# Pointer Analysis Lecture 2 - PowerPoint PPT Presentation

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Pointer Analysis Lecture 2. G. Ramalingam Microsoft Research, India. Andersen’s Analysis. A flow-insensitive analysis computes a single points-to solution valid at all program points ignores control-flow – treats program as a set of statements

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Pointer Analysis Lecture 2

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## Pointer AnalysisLecture 2

G. Ramalingam

Microsoft Research, India

### Andersen’s Analysis

• A flow-insensitive analysis

• computes a single points-to solution valid at all program points

• ignores control-flow – treats program as a set of statements

• equivalent to merging all vertices into one (and applying algorithm A)

• equivalent to adding an edge between every pair of vertices (and applying algo. A)

• a solution R such that

R IdealMayPT(u) for every vertex u

1

x = &a;

y = x;

x = &b;

z = x;

x = &a

2

y = x

3

x = &b

4

z = x

5

1

x = &a;

y = x;

x = &b;

z = x;

x = &a

2

y = x

3

x = &b

4

z = x

5

### Andersen’s Analysis

• Initial state?

### Why Flow-Insensitive Analysis?

• Reduced space requirements

• a single points-to solution

• Reduced time complexity

• no copying

• no need for joins

• number of iterations?

• a cubic-time algorithm

• Scales to millions of lines of code

• most popular points-to analysis

### Andersen’s AnalysisA Set-Constraints Formulation

• Compute PTx for every variable x

### Steensgaard’s Analysis

• Unification-based analysis

• Inspired by type inference

• an assignment “lhs := rhs” is interpreted as a constraint that lhs and rhs have the same type

• the type of a pointer variable is the set of variables it can point-to

• “Assignment-direction-insensitive”

• treats “lhs := rhs” as if it were both “lhs := rhs” and “rhs := lhs”

• An almost-linear time algorithm

• single-pass algorithm; no iteration required

1

x = &a;

y = x;

y = &b;

b = &c;

x = &a

2

y = x

3

y = &b

4

b = &c

5

1

x = &a;

y = x;

y = &b;

b = &c;

x = &a

2

y = x

3

y = &b

4

b = &c

5

### Steensgaard’s Analysis

• Can be implemented using Union-Find data-structure

• Leads to an almost-linear time algorithm

x = &a;

y = x;

y = &b;

b = &c;

*x = &d;

### May-Point-To Analyses

Ideal-May-Point-To

???

Algorithm A

more efficient / less precise

Andersen’s

more efficient / less precise

Steensgaard’s

### Ideal Points-To Analysis:Definition Recap

• A sequence of states s1s2 … snis said to be an execution (of the program) iff

• s1is the Initial-State

• si| si+1for 1 <= I < n

• A state s is said to be a reachable stateiff there exists some execution s1s2 … snis such that sn= s.

• RS(u) = { s | (u,s) is reachable }

• IdealMayPT (u) = { (p,x) | \$ s Î RS(u). s(p) == x }

• IdealMustPT (u) = { (p,x) | " s Î RS(u). s(p) == x }

### Ideal <-> Algorithm A

• Abstract away correlations between variables

• relational analysis vs.

• independent attribute

x: &y

y: &z

x: &b

y: &x

g

x: &b

x: &y

y: &x

y: &x

a

x: {&y,&b}

y: {&x,&z}

x: &y

y: &z

x: &b

y: &z

### Is The Precise Solution Computable?

• Claim: The set RS(u) of reachable concrete states (for our language) is computable.

• Note: This is true for any collecting semantics with a finite state space.

### Precise Points-To Analysis:Decidability

• Corollary: Precise may-point-to analysis is computable.

• Corollary: Precise (demand) may-alias analysis is computable.

• Given ptr-exp1, ptr-exp2, and a program point u, identify if there exists some reachable state at u where ptr-exp1 and ptr-exp2 are aliases.

• Ditto for must-point-to and must-alias

• … for our restricted language!

### Precise Points-To Analysis:Computational Complexity

• What’s the complexity of the least-fixed point computation using the collecting semantics?

• The worst-case complexity of computing reachable states is exponential in the number of variables.

• Can we do better?

• Theorem: Computing precise may-point-to is PSPACE-hard even if we have only two-level pointers.

### May-Point-To Analyses

Ideal-May-Point-To

more efficient / less precise

Algorithm A

more efficient / less precise

Andersen’s

more efficient / less precise

Steensgaard’s

### Precise Points-To Analysis: Caveats

• Theorem: Precise may-alias analysis is undecidable in the presence of dynamic memory allocation.

• Add “x = new/malloc ()” to language

• State-space becomes infinite

• Digression: Integer variables + conditional-branching also makes any precise analysis undecidable.

### May-Point-To Analyses

Ideal (with Int, with Malloc)

Ideal (with Int)

Ideal (with Malloc)

Ideal (no Int, no Malloc)

Algorithm A

Andersen’s

Steensgaard’s

### Dynamic Memory Allocation

• s: x = new () / malloc ()

• Assume, for now, that allocated object stores one pointer

• s: x = malloc ( sizeof(void*) )

• Introduce a pseudo-variable Vs to represent objects allocated at statement s, and use previous algorithm

• treat s as if it were “x = &Vs”

• also track possible values of Vs

• allocation-site based approach

• Key aspect: Vs represents a set of objects (locations), not a single object

• referred to as a summary object (node)

1

x = new;

y = x;

*y = &b;

*y = &a;

x = new

2

y = x

3

*y = &b

4

*y = &a

5

### Dynamic Memory Allocation:Object Fields

• Field-sensitive analysis

class Foo {

A* f;

B* g;

}

s: x = new Foo()

x->f = &b;

x->g = &a;

### Dynamic Memory Allocation:Object Fields

• Field-insensitive analysis

class Foo {

A* f;

B* g;

}

s: x = new Foo()

x->f = &b;

x->g = &a;

### Other Aspects

• Context-sensitivity

• Indirect (virtual) function calls and call-graph construction

• Pointer arithmetic

• Object-sensitivity

### Andersen’s Analysis:Further Optimizations and Extensions

• Fahndrich et al., Partial online cycle elimination in inclusion constraint graphs, PLDI 1998.

• Rountev and Chandra, Offline variable substitution for scaling points-to analysis, 2000.

• Heintze and Tardieu, Ultra-fast aliasing analysis using CLA: a million lines of C code in a second, PLDI 2001.

• M. Hind, Pointer analysis: Haven’t we solved this problem yet?, PASTE 2001.

• Hardekopf and Lin, The ant and the grasshopper: fast and accurate pointer analysis for millions of lines of code, PLDI 2007.

• Hardekopf and Lin, Exploiting pointer and location equivalence to optimize pointer analysis, SAS 2007.

• Hardekopf and Lin, Semi-sparse flow-sensitive pointer analysis, POPL 2009.

### Context-Sensitivity Etc.

• Liang & Harrold, Efficient computation of parameterized pointer information for interprocedural analyses. SAS 2001.

• Lattner et al., Making context-sensitive points-to analysis with heap cloning practical for the real world, PLDI 2007.

• Zhu & Calman, Symbolic pointer analysis revisited. PLDI 2004.

• Whaley & Lam, Cloning-based context-sensitive pointer alias analysis using BDD, PLDI 2004.

• Rountev et al. Points-to analysis for Java using annotated constraints. OOPSLA 2001.

• Milanova et al. Parameterized object sensitivity for points-to and side-effect analyses for Java. ISSTA 2002.

### Applications

• Compiler optimizations

• Verification & Bug Finding

• use in preliminary phases

• use in verification itself

4

*y = &a

5

### Abstract Transformers:Weak/Strong Update

AS[stmt] : AbsDataState -> AbsDataState

AS[ *x = y ] s =

s[z  s(y)] if s(x) = {z}

s[z1 s(z1) s(y)] if s(x) = {z1, …, zk}

[z2s(z2) s(y)] (where k > 1)

[zks(zk) s(y)]

### Correctness & Precision

• How can we formally reason about the correctness & precision of abstract transformers?

• Can we systematically derive a correct abstract transformer?

### Enter: The French Recipe(Abstract Interpretation)

• Concrete Domain

• Concrete states: C

• Semantics: For every statement st,

• CS[st] : C -> C

• AbstractDomain

• Asemi-lattice(A, )

• TransferFunctions

• For every statement st,

• AS[st] : A -> A

• Concrete (Collecting) Domain

• Asemi-lattice(2C, )

• TransferFunctions

• For every statement st,

• CS*[st] : 2C -> 2C

a

g

2Data-State

2Var x Var’

### Points-To Analysis(Abstract Interpretation)

a(Y) = { (p,x) | exists s in Y. s(p) == x }

MayPT(u)

a

Í

a

RS(u)

IdealMayPT(u)

2Data-State

2Var x Var’

IdealMayPT (u) = a ( RS(u) )

### Approximating Transformers:Correctness Criterion

c is said to be correctly

approximated by a

iff

a(c) Ía

c1

correctly

approximated by

a1

f

c2

f#

correctly

approximated by

a2

C

A

### Approximating Transformers:Correctness Criterion

concretization

g

c1

a1

f

abstraction

a

c2

f#

a2

requirement:

f#(a1) ≥a (f( g(a1))

C

A

### Concrete Transformers

• CS[stmt] : Data-State -> Data-State

• CS[ x = y ] s = s[x s(y)]

• CS[ x = *y ] s = s[x s(s(y))]

• CS[ *x = y ] s = s[s(x) s(y)]

• CS[ x = null ] s = s[x null]

• CS*[stmt] : 2Data-State -> 2Data-State

• CS*[st] X = { CS[st]s | s Î X }

### Abstract Transformers

• AS[stmt] : AbsDataState -> AbsDataState

• AS[ x = y ] s = s[x s(y)]

• AS[ x = null ] s = s[x {null}]

• AS[ x = *y ] s = s[x s*(s(y))]

where s*({v1,…,vn}) = s(v1) È … È s(vn)

• AS[ *x = y ] s = ???

x: &y

y: &x

z: &a

g

x: {&y}

y: {&x,&z}

z: {&a}

x: &y

y: &z

z: &a

f

f#

*y = &b;

*y = &b;

x: &b

y: &x

z: &a

x: {&y,&b}

y: {&x,&z}

z: {&a,&b}

x: &y

y: &z

z: &b

a

x: &y

y: &x

z: &a

g

x: {&y}

y: {&x,&z}

z: {&a}

x: &y

y: &z

z: &a

f

f#

*x = &b;

*x = &b;

x: &y

y: &b

z: &a

x: {&y}

y: {&b}

z: {&a}

x: &y

y: &b

z: &a

a

4

*y = &a

5