Dataflow Analysis

Challenge the future
Delft
University of
Technology
Course IN4303, 2016-2017
Compiler Construction
Guido Wachsmuth, Eelco Visser
Dataflow Analysis

today’s lecture
Dataflow Analysis
Overview
Control flow graphs
• intermediate representation
2

today’s lecture
Dataflow Analysis
Overview
Control flow graphs
Non-local optimizations
• dead code elimination
• constant & copy propagation
• common subexpression elimination
2

today’s lecture
Dataflow Analysis
Overview
Control flow graphs
Data flow analyses
• liveness analysis
• reaching definitions
• available expressions
2

today’s lecture
Dataflow Analysis
Overview
Control flow graphs
Data flow analyses
• liveness analysis
Optimizations (again)
2

Term Rewriting
Optimizations
I
3

optimization
Dataflow Analysis
Optimizing Compilers
Fully optimizing compiler
• Opt(P) produces smallest program with same I/O behavior
• smallest program for non-terminating programs without I/O: 
Loop = (L : goto L)
• solves halting problem: Opt(Q) = Loop iff Q halts
Optimizing compiler
• produces program with same I/O behavior
• that is smaller or faster
Full employment theorem for compiler writers
• there is always a better optimizing compiler
4

optimization
Dataflow Analysis
Local vs Non-Local Optimizations
Local optimizations
• peephole optimization
• constant folding
• pattern match + rewrite
• constant propagation
• dead-code elimination
• require information from different parts of program
5
iload 1
iload 1
mul
iload 1
dup
mul
(3 + 4) * x 7 * x

optimization
Dataflow Analysis
Program Optimizations
Register allocation
• keep non-overlapping temporaries in same register
Common-subexpression elimination
• if expression is computed more than once, eliminate one computation
Dead-code elimination
• delete computation whose result will never be used
Constant folding
• if operands of expression are constant, do computation at compile-time
And many more possible optimizations
6

example
Dataflow Analysis
Dead Code Elimination
7
delete computation whose result will never be used
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
return c

example
Dataflow Analysis
7
a ← 0
b ← a + 1
c ← c + b
return c
delete computation whose result will never be used
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
return c

example
Dataflow Analysis
Constant Propagation
8
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
return c
substitute reference to constant valued variable

example
Dataflow Analysis
8
a ← 0
b ← 0 + 1
c ← c + b
a ← 2 * b
return c
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
return c
substitute reference to constant valued variable

example
Dataflow Analysis
Constant Propagation + Folding
9
a ← 1
b ← a + 1
c ← c + b
a ← 2 * b
return c

example
Dataflow Analysis
Constant Propagation + Folding
9
a ← 0
b ← 2
c ← c + 2
a ← 4
return c
a ← 1
b ← a + 1
c ← c + b
a ← 2 * b
return c

example
Dataflow Analysis
Copy Propagation
10
a ← e
b ← a + 1
c ← c + b
a ← 2 * b
return c

example
Dataflow Analysis
Copy Propagation
10
a ← e
b ← e + 1
c ← c + b
a ← 2 * b
return c
a ← e
b ← a + 1
c ← c + b
a ← 2 * b
return c

example
Dataflow Analysis
Common Subexpression Elimination
11
c ← a + b
d ← 1
e ← a + b
if expression is computed more than once, eliminate one computation

example
Dataflow Analysis
11
c ← a + b
d ← 1
e ← a + b
x ← a + b
c ← x
d ← 1
e ← x
if expression is computed more than once, eliminate one computation

optimization
Dataflow Analysis
Intraprocedural Global Optimization
Terminology
• Intraprocedural: within a single procedure or function
• Global: spans all statements with procedure
• Interprocedural: operating on several procedures at once
Recipe
• Dataflow analysis: traverse flow graph, gather information
• Transformation: modify program using information from analysis
12

Dataflow Analysis
Control-flow Graphs
II
13

quadruples
Dataflow Analysis
Intermediate Language
Store
a ← b ⊕ c
a ← b
Memory access
a ← M[b]
M[a] ← b
Functions
f(a1, …, an)
b ← f(a1, …, an)
Jumps
L:
goto L
if a ⊗ b
goto L1
else
goto L2
14

example
Dataflow Analysis
Control-Flow Graphs
15
a ← 0
L1: b ← a + 1
c ← c + b
a ← 2 * b
if a < N
goto L1
else
goto L2
L2: return c

example
Dataflow Analysis
Control-Flow Graphs
15
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
a ← 0
L1: b ← a + 1
c ← c + b
a ← 2 * b
if a < N
goto L1
else
goto L2
L2: return c

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
node

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
node
in-edge

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
node
in-edge
in-edge

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
node
predecessor
predecessor
in-edge
in-edge

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
node
predecessor
predecessor
in-edge
out-edge
in-edge

terminology
Dataflow Analysis
Control-Flow Graphs
16
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
node
predecessor
predecessor
successor
in-edge
out-edge
in-edge

Dataflow Analysis
Liveness Analysis
III
17

definition
Dataflow Analysis
Liveness Analysis
Motivation: register allocation
• intermediate code with unbounded number of temporary variables
• map to bounded number of registers
• if two variables are not ‘live’ at the same time, store in same register
Liveness
• a variable is live if it holds a value that may be needed in the future
18

example
Dataflow Analysis
Register Allocation
19
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

example
Dataflow Analysis
Register Allocation
19
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
{a, b} ! r
{c} ! q

example
Dataflow Analysis
Register Allocation
19
r ← 0
r ← r + 1
q ← q + r
r ← 2 * r
if r < N
return q
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
{a, b} ! r
{c} ! q

a ← 0
return c
if a < N
b ← a + 1
a ← 2 * b
c ← c + b
example
Dataflow Analysis
Liveness Analysis
20

definitions
Dataflow Analysis
Liveness Analysis
Def and use
• an assignment to a variable defines a variable
• an occurrence of a variable in an expression uses that variable
• def[x] = {n | n defines x}, def[n] = {x | n defines x}
• use[x] = {n | n uses x}, use[n] = {x | n uses x}
A variable is live on an edge if
• there is a directed path from that edge to a use of the variable
• that does not go through any def
A variable is live in/out at a node
• live-in: it is live on any of the in-edges of that node
• live-out: it is live on any of the out-edges of the node
21

terminology
Dataflow Analysis
Liveness Analysis
22
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
definition a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
definition
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
usage
definition
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
usage
definition
live-in
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
usage
definition
live-in
live-out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
usage
definition
live-in
live-in
live-out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
usage
definition
live-in
live-out
live-in
live-out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

terminology
Dataflow Analysis
Liveness Analysis
22
usage
definition
usage
definition
live-in
live-out
live-out
live-in
live-out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

formalization
Dataflow Analysis
Liveness Analysis
in[n] = use[n] ∪ (out[n] - def[n])
out[n] = ∪s∈succ[n] in[s]
1. If a variable is used at node n, then it is live-in at n
2. If a variable is live-out but not defined at node n, it is live-in at n
3. If a variable is live-in at node n, then it is live-out at its predecessors
23

algorithm
Dataflow Analysis
Liveness Analysis
for each n
in[n] ← {} ; out[n] ← {}
repeat
for each n
in'[n] = in[n] ; out'[n] = out[n]
in[n] = use[n] ∪ (out[n] - def[n])
for each s in succ[n]
out[n] = out[n] ∪ in[s]
until
for all n: in[n] = in'[n] and out[n] = out'[n]
24

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a
2
i
a
bc
b
a
c
o
a
bc
b
a
ac

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a
2
i
a
bc
b
a
c
o
a
bc
b
a
ac
3
i
ac
bc
b
ac
c
o
a
bc
b
a
ac

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a
2
i
a
bc
b
a
c
o
a
bc
b
a
ac
3
i
ac
bc
b
ac
c
o
a
bc
b
a
ac
4
i
ac
bc
b
ac
c
o
ac
bc
b
ac
ac

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a
2
i
a
bc
b
a
c
o
a
bc
b
a
ac
3
i
ac
bc
b
ac
c
o
a
bc
b
a
ac
4
i
ac
bc
b
ac
c
o
ac
bc
b
ac
ac
5
i
c
ac
bc
bc
ac
c
o
ac
bc
b
ac
ac

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a
2
i
a
bc
b
a
c
o
a
bc
b
a
ac
3
i
ac
bc
b
ac
c
o
a
bc
b
a
ac
4
i
ac
bc
b
ac
c
o
ac
bc
b
ac
ac
5
i
c
ac
bc
bc
ac
c
o
ac
bc
b
ac
ac
6
i
c
ac
bc
bc
ac
c
o
ac
bc
bc
ac
ac

example
Dataflow Analysis
Liveness Analysis
25
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
1
2
3
4
5
6
u
a
bc
b
a
c
d
a
b
c
a
1
i
a
bc
b
a
c
o
a
2
i
a
bc
b
a
c
o
a
bc
b
a
ac
3
i
ac
bc
b
ac
c
o
a
bc
b
a
ac
4
i
ac
bc
b
ac
c
o
ac
bc
b
ac
ac
5
i
c
ac
bc
bc
ac
c
o
ac
bc
b
ac
ac
6
i
c
ac
bc
bc
ac
c
o
ac
bc
bc
ac
ac
7
i
c
ac
bc
bc
ac
c
o
ac
bc
bc
ac
ac

optimization
Dataflow Analysis
Liveness Analysis
26
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
6
5
4
3
2
1
u
c
a
b
bc
a

optimization
Dataflow Analysis
Liveness Analysis
26
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
6
5
4
3
2
1
u
c
a
b
bc
a
d
a
c
b
a

optimization
Dataflow Analysis
Liveness Analysis
26
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
6
5
4
3
2
1
u
c
a
b
bc
a
d
a
c
b
a
1
o
c
ac
bc
bc
ac
i
c
ac
bc
bc
ac
a

optimization
Dataflow Analysis
Liveness Analysis
26
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
6
5
4
3
2
1
u
c
a
b
bc
a
d
a
c
b
a
1
o
c
ac
bc
bc
ac
i
c
ac
bc
bc
ac
a
2
o
ac
ac
bc
bc
ac
i
c
ac
bc
bc
ac
c

optimization
Dataflow Analysis
Liveness Analysis
26
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c
1
2
3
4
5
6
6
5
4
3
2
1
u
c
a
b
bc
a
d
a
c
b
a
1
o
c
ac
bc
bc
ac
i
c
ac
bc
bc
ac
a
2
o
ac
ac
bc
bc
ac
i
c
ac
bc
bc
ac
c
3
o
ac
ac
bc
bc
ac
i
c
ac
bc
bc
ac
c

generalisation
Dataflow Analysis
Liveness Analysis
27
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
kill a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
kill
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
gen
kill
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
gen
kill
in
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
gen
kill
in
out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
gen
kill
in
out
out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
gen
kill
in
out
out
in
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

generalisation
Dataflow Analysis
Liveness Analysis
27
gen
kill
gen
kill
in
out
out
in
out
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
if a < N
return c

gen & kill
Dataflow Analysis
Liveness Analysis
28
a ← b ⊕ c
a ← b
a ← M[b]
M[a] ← b
f(a1, …, an)
a ← f(a1, …, an)
goto L
if a ⊗ b
gen
{b,c}
{b}
{b}
{a,b}
{a1,…,an}
{a1,…,an}
{a,b}

gen & kill
Dataflow Analysis
Liveness Analysis
28
a ← b ⊕ c
a ← b
a ← M[b]
M[a] ← b
f(a1, …, an)
a ← f(a1, …, an)
goto L
if a ⊗ b
gen
{b,c}
{b}
{b}
{a,b}
{a1,…,an}
{a1,…,an}
{a,b}
kill
{a}
{a}
{a}
{a}

generalisation
Dataflow Analysis
Liveness Analysis
in[n] = gen[n] ∪ (out[n] - kill[n])
out[n] = ∪s∈succ[n] in[s]
29

Dataflow Analysis
More Analyses
IV
30

definition
Dataflow Analysis
Reaching Definitions
Unambiguous definition of a
• a statement of the form (d : a ← b ⊕ c) or (d : a ← M[x])
Definition d reaches statement u if
• there is some path in control-flow graph from d to u
• that does not contain an unambiguous definition of a
Used in
• constant propagation
31

example
Dataflow Analysis
32
a ← 5
c ← 1
if c > a
c ← c + c
goto 3
a ← c - a
1
2
3
4
5
6

gen & kill
Dataflow Analysis
33
d: a ← b ⊕ c
d: a ← b
d: a ← M[b]
M[a] ← b
f(a1, …, an)
d: a ← f(a1, …, an)
goto L
if a ⊗ b
gen
{d}
{d}
{d}
{d}
defs(a) : all definitions of a

gen & kill
Dataflow Analysis
33
d: a ← b ⊕ c
d: a ← b
d: a ← M[b]
M[a] ← b
f(a1, …, an)
d: a ← f(a1, …, an)
goto L
if a ⊗ b
gen
{d}
{d}
{d}
{d}
kill
defs(a)-{d}
defs(a)-{d}
defs(a)-{d}
defs(a)-{d}
defs(a) : all definitions of a

formalisation
Dataflow Analysis
in[n] = ∪p∈pred[n] out[p]
out[n] = gen[n] ∪ (in[n] - kill[n])
34

definition
Dataflow Analysis
Available Expressions
An expression (b ⊕ c) is available at node n if
• on every path from the entry node to node n
• (b ⊕ c) is computed at least once, and
• there are no definitions of b or c since most recent occurrence  
of (b ⊕ c) on that path
Used in
• common-subexpression elimination
35

example
Dataflow Analysis
36
c ← a + b
d ← 1
e ← a + b
1
2
3

gen & kill
Dataflow Analysis
37
d: a ← b ⊕ c
d: a ← b
d: a ← M[b]
M[a] ← b
f(a1, …, an)
d: a ← f(a1, …, an)
goto L
if a ⊗ b
gen
{b⊕c}-kill
{M[b]}-kill

gen & kill
Dataflow Analysis
37
d: a ← b ⊕ c
d: a ← b
d: a ← M[b]
M[a] ← b
f(a1, …, an)
d: a ← f(a1, …, an)
goto L
if a ⊗ b
gen
{b⊕c}-kill
{M[b]}-kill
kill
exps(a)
exps(a)
exps(M[_])
exps(M[_])
exps(M[_]) ∪ exps(a)

formalisation
Dataflow Analysis
in[n] = ∩p∈pred[n] out[p]
out[n] = gen[n] ∪ (in[n] - kill[n])
38

definition
Dataflow Analysis
Reaching Expressions
An expression (s : a ← b ⊕ c) reaches node n if
• there is a path from s to n
• that does not go through assignment to b or c
• or through any computation of (b ⊕ c)
Used in
• common-subexpression elimination
39

Dataflow Analysis
Optimizations
V
40

example
Dataflow Analysis
41
a ← 0
b ← a + 1
c ← c + b
return c
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
return c
consider:
(s : a ← b ⊕ c) or (s : a ← M[x])
if a is not live-out at s
transform:
remove s

example
Dataflow Analysis
42
c ← a + b
d ← 1
e ← a + b
x ← a + b
c ← x
d ← 1
e ← x
consider:
(n : a ← b ⊕ c) reaches (s : d ← b ⊕ c)
path from n to s does not compute b ⊕ c or define b or c
e is a fresh variable
transform:
n : a ← b ⊕ c
n’ : e ← a
…
s : d ← e

example
Dataflow Analysis
43
a ← 0
b ← 0 + 1
c ← c + b
a ← 2 * b
return c
a ← 0
b ← a + 1
c ← c + b
a ← 2 * b
return c
consider: (d : a ← c) & (n : e ← a ⊕ b)
if c is constant
& (d reaches n)
& (no other def of a reaches n)
transform:
(n : e ← a ⊕ b) => (n : e ← c ⊕ b)

example
Dataflow Analysis
Copy Propagation
44
a ← e
b ← e + 1
c ← c + b
a ← 2 * b
return c
a ← e
b ← a + 1
c ← c + b
a ← 2 * b
return c
consider: (d : a ← z) & (n : e ← a ⊕ b)
if z is a variable
& (d reaches n)
& (no other def of a reaches n)
& (no def of z on path from d to n)
transform:
(n : e ← a ⊕ b) => (n : e ← z ⊕ b)

Dataflow Analysis
Summary
Liveness analysis
• intermediate language
• control-flow graphs
• definition & algorithm
More dataflow analyses
Optimizations
46

learn more
Dataflow Analysis
Literature
Andrew W. Appel, Jens Palsberg: Modern Compiler Implementation in
Java, 2nd edition. 2002
47

Dataflow Analysis
Copyrights & Credits
48

copyrights
Dataflow Analysis
Pictures
Slide 1:
ink swirl by Graham Richardson, some rights reserved
Slide 25:
Seattle's Best Coffee by Dominica Williamson, some rights reserved
Slide 32:
Animal Ark Reno by Joel Riley, some rights reserved
50

Dataflow Analysis

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