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 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

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
Exercise

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)
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;

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
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 = ???
Algorithm A: TranformersWeak/Strong Update

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

Algorithm A: TranformersWeak/Strong Update

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