshaveavalueonsomeotherpath,wecanjustpretenditalwayshadtheassignedvalue)

Asemilatticeforconstantpropagation• Onepossiblesemilatticeforthisanalysisisshownhere(foreachvariable):

Undefined

0-1-2 1 2 ......

Not-a-constant

The lattice is infinitely wide

Asemilatticeforconstantpropagation• Onepossiblesemilatticeforthisanalysisisshownhere(foreachvariable):

Undefined

0-1-2 1 2 ......

Not-a-constant

• Note:• ThejoinofanytwodifferentconstantsisNot-a-Constant• ThejoinofNotaConstantandanyothervalueisNot-a-Constant• ThejoinofUndefined andanyothervalueisthatothervalue

exit x = 4;Undefined

z = x;Undefined

w = x;

y = x; z = y;

x = 6;entry

z = x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

x = 6;Undefined

entryUndefined

x=Undefinedy=Undefinedz=Undefinedw=Undefined

z = x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

x = 6;Undefined

entryUndefined

z = x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

Undefinedx = 6;Undefined

entryUndefined

z = x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

Undefinedx = 6;x = 6, y=z=w=Ω

entryUndefined

z = x;Undefined

w = x;Undefined

y = x;Undefined

z = y;Undefined

Undefinedx = 6;x = 6, y=z=w=Ω

entryUndefined

z = x;Undefined

w = x;Undefined

x=6y = x;Undefined

z = y;Undefined

Undefinedx = 6;x = 6

entryUndefined

z = x;Undefined

w = x;Undefined

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

z = x;Undefined

w = x;Undefined

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

y=67 y=Undefinedgiveswhat?

z = x;Undefined

x=6,y=6w = x;Undefined

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

z = x;Undefined

x=6,y=6w = x;Undefined

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

z = x;Undefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

z = x;Undefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

x=y=w=6z = x;Undefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

x=y=w=6z = x;x=y=w=z=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

exitx=y=w=z=6x = 4;Undefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

exitx=y=w=z=6x = 4;x=4, y=w=z=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

z = y;Undefined

entryUndefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;Undefined

entryUndefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;Undefined

entryUndefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

x=67 x=4giveswhat?

y=w=6, x=ºz = x;x=y=w=z=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

y=w=6z = x;y=w=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

y=w=6z = x;y=w=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

exity=w=6 x = 4;x=4, y=w=6

y=w=6z = x;y=w=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

exity=w=6 x = 4;x=4, y=w=6

y=w=6z = x;y=w=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

Globalanalysisreachedfixpoint

exity=w=6x = 4;y=w=6

y=w=6z = x;y=w=6

x=6,y=6w = x;x=y=w=6

x=6y = x;x=6,y=6

x = 6z = y;x = 6

entryUndefined

exity=w=6x = 4;y=w=6

y=w=6z = x;y=w=6

x=6,y=6w = 6;x=y=w=6

x=6y = 6;x=6,y=6

x = 6z = y;x = 6

entryUndefined

Dataflowforconstantpropagation

• Direction:Forward• Semilattice:Varst Undefined,0,1,-1,2,-2,…,Not-a-Constant– Joinmappingforvariablespoint-wisexh1,yh1,zh17 xh1,yh2,zhNot-a-Constant=xh1,yhNot-a-Constant,zhNot-a-Constant

• Transferfunctions:– fx=k(V)=V|xhk (updateVbymappingxtok)– fx=a+b(V)=V|xhNot-a-Constant (assignNot-a-Constant)

• Initialvalue:xisUndefined– (Whenmightweusesomeothervalue?)

Provingtermination

• Ouralgorithmforrunningtheseanalysescontinuouslyloopsuntilnochangesaredetected

• Giventhis,howdoweknowtheanalyseswilleventuallyterminate?– Ingeneral,wedon‘t

Terminates?

LivenessAnalysis

• Avariableislive atapointinaprogramiflaterintheprogramitsvaluewillbereadbeforeitiswrittentoagain

Joinsemilatticedefinition

• Ajoinsemilatticeisapair(V,7),where• Visadomainofelements• 7 isajoinoperatorthatis

– commutative:x7 y=y7 x– associative:(x7 y)7 z=x7 (y7 z)– idempotent:x7 x=x

• Ifx7 y=z,wesaythatzisthejoinor(LeastUpperBound)ofxandy

• Everyjoinsemilatticehasabottomelementdenotedz suchthatz 7 x=xforallx

Partialorderinginducedbyjoin

• Everyjoinsemilattice(V,7)inducesanorderingrelationshipb overitselements

• Definexb yiffx7 y=y• Needtoprove

– Reflexivity:xb x– Antisymmetry:Ifxb yandyb x,thenx=y– Transitivity:Ifxb yandyb z,thenxb z

Ajoinsemilatticeforliveness

• Setsoflivevariablesandthesetunionoperation• Idempotent:

– x4 x=x• Commutative:

– x4 y=y4 x• Associative:

– (x4 y)4 z=x4 (y4 z)• Bottomelement:

– Theemptyset:Ø4 x=x• Orderingoverelements=subsetrelation

Joinsemilatticeexampleforliveness

a, b a, c b, c

a, b, c

Bottomelement

Dataflowframework

• Aglobalanalysisisatuple(D,V,7,F,I),where– D isadirection(forwardorbackward)

• Theordertovisitstatementswithinabasicblock,NOT theorderinwhichtovisitthebasicblocks

– V isasetofvalues(sometimescalleddomain)– 7 isajoinoperatoroverthosevalues– F isasetoftransferfunctionsfs :Vt V(foreverystatements)– I isaninitialvalue

Runningglobalanalyses• Assumethat(D,V,7,F,I)isaforwardanalysis• Foreverystatementsmaintainvaluesbefore- IN[s]- andafter

- OUT[s]• SetOUT[s]=z forallstatementss• SetOUT[entry]=I• Repeatuntilnovalueschange:

– Foreachstatements withpredecessorsPRED[s]=p1,p2,…,pn• SetIN[s]=OUT[p1]7 OUT[p2]7 …7 OUT[pn]• SetOUT[s]=fs(IN[s])

• Theorderofthisiterationdoesnotmatter– Chaoticiteration

Provingtermination

• Ouralgorithmforrunningtheseanalysescontinuouslyloopsuntilnochangesaredetected

• Problem: howdoweknowtheanalyseswilleventuallyterminate?

Anon-terminatinganalysis

• ThefollowinganalysiswillloopinfinitelyonanyCFGcontainingaloop:

• Direction: Forward• Domain: ℕ• Joinoperator:max• Transferfunction: f(n) = n+1• Initialvalue:0

Initialization

x = y0

Fixed-pointiteration

x = y0

Chooseablock

x = y0

Iteration1

x = y0

Iteration1

x = y1

Chooseablock

x = y1

Iteration2

x = y1

Iteration2

x = y1

Iteration2

x = y2

Chooseablock

x = y2

Iteration3

x = y2

Iteration3

x = y2

Iteration3

x = y3

Whydoesn’tthisterminate?• Valuescanincreasewithoutbound• Notethat“increase”referstothelatticeordering,nottheorderingonthenaturalnumbers

• Theheight ofasemilatticeisthelengthofthelongestincreasingsequenceinthatsemilattice

• Thedataflowframeworkisnotguaranteedtoterminateforsemilatticesofinfiniteheight

• Notethatasemilatticecanbeinfinitelylargebuthavefiniteheight– e.g.constantpropagation

Heightofalattice

• Anincreasingchainisasequenceofelementsz a a1 a a2 a …a ak– Thelengthofsuchachainisk

• Theheightofalatticeisthelengthofthemaximalincreasingchain

• Forlivenesswithn programvariables:– _ v1_ v1,v2_ …_ v1,…,vn

• Foravailableexpressionsitisthenumberofexpressionsoftheforma=bopc– Forn programvariablesandm operatortypes:m$n3

Anothernon-terminatinganalysis

• Thisanalysisworksonafinite-heightsemilattice,butwillnotterminateoncertainCFGs:

• Direction: Forward• Domain: Booleanvaluestrue andfalse• Joinoperator:LogicalOR• Transferfunction:LogicalNOT• Initialvalue:false

Initialization

x = yfalse

Fixed-pointiteration

x = yfalse

Chooseablock

x = yfalse

Iteration1

x = yfalse

Iteration1

x = ytrue

Iteration2

x = ytrue

Iteration2

x = yfalse

Iteration3

x = yfalse

Iteration3

x = ytrue

Whydoesn’titterminate?• Valuescanloopindefinitely• Intuitively,thejoinoperatorkeepspullingvaluesup

• Ifthetransferfunctioncankeeppushingvaluesbackdownagain,thenthevaluesmightcycleforever

Whydoesn’titterminate?• Valuescanloopindefinitely• Intuitively,thejoinoperatorkeepspullingvaluesup

• Ifthetransferfunctioncankeeppushingvaluesbackdownagain,thenthevaluesmightcycleforever

• Howcanwefixthis?

Monotonetransferfunctions

• Atransferfunctionf ismonotone iffifxb y,thenf(x)b f(y)

• Intuitively,ifyouknowlessinformationaboutaprogrampoint,youcan't“gainback”moreinformationaboutthatprogrampoint

• Manytransferfunctionsaremonotone,includingthoseforlivenessandconstantpropagation

• Note:Monotonicity doesnotmeanthatxb f(x)– (Thisisadifferentpropertycalledextensivity)

Livenessandmonotonicity

• Atransferfunctionf ismonotone iffifxb y,thenf(x)b f(y)

• Recallourtransferfunctionfora=b+cis– fa=b+c(V)=(V– a)4 b,c

• Recallthatourjoinoperatorissetunionandinducesanorderingrelationship

Xb Yiff X`Y• Isthismonotone?

Isconstantpropagationmonotone?• Atransferfunctionf ismonotone iff

ifxb y,thenf(x)b f(y)• Recallourtransferfunctions

– fx=k(V)=V|xhk (updateVbymappingxtok)– fx=a+b(V)=V|xhNot-a-Constant (assignNot-a-Constant)

• Isthismonotone?

282Undefined

0-1-2 1 2 ......

Not-a-constant

Thegrandresult

• Theorem: Adataflowanalysiswithafinite-heightsemilattice andfamilyofmonotonetransferfunctions alwaysterminates

• Proofsketch:– Thejoinoperatorcanonlybringvaluesup– Transferfunctionscanneverlowervaluesbackdownbelowwheretheywereinthepast(monotonicity)

– Valuescannotincreaseindefinitely(finiteheight)

An“optimality”result

• Atransferfunctionf isdistributiveiff(a 7 b)=f(a)7 f(b)

foreverydomainelementsa andb• Ifalltransferfunctionsaredistributivethenthefixed-pointsolutionisthesolutionthatwouldbecomputedbyjoiningresultsfromall(potentiallyinfinite)control-flowpaths– Joinoverallpaths

• Optimalifweignoreprogramconditions

An“optimality”result

• Atransferfunctionf isdistributiveiff(a 7 b)=f(a)7 f(b)

foreverydomainelementsa andb• Ifalltransferfunctionsaredistributivethenthefixed-pointsolutionisequaltothesolutioncomputedbyjoiningresultsfromall(potentiallyinfinite)control-flowpaths– Joinoverallpaths

• Optimalifwepretendallcontrol-flowpathscanbeexecutedbytheprogram

• Whichanalysesusedistributivefunctions?

Loopoptimizations• Mostofaprogram’scomputationsaredoneinside

loops– Focusoptimizationseffortonloops

• Theoptimizationswe’veseensofarareindependentofthecontrolstructure

• Someoptimizationsarespecializedtoloops– Loop-invariantcodemotion– (Strengthreductionviainductionvariables)

• Requireanothertypeofanalysistofindoutwhereexpressionsgettheirvaluesfrom– Reachingdefinitions

• (Alsousefulforimprovingregisterallocation)

Loopinvariantcomputation

y = t * 4x < y + z

endx = x + 1

y = …t = …z = …

Loopinvariantcomputation

y = t * 4x < y + z

endx = x + 1

y = …t = …z = …

t*4andy+zhavesamevalueoneachiteration

Codehoisting

endx = x + 1

y = …t = …z = …y = t * 4w = y + z

Whatreasoningdidweuse?

y = t * 4x < y + z

endx = x + 1

y = …t = …z = …

yisdefinedinsideloopbutitisloopinvariantsincet*4isloop-invariant

Bothtandzaredefinedonlyoutsideofloop

constantsaretriviallyloop-invariant

Whataboutnow?

y=t*4x<y+z

endx=x+1t=t+1

y=…t=…z=…

Nowtisnotloop-invariantandsoaret*4andy

Loop-invariantcodemotion• d:t=a1 opa2

– d isaprogramlocation• a1 opa2loop-invariant (foraloopL)ifcomputesthe

samevalueineachiteration– Hardtoknowingeneral

• Conservativeapproximation– Eachai isaconstant,or– Alldefinitionsofai thatreachd areoutsideL,or– Onlyonedefinitionofof ai reachesd,andisloop-invariant

itself• Transformation:hoisttheloop-invariantcodeoutside

oftheloop

Reachingdefinitionsanalysis• Adefinitiond:t=…reaches aprogramlocationifthereisa

pathfromthedefinitiontotheprogramlocation,alongwhichthedefinedvariableisneverredefined

Reachingdefinitionsanalysis• Adefinitiond:t=…reaches aprogramlocationifthereisa

pathfromthedefinitiontotheprogramlocation,alongwhichthedefinedvariableisneverredefined

• Direction: Forward• Domain: setsofprogramlocationsthataredefinitions`• Joinoperator: union• Transferfunction:

fd:a=bopc(RD) =(RD- defs(a))4 dfd:not-a-def(RD) =RD

– Wheredefs(a)isthesetoflocationsdefininga (statementsoftheforma=...)

• Initialvalue:

Reachingdefinitionsanalysis

d4: y = t * 4

d4:x < y + z

d6: x = x + 1

d1: y = …

d2: t = …

d3: z = …

Reachingdefinitionsanalysis

d4: y = t * 4

d4:x < y + z

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

Initialization

d4: y = t * 4

d4:x < y + z

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

Iteration1

d4: y = t * 4

d4:x < y + z

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

Iteration1

d4: y = t * 4

d4:x < y + z

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2

d1, d2, d3

Iteration2

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2

d1, d2, d3

Iteration2

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3

d1, d2

d1, d2, d3

Iteration2

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3

d1, d2

d1, d2, d3

d2, d3, d4

Iteration2

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3

d1, d2

d1, d2, d3

d2, d3, d4

Iteration3

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3

d2, d3, d4

d1, d2

d1, d2, d3

d2, d3, d4

Iteration3

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3

d2, d3, d4

d1, d2

d1, d2, d3

d2, d3, d4

d2, d3, d4, d5

Iteration4

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3

d2, d3, d4

d1, d2

d1, d2, d3

d2, d3, d4

d2, d3, d4, d5

Iteration4

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3, d4, d5

d2, d3, d4

d1, d2

d1, d2, d3

d2, d3, d4

d2, d3, d4, d5

Iteration4

d4: y = t * 4

x < y + z end

d5: x = x + 1

d1: y = …

d2: t = …

d3: z = …

d1, d2, d3, d4, d5

d2, d3, d4

d1, d2

d1, d2, d3

d2, d3, d4, d5

Iteration5

d1: y = …

d2: t = …

d3: z = …

d2, d3, d4, d5

d1, d2

d1, d2, d3

d5: x = x + 1d2, d3, d4

d2, d3, d4, d5

d4: y = t * 4

x < y + z

d1, d2, d3, d4, d5

d2, d3, d4, d5

Iteration6

d1: y = …

d2: t = …

d3: z = …

d2, d3, d4, d5

d1, d2

d1, d2, d3

d5: x = x + 1d2, d3, d4, d5

d2, d3, d4, d5

d4: y = t * 4

x < y + z

d1, d2, d3, d4, d5

d2, d3, d4, d5

Whichexpressionsareloopinvariant?

tisdefinedonlyind2– outsideofloop

zisdefinedonlyind3– outsideofloop

yisdefinedonlyind4– insideofloopbutdependsontand4,bothloop-invariant

d1: y = …

d2: t = …

d3: z = …

d1, d2

d1, d2, d3

endd2, d3, d4, d5

d5: x = x + 1d2, d3, d4, d5

d2, d3, d4, d5

d4: y = t * 4

x < y + z

d1, d2, d3, d4, d5

d2, d3, d4, d5

d2, d3, d4, d5xisdefinedonlyind5–insideofloopsoisnotaloop-invariant

Inferringloop-invariantexpressions

• Forastatements oftheformt=a1 opa2• Avariableai isimmediatelyloop-invariantifallreachingdefinitionsIN[s]=d1,…,dkforai areoutsideoftheloop

• LOOP-INV=immediatelyloop-invariantvariablesandconstantsLOOP-INV=LOOP-INV4 x|d:x=a1 opa2, disintheloop,andbotha1 anda2areinLOOP-INV– Iterateuntilfixed-point

• Anexpressionisloop-invariantifalloperandsareloop-invariants

ComputingLOOP-INV

d1:y=…

d2:t=…

d3:z=…

d2,d3,d4

d1,d2,d3

d4:y=t*4

d5:x=x+1

d1,d2,d3,d4,d5

d2,d3,d4,d5

(immediately)LOOP-INV=T

ComputingLOOP-INV

d1:y=…

d2:t=…

d3:z=…

d2,d3,d4

d1,d2,d3

d4:y=t*4

d5:x=x+1

d1,d2,d3,d4,d5

d2,d3,d4,d5

(immediately)LOOP-INV=t

ComputingLOOP-INV

d1:y=…

d2:t=…

d3:z=…

d2,d3,d4

d1,d2,d3

d4:y=t*4

d5:x=x+1

d1,d2,d3,d4,d5

d2,d3,d4,d5

(immediately)LOOP-INV=t,z

ComputingLOOP-INV

d1:y=…

d2:t=…

d3:z=…

d2,d3,d4

d1,d2,d3

d4:y=t*4

d5:x=x+1

d1,d2,d3,d4,d5

d2,d3,d4,d5

ComputingLOOP-INV

d1:y=…

d2:t=…

d3:z=…

d2,d3,d4

d1,d2,d3

d4:y=t*4

d5:x=x+1

d1,d2,d3,d4,d5

d2,d3,d4,d5

d1:y=…

d2:t=…

d3:z=…

d2,d3,d4

d1,d2,d3LOOP-INV=t,z

d4:y=t*4

d5:x=x+1

d1,d2,d3,d4,d5

d2,d3,d4,d5

ComputingLOOP-INV

d4:y=t*4

x<y+z end

d5:x=x+1

d1:y=…

d2:t=…

d3:z=…

d1,d2,d3,d4,d5

d2,d3,d4,d5

d2,d3,d4

d1,d2,d3

d2,d3,d4,d5

LOOP-INV=t,z,y

Inductionvariables

while (i < x) j = a + 4 * ia[j] = ji = i + 1

i isincrementedbyaloop-invariantexpressiononeachiteration– thisiscalledaninductionvariable

jisalinearfunctionoftheinductionvariablewithmultiplier4

Strength-reduction

j = a + 4 * iwhile (i < x)

j = j + 4a[j] = ji = i + 1

Prepareinitialvalue

Incrementbymultiplier

TheEnd

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