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## Introduction to Compilers

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### Introduction to Compilers

Ben Livshits

Based in part of Stanford class slides from

http://infolab.stanford.edu/~ullman/dragon/w06/w06.html

My Lectures

reliability & bug finding

performance

security

Lecture 1: Introduction to staticanalysis

Lecture 2: Introduction to runtimeanalysis

Lecture 3: Applicationsof static and runtime analysis

L1: Lecture Organization

- Really basic stuff
- Flow Graphs
- Constant Folding
- Global Common Subexpression Elimination

- Data-flow analysis
- Proving Little Theorems
- Data-Flow Equations
- Major Examples: reaching definitions

- Looking forward…

Dawn of Code Optimization

A never-published Stanford technical report by Fran Allen in 1968

Flow graphs of intermediate code

if i>=n goto …

t1 = 8*i

A[t1] = 1

i = i+1

Intermediate Codefor (i=0; i<n; i++)

A[i] = 1;

Make flow explicit by breaking into basic blocks= sequences of steps with entry at beginning, exit at end

Intermediate code exposes optimizable constructs we cannot see at source-code level

Constant Folding

Sometimes a variable has a known constant value at a point

If so, replacing the variable by the constant simplifies and speeds-up the code

Easy within a basic block; harder across blocks

n = 100

if i>=n goto …

t1 = 8*i

A[t1] = 1

i = i+1

Constant Folding Examplet1 = 0

if t1>=800 goto …

A[t1] = 1

t1 = t1+8

Global Common Subexpressions

- Suppose block B has a computation of x+y
- Suppose we are sure that when we reach this computation, we are sure to have:
- Computed x+y, and
- Not subsequently reassigned x or y
- Then we can hold the value of x+yand use it in B

Data-Flow Analysis

- Proving Little Theorems
- Data-Flow Equations
- Major Examples: reaching definitions

An Obvious Theorem

boolean x = true;

while (x) {

. . . // no change to x

}

Let’s prove that this loop doesn’t terminate

Proof: only assignment to x is at top, so x is always true

Formulation: Reaching Definitions

Each place some variable x is assigned is a definition

Ask: for this use of x, where could x last have been defined

In our example: only at x=true

Clincher

Since at x == true, d1is the only definition of x that reaches, it must be that x is true at that point

The conditional is not really a conditional and can be replaced by a branch

Not Always That Easy

int i = 2;

intj = 3;

while (i != j) {

if (i < j) i += 2;

else j += 2;

}

We’ll develop techniques for this problem, but later …

d2

d3

d4

d2, d3, d4

d1, d3, d4

d1, d2, d3, d4

d1, d2, d3, d4

The Flow Graphd1: i = 2

d2: j = 3

if i != j

d1, d2, d3, d4

if i < j

d3: i = i+2

d4: j = j+2

DFA Is Sometimes Insufficient

In this example, i can be defined in two places, and j in two places

No obvious way to discover that i!=jis always true

But OK, because reaching definitions is sufficient to catch most opportunities for constant folding(replacement of a variable by its only possible value)

Be Conservative!

It’s OK to discover a subset of the opportunities to make some code-improving transformation.

It’s not OK to think you have an opportunity that you don’t really have.

Must we always be conservative?

Example: Be Conservative

boolean x = true;

while (x) {

. . . *p = false; . . .

}

Is it possible that p points to x?

Possible Resolution

Just as data-flow analysis of “reaching definitions” can tell what definitions of x might reach a point, another DFA can eliminate cases where p definitely does not point to x

Example: in a C program the only definition of p is p = &yand there is no possibility that y is an alias of x

Reaching Definitions Formalized

- A definition d of a variable x is said to reach a point p in a flow graph if:
- Every path from the entry of the flow graph to p has d on the path, and
- After the last occurrence of d there is no possibility that x is redefined

Data-Flow Equations: (1)

- A basic block can generate a definition
- A basic block can either
- Killa definition of x if it surely redefines x
- Transmit a definition if it may not redefine the same variable(s) as that definition

Data-Flow Equations: (2)

- Variables:
- IN(B) = set of definitions reaching the beginning of block B.
- OUT(B) = set of definitions reaching the end of B.

Data-Flow Equations: (3)

- Two kinds of equations:
- Confluence equations: IN(B) in terms of outs of predecessors of B
- Transfer equations: OUT(B) in terms of of IN(B) and what goes on in block B.

Transfer Equations

Generate a definition in the block if its variable is not definitely rewritten later in the basic block.

Killa definition if its variable is definitely rewritten in the block.

An internal definition may be both killed and generated.

Example: Gen and Kill

IN = {d2(x), d3(y), d3(z), d5(y), d6(y), d7(z)}

Kill includes {d1(x), d2(x),

d3(y), d5(y), d6(y),…}

d1: y = 3

d2: x = y+z

d3: *p = 10

d4: y = 5

Gen = {d2(x), d3(x),

d3(z),…, d4(y)}

OUT = {d2(x), d3(x), d3(z),…, d4(y), d7(z)}

Iterative Solution to Equations

For an n-block flow graph, there are 2n equations in 2n unknowns

Alas, the solution is not unique

Use iteration to get the least fixed-point

Has nice performance guarantees

Iterative Solution: (2)

IN(entry) = ∅;

for each block B do OUT(B)= ∅;

while (changes occur) do

for each block B do {

IN(B) = ∪predecessors P of B OUT(P);

OUT(B) = (IN(B) – Kill(B)) ∪Gen(B);

}

OUT(B1) = {

IN(B2) = {

d1,

OUT(B2) = {

IN(B3) = {

d1,

OUT(B3) = {

Example: Reaching Definitionsd1: x = 5

B1

d1}

d2}

if x == 10

B2

d1,

d2}

d2}

d2: x = 15

B3

d2}

Notice the Conservatism

Not only the most conservative assumption about when a def is killed or gen’d

Also the conservative assumption that any path in the flow graph can actually be taken

Must we always be conservative?

Where are we Going From Here?

- Inter-procedural analysis
- Context-sensitive analysis
- Object-sensitive analysis

- Pointers or references
- Reflection
- Dynamic code generation

research

Some is covered in Yannis’s lectures…

A Little JavaScript Program

f = function(x){ return x+1; }

function compose(){

return function(x){

return f(f(x)) }; }

compose(f)(3);

this[‘compose’](f)(3)

this['comp' + 'os' + ("E".toLowerCase())](f)(3)

function(f){ return ;

eval("compose(f)(" + arguments[0] + ");"); }()(3);

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