263-2810:AdvancedCompilerDesign

3.0SSAformforarbitrarycontrolflowgraphs

RudiEigenmannThomasR.Gross

ComputerScienceDepartmentETHZurich,Switzerland

SSAformat

§ f functionsallowsustodealwithabasicblockthathasmultiplepredecessors

§ ToturnaprogramintoSSAformat§ Insertf functions§ Determineversiontobeusedasoperand§ Producenewversionswhenthereisanassignment

§ Oraf functiondeliversthecorrectversion

§ Sofar:syntacticapproach§ Worksforwell-structuredprograms§ Next:anapproachforgeneralcontrolflowgraphs§ Butfirst:reviewofconceptsforgraphs

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Outline

§ 3.1Graphs

§ 3.2Approachestoinsertionoff functions

§ 3.3Dominancefrontier

§ 3.4Algorithmforinsertionoff functions

§ 3.5Algorithmforvariablerenaming

§ 3.6Example

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3.1Graphs

§ Controlflowgraph:centraldatastructure§ Nodes:basicblocks(orsequencesofstraight-lineprogramsegments)§ Edges:directed,possiblecontrolflow

§ Specialnodes§ ENTRY:specialwithoutpredecessor§ EXIT:specialwithoutsuccessor

§ EdgefromENTRYtoEXIT

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Dominance

§ GivenaCFG.AnodeA(orbasicblockA)dominates nodeB(basicblockB)ifAisoneverypathfromENTRYtoB.

Adom B

Adom A

X¬dom Y:XdoesnotdominateY

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Immediatedominator

§ Aidom B§ Adom B§ ∄ XsuchthatAdom XandXdom B(X≠A,X≠B)

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Dominatortree

§ Thedominatortree(DT)capturestheimmediatedominatorrelationship§ Xidom Y:edgeX➞ YinDT

§ Thereisexactlyoneimmediatedominator§ That’swhyit’satree…§ LastdominatoronanypathfromENTRY

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Example2(Dominatortree)

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ENTRY

A

B

C D

EXIT

Terminology

§ Controlflowgraph§ predecessor§ successor§ direct(predecessor/successor):Thereisanedge--- indicatespossible

controltransfer

§ Example§ BdirectsuccessorofA§ CdirectsuccessorofB§ AdirectpredecessorofB§ DsuccessorofA§ ApredecessorofC§ CpredecessorofB

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Terminology2

§ Dominatortree§ child§ parent§ ancestor:parent,grandparent,great-grandparent,…§ descendant:child,grandchild,great-grandchild,…

§ Example§ AparentofB§ BchildofA§ CchildofB

§ Nostatementaboutactualcontrolflow§ ENTRY→A→B→D→EXIT 23

Dominatorrelationship

§ dom reflexive§ Xdom X

§ Sometimeswewantanirreflexive (anti-reflexive)relationship§ WanttobesurethatAdom BimpliesA≠B

§ XstrictlydominatesY:Xdom YandX≠Y

X⨠Y

§ Distinguishfrom(weak)domination:Xdom Y

X⨠Y

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3.2Insertionoff functions

§ Wherecouldweinsertf functions?

§ Whereshouldweinsertf functions?

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Examplegraph

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ENTRY

p =

EXIT

p =

= p BB6

§ Programnotwell-structured

§ Insertf functions– OptionA:§ Insertaf functionintotheblockthatusesavariable§ Example:p isusedinbasicblockBB6

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§ Insertf functions– OptionB:§ Insertaf functionasearlyaspossible§ Evenifthereisnouseofavariableinbasicblock§ Example:functionsinBB3,BB5,andBB6

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§ OptionB:Numberofargumentstof functiondependsonthenumberofdirectpredecessorsinCFG§ Fixedforagivenprogramminglanguageresp.implementation§ case/switchstatementmayresultinunlimitednumberof

predecessorsunlesscompiledintocascadingsetofif-statements

§ OptionA:Numberofargumentstof functiondependsonnumberofpathsthatreachagivenpoint§ Unbounded

§ OptionBpreferredbycompilerdesigners

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Setup

§ Letusassumethereisaninitial(dedicated)assignmenttoeachvariable§ Forp,q,r,…wehavep0,q0,r0,…§ “Placed”intotheENTRYnode§ “pseudo-assignment”

§ Benefits§ Thereisatleastoneassignmentoneverypath§ IftheRHSofanassignmentorexpressionreadsv0 thereispossiblya

useofanuninitializedvariable

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CFGnodeswithf functions

§ ConsidertwonodesA,B

§ Adefinesp,Busesp

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p =

B

A

§ No functionisneededalongpathfromAtoBifA⨠B

= p

§ ButA=Bispossible.Needtodealwith

§ Extenddefinitionofdominancefrombasicblocks(nodesofCFG)tooperationsinbasicblock§ definitions§ uses

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= p

p =

p =

= p

§ Definition(def)di inblockA

§ Use(use)uj inblockB

di⨠ujiff di isinblockA,uj isinblockB,anddi isthelastdefinitionalonganypathfromENTRYtouj

§ Corrollary:A⨠B

§ Candefinedi ⨠ uj aswell

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