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Another example p := &x; *p := 5 y := x + 1;

Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

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Page 1: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Another example

p := &x;*p := 5y := x + 1;

Page 2: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Another example

p := &x;*p := 5y := x + 1;

x := 5;*p := 3y := x + 1; ???

Page 3: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Another example

for j := 1 to N for i := 1 to M a[i] := a[i] + b[j]

Page 4: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Another example

for j := 1 to N for i := 1 to M a[i] := a[i] + b[j]

Page 5: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Another example

area(h,w) { return h * w }

h := ...;w := 4;a := area(h,w)

Page 6: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Another example

area(h,w) { return h * w }

h := ...;w := 4;a := area(h,w)

Page 7: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Optimization themes

• Don’t compute if you don’t have to– unused assignment elimination

• Compute at compile-time if possible– constant folding, loop unrolling, inlining

• Compute it as few times as possible– CSE, PRE, PDE, loop invariant code motion

• Compute it as cheaply as possible– strength reduction

• Enable other optimizations– constant and copy prop, pointer analysis

• Compute it with as little code space as possible– unreachable code elimination

Page 8: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Dataflow analysis

Page 9: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Dataflow analysis: what is it?

• A common framework for expressing algorithms that compute information about a program

• Why is such a framework useful?

Page 10: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Dataflow analysis: what is it?

• A common framework for expressing algorithms that compute information about a program

• Why is such a framework useful?

• Provides a common language, which makes it easier to:– communicate your analysis to others– compare analyses– adapt techniques from one analysis to another– reuse implementations (eg: dataflow analysis

frameworks)

Page 11: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Control Flow Graphs

• For now, we will use a Control Flow Graph representation of programs– each statement becomes a node– edges between nodes represent control flow

• Later we will see other program representations– variations on the CFG (eg CFG with basic blocks)– other graph based representations

Page 12: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

x := ...

x := ...y := ...y := ...p := ...if (...) { ... x ... x := ... ... y ...}else { ... x ... x := ... *p := ...}... x ...... y ...y := ...

y := ...

y := ...

p := ...

... x ...

x := ...

... y ...

... x ...

x := ...

*p := ...

... x ...

... x ...

y := ...

if (...)

Example CFG

Page 13: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

An example DFA: reaching definitions

• For each use of a variable, determine what assignments could have set the value being read from the variable

• Information useful for:– performing constant and copy prop– detecting references to undefined variables– presenting “def/use chains” to the programmer– building other representations, like the DFG

• Let’s try this out on an example

Page 14: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...

x := ...

y := ...

y := ...

p := ...

... x ...

x := ...

... y ...

... x ...

x := ...

*p := ...

... x ...

... x ...

y := ...

if (...)

Visual sugar

Page 15: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...

Page 16: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...

Page 17: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Safety

• When is computed info safe?

• Recall intended use of this info:– performing constant and copy prop– detecting references to undefined variables– presenting “def/use chains” to the programmer– building other representations, like the DFG

• Safety:– can have more bindings than the “true” answer, but

can’t miss any

Page 18: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Reaching definitions generalized

• DFA framework is geared towards computing information at each program point (edge) in the CFG– So generalize the reaching definitions problem by

stating what should be computed at each program point

• For each program point in the CFG, compute the set of definitions (statements) that may reach that point

• Notion of safety remains the same

Page 19: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Reaching definitions generalized

• Computed information at a program point is a set of var ! stmt bindings– eg: { x ! s1, x ! s2, y ! s3 }

• How do we get the previous info we wanted?– if a var x is used in a stmt whose incoming info is in,

then:

Page 20: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Reaching definitions generalized

• Computed information at a program point is a set of var ! stmt bindings– eg: { x ! s1, x ! s2, y ! s3 }

• How do we get the previous info we wanted?– if a var x is used in a stmt whose incoming info is in,

then: { s | (x ! s) 2 in }

• This is a common pattern– generalize the problem to define what information

should be computed at each program point– use the computed information at the program points

to get the original info we wanted

Page 21: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...

Page 22: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...

Page 23: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Using constraints to formalize DFA

• Now that we’ve gone through some examples, let’s try to precisely express the algorithms for computing dataflow information

• We’ll model DFA as solving a system of constraints

• Each node in the CFG will impose constraints relating information at predecessor and successor points

• Solution to constraints is result of analysis

Page 24: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Constraints for reaching definitions

s: x := ...

in

out

s: *p := ...

in

out

Page 25: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Constraints for reaching definitions

• Using may-point-to information:

out = in [ { x ! s | x 2 may-point-to(p) }

• Using must-point-to aswell:

out = in – { x ! s’ | x 2 must-point-to(p) Æ s’ 2 stmts } [ { x ! s | x 2 may-point-to(p) }

s: x := ...

in

out

s: *p := ...

in

out

out = in – { x ! s’ | s’ 2 stmts } [ { x ! s }

Page 26: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Constraints for reaching definitions

s: if (...)

in

out[0] out[1]

merge

out

in[0] in[1]

Page 27: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Constraints for reaching definitions

s: if (...)

in

out[0] out[1] more generally: 8 i . out [ i ] = in

out [ 0 ] = in Æout [ 1 ] = in

merge

out

in[0] in[1]

more generally: out = i in [ i ]

out = in [ 0 ] [ in [ 1 ]

Page 28: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Flow functions

• The constraint for a statement kind s often have the form: out = Fs(in)

• Fs is called a flow function– other names for it: dataflow function, transfer function

• Given information in before statement s, Fs(in) returns information after statement s

• Other formulations have the statement s as an explicit parameter to F: given a statement s and some information in, F(s,in) returns the outgoing information after statement s

Page 29: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Flow functions, some issues

• Issue: what does one do when there are multiple input edges to a node?

• Issue: what does one do when there are multiple outgoing edges to a node?

Page 30: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Flow functions, some issues

• Issue: what does one do when there are multiple input edges to a node?– the flow functions takes as input a tuple of values, one

value for each incoming edge

• Issue: what does one do when there are multiple outgoing edges to a node?– the flow function returns a tuple of values, one value

for each outgoing edge– can also have one flow function per outgoing edge

Page 31: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Flow functions

• Flow functions are a central component of a dataflow analysis

• They state constraints on the information flowing into and out of a statement

• This version of the flow functions is local– it applies to a particular statement kind– we’ll see global flow functions shortly...

Page 32: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Summary of flow functions

• Flow functions: Given information in before statement s, Fs(in) returns information after statement s

• Flow functions are a central component of a dataflow analysis

• They state constraints on the information flowing into and out of a statement

Page 33: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

if(...)

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

merge

... x ...

... y ...

8: y := ...

d0

d1

d2

d3

d5

d6

d7

d9

d10

d11

d13

d14

d15

d16

d12

d4

d8

Back to exampled1 = Fa(d0)

d2 = Fb(d1)

d3 = Fc(d2)

d4 = Fd(d3)

d5 = Fe(d4)

d6 = Fg(d5)

d7 = Fh(d6)

d8 = Fi(d7)

d10 = Fj(d9)

d11 = Fk(d10)

d12 = Fl(d11)

d9 = Ff(d5)

d13 = Fm(d12, d8)

d14 = Fn(d13)

d15 = Fo(d14)

d16 = Fp(d15)

How to find solutions for di?

Page 34: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

How to find solutions for di?

• This is a forward problem– given information flowing in to a node, can determine

using the flow function the info flow out of the node

• To solve, simply propagate information forward through the control flow graph, using the flow functions

• What are the problems with this approach?

Page 35: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

if(...)

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

merge

... x ...

... y ...

8: y := ...

d0

d1

d2

d3

d5

d6

d7

d9

d10

d11

d13

d14

d15

d16

d12

d4

d8

First problemd1 = Fa(d0)

d2 = Fb(d1)

d3 = Fc(d2)

d4 = Fd(d3)

d5 = Fe(d4)

d6 = Fg(d5)

d7 = Fh(d6)

d8 = Fi(d7)

d10 = Fj(d9)

d11 = Fk(d10)

d12 = Fl(d11)

d9 = Ff(d5)

d13 = Fm(d12, d8)

d14 = Fn(d13)

d15 = Fo(d14)

d16 = Fp(d15)

What about the incoming information?

Page 36: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

First problem

• What about the incoming information?– d0 is not constrained

– so where do we start?

• Need to constrain d0

• Two options:– explicitly state entry information– have an entry node whose flow function sets the

information on entry (doesn’t matter if entry node has an incoming edge, its flow function ignores any input)

Page 37: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Entry node

s: entry

out

out = { x ! s | x 2 Formals }

Page 38: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

if(...)

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

merge

... x ...

... y ...

8: y := ...

d0

d1

d2

d3

d5

d6

d7

d9

d10

d11

d13

d14

d15

d16

d12

d4

d8

Second problemd1 = Fa(d0)

d2 = Fb(d1)

d3 = Fc(d2)

d4 = Fd(d3)

d5 = Fe(d4)

d6 = Fg(d5)

d7 = Fh(d6)

d8 = Fi(d7)

d10 = Fj(d9)

d11 = Fk(d10)

d12 = Fl(d11)

d9 = Ff(d5)

d13 = Fm(d12, d8)

d14 = Fn(d13)

d15 = Fo(d14)

d16 = Fp(d15)

d0 = Fentry()

Which order to process nodes in?

Page 39: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Second problem

• Which order to process nodes in?

• Sort nodes in topological order– each node appears in the order after all of its

predecessors

• Just run the flow functions for each of the nodes in the topological order

• What’s the problem now?

Page 40: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

Second problem, prime

• When there are loops, there is no topological order!

• What to do?

• Let’s try and see what we can do

Page 41: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...

Page 42: Another example p := &x; *p := 5 y := x + 1;. Another example p := &x; *p := 5 y := x + 1; x := 5; *p := 3 y := x + 1; ???

1: x := ...

2: y := ...

3: y := ...

4: p := ...

... x ...

5: x := ...

... y ...

... x ...

6: x := ...

7: *p := ...

... x ...

... y ...

8: y := ...


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