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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

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

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: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}

[z2s(z2) s(y)] (where k > 1)

…

[zks(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

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