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Approximating Inclusion-based Points-to Analysis

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Rupesh Nasre.

Department of Computer Science and Automation,

Indian Institute of Science, Bangalore, India

MSPC 2011

June 05, 2011

Improved runtime.

Parallelizing compiler.

Lock synchronizer.

Memory leak detector.

Secure code.

Pointer Analysis.

Data flow analyzer.

String vulnerability finder.

Better compile time.

Affine expression analyzer.

Type analyzer.

Program slicer.

Better debugging.

p

p

q

q

- p = &q address-of
- p = q copy
- p = *q load
- *p = q store

p

q

p

q

p

q

p

q

p

q

p

q

Points-to Analysis

...

a = &x

c = b

d = *b

*b = a

...

Program

Points-to Sets

main ( ) {

if (...) {

...

}

}

...

a → {x,y}

b → {a,z}

c → {a,z,x}

...

- Online cycle elimination (Fahndrich et al., 1998)
- Offline variable substitution(Rountev and Chandra, 2000)
- Pointer and location equivalence (Hardekopf and Lin, 2007)

These optimizations preserve the precision of the underlying analysis.

x1

x1

x2

x2

x3

x3

P1

P2

P1, P2

x4

x4

x5

x5

x6

x6

x7

x7

Original points-to sets

Modified points-to sets

P1

P1

P2

P2

P3

P3

x1

x2

X1, X2

P4

P4

P5

P5

P6

P6

P7

P7

Original points-to sets

Modified points-to sets

- Cubic time complexity.
- High absolute running times.
- Approximations are inevitable for scalability.

Approximate Pointer Equivalence (APE)

x1

x1

x2

x2

x3

x3

P1

P2

P1, P2

x4

x4

x5

x5

x6

x6

x7

x7

Original points-to sets

Approximate points-to sets

Approximate Location Equivalence (ALE)

P1

P1

P2

P2

P3

P3

x1

x2

X1, X2

P4

P4

P5

P5

P6

P6

P7

P7

Original points-to sets

Approximate points-to sets

- Approximate pointer and location equivalence
- Sound algorithm to compute APE and ALE online
- Optimizations:
- Proximity merge
- Eager/lazy merging
- Merge order
- Equivalence identification frequency

- Extensive empirical evaluation

Pointers P1 and P2 are approximately pointer equivalent with similarity αif sim(ptsto(P1), ptsto(P2)) ≥α.

Objects x1 and x2 are approximately location equivalent with similarity βif sim(ptdby(x1), ptdby(x2)) ≥ β.

s1 Ո s2

sim(s1, s2) =

s1 Ս s2

ptsto(p1) = {x,y,z}ptdby(x) = {p1, p3}

ptsto(p2) = {y,z,w}ptdby(y) = {p1, p2}

ptsto(p3) = {x,w}ptdby(z) = {p1, p2}

ptdby(w) = {p2, p3}

α= 0.5

p1 and p2 are APE with similarity 2/4 = 0.5

p1 and p3 are not APE with similarity 1/4 = 0.25

β= 0.7

y and z are ALE with similarity 2/2 = 1.0

x and w are not ALE with similarity 1/3 = 0.33

Input: set of constraints, α, β.

Process address-of constraints

Add edges to constraint graph G using copy edges

repeat

Propagate points-to information in G

for each variable pair (x,y) do

simα = sim(ptsto(x), ptsto(y))

simβ = sim(ptdby(x), ptdby(y))

if simα ≥ αor simβ ≥ βthen

merge(x,y)

end if

end for

Add edges to G using load and store constraints

until fixed point

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

a {u,v,w,x}

0: Processing address-of constraints

p {a}

b {y}

q {b}

c {z}

d { }

r {a}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

a {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

p {a}

b {y}

q {b}

c {z}

d { }

r {a}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

a {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

p {a}

b {y}

q {b}

c {z}

d {u,v,w,x}

r {a}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

p {a}

b {y}

q {b}

c {z}

r {a}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,w,x}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,[w,x]}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x]}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x]}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

[p,r] {a}

b {y}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

[p,r] {a}

b {y,z}

q {b}

c {z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

α = 0.5, β = 0.7

[a,d] {u,v,[w,x],z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

2: merge(b, c)

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],y,z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

2: merge(b, c)

3: Propagate points-to information

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],y,z}

0: Processing address-of constraints

0: Processing copy constraints

1: Propagate points-to information

1: merge(a, d)

1: merge(p, r)

1: merge(w, x)

1: Processing load/store constraints

2: Propagate points-to information

2: merge(b, c)

3: Propagate points-to information

Fixed point

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Input constraints:

a = &u, a = &v, a = &w, a = &x, b = &y, c = &z, u = &v, v = &u,

p = &a, q = &b, r = &a, d = a, *p = c, *q = c

[a,d] {u,v,[w,x],y,z}

Exact analysis: 18 points-to pairs

Approximate analysis: 19 points-to pairs

[p,r] {a}

q {b}

[b,c] {y,z}

v {u}

u {v}

Constraint graph

Similarity of a node is checked against another that is at most k-reachable.

0

1

1

2

2

2

3

3

A proximity of k=10 gives a 38% improvement in analysis time and the precision loss is only 2.6%.

When to merge matters.

Example:

α = 0.5

ptsto(p1) = {a,b}, ptsto(p2) = {b,c}, ptsto(p3) = {c,d},

ptsto(p4) = {d,e}, ptsto(p5) = {e,f}, ptsto(p6) = {f,g}

Eager merging: [p1,p2], [p3,p4], [p5,p6]

Lazy merging: [p1,p2,p3,p4,p5,p6]

Lazy merging improves analysis time over eager merging by 8%, but reduces precision.

Merge order matters.

Example:

α = 0.5

ptsto(a) = {x,y,z}, ptsto(b) = {w,y,z}, ptsto(c) = {w,x}

Order (a,c), (b,c), (a,b) merges a and b.

Order (a,b), (a,c), (b,c) merges all a, b and c.

We arrange nodes in non-increasing and non-decreasing similarities and find that the former reduces analysis time by 10% while the latter requires 8% more time than not ordering the variables.

Precision improves non-linearly with α.

Precision improves almost linearly with β.

A client provides MAXMEM and the analysis updates α, β to choose the maximum amount of precision possible.

Note: observations are averaged across benchmarks.

- APE is more important that ALE
- Proximity merging helps the analysis scale.
- Lazy vs eager merging, order of merging offer nice trade-offs.
- Client can specify a MAXMEM to make its judicious use.

Rupesh Nasre.

Department of Computer Science and Automation,

Indian Institute of Science, Bangalore, India

MSPC 2011

June 05, 2011