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# Pointer Analysis Lecture 2 - PowerPoint PPT Presentation

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 AnalysisLecture 2

G. Ramalingam

Microsoft Research, India

• 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: Vars -> 2Vars’ such that

R IdealMayPT(u) for every vertex u

Example(Flow-Sensitive Analysis)

1

x = &a;

y = x;

x = &b;

z = x;

x = &a

2

y = x

3

x = &b

4

z = x

5

Example:Andersen’s Analysis

1

x = &a;

y = x;

x = &b;

z = x;

x = &a

2

y = x

3

x = &b

4

z = x

5

• Initial state?

• 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

Example:Andersen’s Analysis

1

x = &a;

y = x;

y = &b;

b = &c;

x = &a

2

y = x

3

y = &b

4

b = &c

5

Example:Steensgaard’s Analysis

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

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 }

• 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

• 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.

Ideal-May-Point-To

more efficient / less precise

Algorithm A

more efficient / less precise

Andersen’s

more efficient / less precise

Steensgaard’s

• 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.

Ideal (with Int, with Malloc)

Ideal (with Int)

Ideal (with Malloc)

Ideal (no Int, no Malloc)

Algorithm A

Andersen’s

Steensgaard’s

• 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)

Dynamic Memory Allocation:Example

1

x = new;

y = x;

*y = &b;

*y = &a;

x = new

2

y = x

3

*y = &b

4

*y = &a

5

Dynamic Memory Allocation:Summary Object Update

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;

Conditional Control-Flow(In The Concrete Semantics)

• Encoding conditional-control-flow

• using “assume” statements

1

if (P) then

S1;

else

S2;

endif

assume !P

assume P

2

4

S1

S2

3

5

Conditional Control-Flow(In The Concrete Semantics)

• Semantics of “assume” statements

• DataState -> {true,false}

1

if (P) then

S1;

else

S2;

endif

assume !P

assume P

2

4

S1

S2

3

5

Abstracting “assume” statements

1

if (x != null) then

y = x;

else

endif

assume (x != null)

assume (x == null)

2

4

y = x

S2

3

5

Abstracting “assume” statements

2

assume x == y

3

• 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.

Andersen’s Analysis:Further Optimizations

• Cycle Elimination

• Offline

• Online

• Pointer Variable Equivalence

• 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.

• Compiler optimizations

• Verification & Bug Finding

• use in preliminary phases

• use in verification itself