Class Data Flow Analysis

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Data Flow Analysis

October 5, 2004


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Compiler Structure

Source code parsed to produce abstract syntax tree.

Abstract syntax tree transformed to control flow graph.

Data flow analysisoperates on the control flow graph(and other intermediate representations).

Source

code

Abstract

Syntax

tree

Control

Flow

Graph

Object

code


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Abstract Syntax Tree (AST)

Programs are written in text

as seuences of characters

may be aw!ward to wor! with.

"irst step# Con$ert to structured representation.

%se lexer (li!e lex) to recogni&e to!ens

%se parser (li!e yacc) to group to!ens structurally

often produce to produce AST


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Abstract Syntax Tree 'xample

x # a b*

y # a + b

,hile (y - a)

a # a /*

x # a b

0

program

while

block

=

x +

a b

>

=

a +

a 1

y a


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ASTs

ASTs are abstract

don1t contain all information in the program

e.g.2 spacing2 comments2 brac!ets2 parenthesis.

Any ambiguity has been resol$ed

e.g.2 a b c produces the same AST as

(a b) c.


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3isad$antages of ASTs

ASTs ha$e many similar forms

e.g.2 for while2 repeat 2 until2 etc

e.g.2 if2 42 switch

'xpressions in AST may be complex2 nested

(56 + y) ( & - 7 4 /6 + & # & 68)

,ant simpler representation for analysis

9 at least for dataflow analysis.


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Control:"low ;raph (C";)

A directed graphwhere

'ach node represents a statement

'dges represent control flow

Statements may be

Assignments x y op &or x op &

Copy statements x y


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>ariations on C";s

%sually don1t include declarations (e.g. int x*).

?ay want a uniue entry and exit point.

?ay group statements into basic blocks.

A basic blockis a seuence of instructions with nobranches into or out of the bloc!.


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Control:"low ;raph with


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C"; $s. AST

C";s are much simpler than ASTs

"ewer forms2 less redundancy2 only simpleexpressions


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3ata "low Analysis

A framewor! for pro$ing facts about program

Beasons about lots of little facts

=ittle or no interaction between facts

,or!s best on properties about how programcomputes


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A$ailable 'xpressions

An expression e x op y is availableat a programpoint p2 if

on e$ery path from the entry node of the graph to node p2 eis computed at least once2 and

And there are no definitions of x or y since the most recentoccurance of e on the path

ptimi&ation Df an expression is a$ailable2 it need not be recomputed At least2 if it is in a register somewhere


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3ata "low "acts

Ds expression e a$ailable4

"acts#

a b is a$ailablea + b is a$ailable

a / is a$ailable


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;en and Eill

,hat is the effect of each

x = a + b

stmt gen kill

y = a * b

a = a + 1

a + b

a * b

a + ba * b

a + 1

statement on the set of facts?


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{a+b}

{a + b,a * b}

{a + b,a * b}

{a + b}

{a+b}{a+b}{a+

b}

Computing A$ailable 'xpressions


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Terminology

Ajoin pointis a program point where two branchesmeet

A$ailable expressions is a forward, mstproblem

!orward 3ata "low from in to out

"st At Foint point2 property must hold on allpaths that are Foined.


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3ata "low 'uations

=et s be a statement

succ(s) immediate successor statements of s0

Pred(s) immediate predecessor statements of s0

Dn(s) program point Fust before executing s

ut(s) program point Fust after executing s

Dn(s) s 2preds!ut(s1)

ut(s) ;en(s) [(Dn(s) G Eill(s))

Hote these are also called transfer functions


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=i$eness Analysis

A $ariable $ is live at a programpoint p if

$ will be used on some execution pathoriginating from p before $ is o$erwritten

ptimi&ation

Df a $ariable is not li$e2 no need to !eep it

in a register Df a $ariable is dead at assignment2 caneliminate assignment.


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3ata "low 'uations

A$ailable expressions is a forward must analysis

3ata flow propagate in same direction as C"; edges

'xpression is a$ailable if a$ailable on all paths

=i$eness is a bac!ward may problem

to !ow if $ariable is li$e2 need to loo! at future uses

>ariable is li$e if a$ailable on some path

Dn(s) ;en(s) [(ut(s) G Eill(s))

ut(s) s 2succs!Dn(s1)


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;en and Eill

,hat is the effect of each

x = a + b

y = a * b

a = a + 1

stmt gen kill

y > a

a, b

a, b

a, y

a

x

y

a

statement on the set of facts?


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{x

}

Computing =i$e >ariables


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{x, y, a}

{x

}



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{x, y, a}

{x

}

{x, y, a}



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{x, y, a}

{x

}

{x,y, a}

{y, a,b}



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{x, y, a}

{x}

{x,y, a}

{y, a,b}

{y, a,

b}



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{x,y, a,b}

{x

}

{x,y, a}

{y, a,b}

{y, a,

b}



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{x,y, a,b}

{x

}

{x, y, a,b}

{y, a,b}

{y, a,

b}



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{x, y, a,b}

{x

}

{x, y, a,b}

{y, a,b}

{y, a,

b}


{x, a, b}


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{x, y, a,b}

{x

}

{x, y, a,b}

{y, a,b}

{y, a,

b}


{x, a, b}

{a, b}


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>ery


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Code Ioisting

Code hoisting finds expressions that are alwayse$aluated following some point in a program2regardless of the execution path and mo$es them tothe latest point beyond which they would always be

e$aluated.

Dt is a transformation that almost always reducesthe space occupied but that may affect its executiontime positi$ely or not at all.


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Beaching 3efinitions

A definition of a variable$ is an assignment to $

A definition of $ariable $ reachespoint p if

There is no inter$ening assignment to $

Also called def#seinformation

,hat !ind of problem4

"orward or bac!ward4 "orward

?ay or must4 may


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Space of 3ata "low Analyses

?ost data flow analyses can be classified this way

A few don1t fit# bidirectional

=ots of literature on data flow analysis

May Must

Forward

Backward

eachi!g

de"i!itio!s

#$ailable

expressio!s

%i$e

&ariables

&ery busy

expressio!s


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3ata "low "acts and lattices

Typically2 data flow facts form a lattice

'xample2 A$ailable expressions

top

bottom


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Partial rders

Apartial orderis a pair (P2 ) such that

P P

is refle$ive# x x is anti#symmetric# x y and y x implies x y

is transitive# x y and y & implies x &


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=attices

A partial order is a lattice if uandtare defined so that

uis the meetor greatest lower boundoperation

x uy x and x uy y Df & x and & y then & x uy

tis theFoinor least upper boundoperation x x ty and y x ty Df x & and y &2 then x ty &


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=attices (cont.)

A finite partial order is a lattice if meet and Foin exist fore$ery pair of elements

A lattice has uniue elements botand topsuch that

x u? ? x t?x

x u> x x t> >

Dn a lattice

x y iff x uy x

x y iff x ty y


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%seful =attices

(6S2 ) forms a lattice for any set S. 6Sis the powerset of S (set of all subsets)

Df (S2 ) is a lattice2 so is (S2)

i.e.2 lattices can be flipped

The lattice for constant propagation

1 ' (

>

?


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"orward ?ust 3ata "low Algorithm

ut(s) ;en(s) for all statements s

, all statements0 (wor!list)

Bepeat

Ta!e s from ,

Dn(s)

s 2preds!ut(s1)

Temp ;en(s) [(Dn(s) G Eill(s))

Df (temp J ut (s))

ut(s) temp

, , [succ(s)

0

%ntil ,


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?onotonicity

A function f on a partial order is monotonic if

x y implies f(x) f(y)

'asy to chec! that operations to compute Dn andut are monotonic

Dn(s) s 2preds!ut(s1)

Temp ;en(s) [(Dn(s) G Eill(s))

Putting the two together

Temp fs(s 2preds!ut(s1))


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Termination

,e !now algorithm terminates because

The lattice has finite height

The operations to compute Dn and ut aremonotonic

n e$ery iteration we remo$e a statement

from the wor!list andKor mo$e down thelattice.

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