Pointer Analysis Lecture 2

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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

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
• 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
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 }
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)
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: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

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