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### Liveness Analysis

Mooly Sagiv

Schrierber 317

03-640-7606

Wed 10:00-12:00

html://www.math.tau.ac.il/~msagiv/courses/wcc01.html

Already Studied

lexical analysis

Tokens

syntax analysis

Abstract syntax tree

semantic analysis

Abstract syntax tree

Translate

Tree IR

Cannon

Cannonical Tree IR

Instruction Selection

Assem (with many reg)

Basic Compiler Phases

lexical analysis

Tokens

syntax analysis

Abstract syntax tree

semantic analysis

Frame

Translate

Intermediate representation

Instruction selection

Assembly

Register Allocation

Fin. Assembly

Register Allocation

- Input:
- Sequence of machine code instructions(assembly)
- Unbounded number of temporary registers

- Sequence of machine code instructions(assembly)
- Output
- Sequence of machine code instructions(assembly)
- Machine registers
- Some MOVE instructions removed
- Missing prologue and epilogue

CJUMP(EQ, TEMP t128, CONST 0, l0, l1)

LABEL( l1)

MOVE(TEMP t131, TEMP t128)

MOVE(TEMP t130, CALL(nfactor, BINOP(MINUS, TEMP t128, CONST 1)))

MOVE(TEMP t129, BINOP(TIMES, TEMP t131, TEMP t130))

LABEL(l2)

MOVE(TEMP t103, TEMP t129)

JUMP(NAME lend)

LABEL(l0)

MOVE(TEMP t129, CONST 1)

JUMP(NAME l2)

Missing updates for static link

l1: or t131, $0, t128

addi t132, t128, -1

or $4, $0, t132

jal nfactor

or t130, $0, $2

or t133, $0, t131

mult t133, t130

mflo t133

or t129, $0, t133

l2: or t103, $0, t129

b lend

l0: addi t129, $0, 1

b l2

l3: beq $25, $0, l0

l1: or $30, $0, $25

addi $4, $25, -1

/* or $4, $0, $4 */

jal nfactor

/* or $2, $0, $2 */

/* or $30, $0, $30 */

mult $30, $2

mflo $30

/* or $30, $0, $30 */

l2: or $2, $0, $30

b lend

l0: addi $30, $0, 1

b l2

.ent nfactor

factor_framesize=40

.frame $sp,nfactor_framesize,$31

nfactor: addiu $sp,$sp,-nfactor_framesize

sw $2,0+nfactor_framesize($sp)

or $25, $0, $4

sw $31,-4+nfactor_framesize($sp)

sw $30,-8+nfactor_framesize($sp)

l3: beq $25, $0, l0

l1: or $30, $0, $25

addi $4, $25, -1

jal nfactor

mult $30, $2

mflo $30

l2: or $2, $0, $30

b lend

l0: addi $30, $0, 1

b l2

lend: lw $30,-8+nfactor_framesize($sp)

lw $31,-4+nfactor_framesize($sp)

addiu $sp,$sp,nfactor_framesize

j $31

.end nfactor

The need for “spilling”

- The number of registers may not be enough
- Spill the content of some registers into memory
- Load when needed

- Increase the number of instructions
- Increase CPU time

The Challenge

- Minimize the number of spills
- Minimize the number of MOVEs
- Minimize CPU time

Outline

- Liveness Analysis
- Motivation
- Static Liveness
- Dataflow Equations
- Solutions
- An Iterative Algorithm
- Liveness in Tiger (Targil)

- Actual Allocation

Liveness Analysis Two variables interfere at a given point

- The same register may be assigned (at compile-time) to two temporaries if their “life-times” do not overlap
- A variable is live a givenprogram point
- its current value is used after this point prior to a definition (assignment)

- v is live at a given program point
- There exists an execution sequence from this point to a use of v that does not assign to v

- they are simultaneously live at this point

b

c

A Simple Example/* c */

L0: a := 0

/* ac */

L1: b := a + 1

/* bc */

c := c + b

/* bc */

a := b * 2

/* ac */

if c < N goto L1

/* c */

return c

Liveness Interference Graph

- For every compiled function
- Nodes
- Pre-colored machine registers
- Temporaries

- Undirected-Edges
- Temporaries that are simultaneously alive
- Different machine registers

- Undirected MOVE edges
- “Correlated” temporaries and registers

b

c

A Simple Example/* c */

L0: a := 0

/* ac */

L1: b := a + 1

/* bc */

c := c + b

/* bc */

a := b * 2

/* ac */

if c < N goto L1

/* c */

return c

t130

t131

t133

t128

t129

$0

$2

$4

t103

l3: beq t128, $0, l0 /* $0, t128 */

l1: or t131, $0, t128 /* $0, t128, t131 */

addi t132, t128, -1 /* $0, t131, t132 */

or $4, $0, t132 /* $0, $4, t131 */

jal nfactor /* $0, $2, t131 */

or t130, $0, $2 /* $0, t130, t131 */

or t133, $0, t131 /* $0, t130, t133 */

mult t133, t130 /* $0, t133 */

mflo t133 /* $0, t133 */

or t129, $0, t133 /* $0, t129 */

l2: or t103, $0, t129 /* $0, t103 */

b lend /* $0, t103 */

l0: addi t129, $0, 1 /* $0, t129 */

b l2 /* $0, t129 */

Undecidabily

- A variable is live at a point in a givenprogram point
- if its current value is used after this point prior to a definition in some execution path

- It is undecidable if a variable is live at a given program location

Conservative

- The compiler need not generate the optimal code
- Can use more registers (“spill code”) than necessary
- Find an upper approximation of the live variables
- A superset of edges in the interference graph
- Not too many superfluous live variables

Control Flow Graph

- Nodes
- Assembly instructions

- Directed-Edges
- If an instruction x can be immediately followed by an instruction y
- A directed edge xy

- If an instruction x can be immediately followed by an instruction y

Static Liveness

- A variable v is statically live at control flow node n
- there exists a directed path p from n to a use of v such that
- p does not include an assignment to v

- there exists a directed path p from n to a use of v such that
- Every live variable is statically live
- Some statically live variables are not live
- since some control flow paths are non-executable

c := a + b ;

c >= b

return a;

return c;

Examplea := b * b ;

c := a + b ;

if (c >= b)

then return c;

else return a;

/* c */

L0: a := 0

/* ac */

L1: b := a + 1

/* bc */

c := c + b

/* bc */

a := b * 2

/* ac */

if c < N goto L1

/* c */

return c

b := a +1 ;

c := c +b ;

a := b*2 ;

c <N goto L1

return c ;

Computing Static Liveness

- Generate a system of equations for every function
- define the set of live variables recursively

- Iteratively compute a minimal solution

The System of Equations

- For every instruction n
- def[n]
- The temporary and physical register(s) assigned by n

- use[n]
- The temporary and physical register used in n

- def[n]
- System of equations
- LiveOut[ex] =
- LiveOut[n] = (n, m) Edges Live[m]
- Live[n] = (LiveOut[n] – def[n]) use[n]

1

b := a +1 ;

2

c := c +b ;

3

a := b*2 ;

4

c <N goto L1

5

return c ;

6

LiveOut[6] = Live[6] = (LiveOut[6] – ) {c}

LiveOut[5] = Live[6] Live[2]

Live[5] = (LiveOut[5] – ) {c}

LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b}

LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b}

LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a}

LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

1

b := a +1 ;

2

c := c +b ;

3

a := b*2 ;

4

c <N goto L1

5

return c ;

6

LiveOut[6] = Live[6] = LiveOut[6] {c}

LiveOut[5] = Live[6] Live[2]

Live[5] = LiveOut[5] {c}

LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b}

LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b}

LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a}

LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

Fixed Points

- A fixed point is a vector solution Live and LiveOut
- for every instruction n
- LiveOut[ex] =
- LiveOut[n] = (n, m) Edges Live[m]
- Live[n] = (LiveOut[n] – def[n]) use[n]

- for every instruction n
- There more than one fixed point
- Every fixed point contains at least the statically live variables
- The least fixed point (in terms of set inclusion) uniquely exists
- it contains exactly the statically live variables

LiveOut[6] = Live[6] = LiveOut[6] {c}

LiveOut[5] = Live[6] Live[2]

Live[5] = LiveOut[5] {c}

LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b}

LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b}

LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a}

LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

a := 0 ;

1

b := a +1 ;

2

c := c +b ;

3

a := b*2 ;

4

c <N goto L1

5

return c ;

6

LiveOut[6] = Live[6] = LiveOut[6] {c}

LiveOut[5] = Live[6] Live[2]

Live[5] = LiveOut[5] {c}

LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b}

LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b}

LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a}

LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

a := 0 ;

1

b := a +1 ;

2

c := c +b ;

3

a := b*2 ;

4

c <N goto L1

5

return c ;

6

LiveOut[6] = Live[6] = LiveOut[6] {c}

LiveOut[5] = Live[6] Live[2]

Live[5] = LiveOut[5] {c}

LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b}

LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b}

LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a}

LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

a := 0 ;

1

b := a +1 ;

2

c := c +b ;

3

a := b*2 ;

4

c <N goto L1

5

return c ;

6

LiveOut[6] = Live[6] = LiveOut[6] {c}

LiveOut[5] = Live[6] Live[2]

Live[5] = LiveOut[5] {c}

LiveOut[4] = Live[5] Live[4] = (LiveOut[4] – {a}) {b}

LiveOut[3] = Live[4] Live[3] = (LiveOut[3] – {c}) {c, b}

LiveOut[2] = Live[3] Live[2] = (LiveOut[2] – {b}) {a}

LiveOut[1] = Live[2] Live[1] = (LiveOut[1] – {a})

a := 0 ;

1

b := a +1 ;

2

c := c +b ;

3

a := b*2 ;

4

c <N goto L1

5

return c ;

6

Computing Least Fixed Points

- Start with an empty set of Live and LiveOut for every instruction
- Repeatedly add new variables according to the equations
- The sets of LiveOut and Live variables must monotonically increase
- The process must terminate
- Unique least solution

An Iterative Algorithm

WL := ;

for each instruction n

LiveOut[n] :=

Live[n] :=

WL := WL {n}

while WL !=

select and remove n from WL

new := (LiveOut[n] –def[n]) use[n]

if new != Live[n] then

Live[n] := new

for all predecessors m of n do

LiveOut[m] := LiveOut[m] Live[n]

WL := WL {m}

Representation of Sets

- Bit-Vectors
- Var bits for every n
- Live[n][v] = 1
- the variable v is live before n

- Cost of set operation is
- O(Vars/word-size)

- Ordered Elements
- Linear time for set operations

Time Complexity

- Parameters
- N number of nodes (instructions)
- Assume that pred[n] is constant
- V Number of variables
- d Number of loop nesting level
- DFS back edges

- Initialization NV
- Inner-Most Iteration V
- For-Loop N
- Repeat
- Worst-Case NV
- Worst-Case-DFS d + 1

- Total-Worst-Case (NV)2
- Total-DFS NVd
- Single-variable N

An Interference Graph

for every instruction n

for every variable a def[n]

for every variable b LiveOut[n]

Create an interference edge

b

a

May introduce too many edges for move instructions

An Interference Graph

for every non move instruction n

for every variable a def[n]

for every variable b LiveOut[n]

Create an interference edge

b

a

for every move instruction n a:= c

for every variable b LiveOut[n] – {c} Create an interference edge

b

a

b

c

A Simple Example/* c */

L0: a := 0

/* ac */

L1: b := a + 1

/* bc */

c := c + b

/* bc */

a := b * 2

/* ac */

if c < N goto L1

/* c */

return c

t130

t131

t133

t128

t129

$0

$2

$4

t103

l3: beq t128, $0, l0 /* $0, t128 */

l1: or t131, $0, t128 /* $0, t128, t131 */

addi t132, t128, -1 /* $0, t131, t132 */

or $4, $0, t132 /* $0, $4, t131 */

jal nfactor /* $0, $2, t131 */

or t130, $0, $2 /* $0, t130, t131 */

or t133, $0, t131 /* $0, t130, t133 */

mult t133, t130 /* $0, t133 */

mflo t133 /* $0, t133 */

or t129, $0, t133 /* $0, t129 */

l2: or t103, $0, t129 /* $0, t103 */

b lend /* $0, t103 */

l0: addi t129, $0, 1 /* $0, t129 */

b l2 /* $0, t129 */

Summary

- The compiler can statically predict liveness of variables
- May be expensive

- Other useful static information
- Constant expressions
- Common sub-expression
- Loop invariant

- Liveness inference graph will be colored next week

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