Pointer analysis lecture 2
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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: 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


Andersen’s Analysis

  • Strong updates?

  • Initial state?


Why Flow-Insensitive Analysis?

  • Reduced space requirements

    • a single points-to solution

  • Reduced time complexity

    • no copying

      • individual updates more efficient

    • 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


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 }


Does Algorithm A Compute The Most Precise Solution?


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


Does Algorithm A Compute The Most Precise Solution?


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.


Computing RS(u)


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)


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;


Interpreting Branch Conditions


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


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.


Andersen’s Analysis:Further Optimizations

  • Cycle Elimination

    • Offline

    • Online

  • Pointer Variable Equivalence


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


Questions?


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