Program AnalysisLast Lesson

Mooly Sagiv

Goals

Show the significance of set constraints forCFA of Object Oriented Programs

Sketch advanced techniques Summarize the course Get some feedback

A Motivating Exampleclass Vehicle Object { int position = 10; void move(x1 : int) { position = position + x1 ;}}class Car extends Vehicle { int passengers;

void await(v : Vehicle) { if (v.position < position) then v.move(position - v.position); else self.move(10); }}class Truck extends Vehicle {

void move(x2 : int) { if (x2 < 55) position = position + x2; }}void main { Car c; Truck t; Vehicle v1;

new c; new t; v1 := c;c.passengers := 2;c.move(60);v1.move(70);c.await(t) ;}

A Motivating Exampleclass Vehicle Object { int position = 10; void move(x1 : int) { position = position + x1 ;}}class Car extends Vehicle { int passengers;

void await(v {Truck} : Vehicle) { if (v {Truck} .position < position) then v {Truck}.move(position - v.position); else self {Car}.move(10); }}class Truck extends Vehicle {

void move(x2 : int) { if (x2 < 55) position = position + x2; }}void main { Car c; Truck t; Vehicle v1;

new c {Car} ; new t {Truck} ; v1 {Car} := c {Car} ;c {Car} .passengers := 2;c {Car} .move(60);v1 {Car}.move(70);c {Car} .await(t {Truck} ) ;}

Flow Insensitive Class Analysis

Determine the set of potential classes of every variable at every program point

Compute a mapping from variables into a set of class names

Combine values of variables at different points Generate a set of constraints for every statement Find a minimal solution

A Motivating Exampleclass Vehicle Object { int position = 10; void move(x1 : int) { position = position + x1 ;}}class Car extends Vehicle { int passengers;

void await(v1 : Vehicle) { if (v1.position < position) then v1.move(position - v1.position); else self.move(10); }}class Truck extends Vehicle {

void move(x2 : int) { if (x2 < 55) position = position + x2; }}void main { Car c; Truck t; Vehicle v2;

new c; new t; v2 := c;c.passengers := 2;c.move(60);v2.move(70);c.await(t) ;

}

{Car} (c){Truck} (t)(c) (v2)

{Car} (c) (t) (v1)

Class Analysis Summary

Resolve called function Can also perform type inference and checking Can be used to warn against programmer errors

at compile-time

Set Constraints Summary Can be used to generate a flow sensitive solution Can also handle sets of “terms”

– Finite set of constructors C={b, c, …}

– Finite set of variables

– Set expressionsE ::= | variable | E1 E2 | E1 E2 | c(E1 , E2 ,…, Ek )| c-i(E)

– Finite set of inequalitiesE1 E2

– Find the least solution (or a symbolic representation)

Advanced Abstract Interpretation Techniques Origin [Cousot&Cousot POPL 1979]

Download from the course homepage Widening & Narrowing Combining dataflow analysis problems Semantic reductions ...

Widening

Accelerate the termination of Chaotic iterations by computing a more conservative solution

Can handle lattices of infinite heights

Example Interval Analysis Find a lower and an upper bound of the value of a

variable Lattice L = (ZZ, , , , ,)

– [a, b] [c, d] if c a and d b– [a, b] [c, d] = [min(a, c), max(b, d)]

– [a, b] [c, d] = [max(a, c), min(b, d)] = =

Programx := 1 ;while x 1000 do x := x + 1;

Widening for Interval Analysis [c, d] = [c, d] [a, b] [c, d] = [

if a cthen aelse if 0 c

then 0 else minint,

if b dthen belse if d 0

then 0else maxint

Chaotic Iterations

for forward problems+ for l Lab* do

DFentry(l) := DFexit(l) :=

DFentry(init(S*)) := WL= Lab*

while WL != do Select and remove an arbitrary l WL

if (temp != DFexit(l))

DFexit(l) := DFexit(l) temp for l' such that (l,l') flow(S*) do DFentry(l') := DFentry(l') DFexit(l) WL := WL {l’}

))(( lDFftemp entryl

Example

[x := 1]1 ;

while [x 1000]2 do [x := x + 1]3;

Requirements on Widening

For all elements l1 l2 l1 l2

For all ascending chains l0 l1 l2 …the following sequence is finite– y0 = l0

– yi+1 = yi li+1

Narrowing

Improve the result of widening

Example

[x := 1]1 ;

while [x 1000]2 do [x := x + 1]3;

Widening and Narrowing Summary

Very simple but produces impressive precision The McCarthy 91 function

Also useful in the finite case Can be used as a methodological tool But not widely accepted

int f(x)if x > 100

then return x -10else return f(f(x+11))

Combining dataflow analysis problems

How to combine different analyses The result can be more precise than both! On some programs more efficient too Many possibly ways to combine (4.4) A simple example sign+parity analysis

x := x - 1

Cartezian Products Analysis 1

– Lattice (L1, 1, 1, 1, 1,1)

– Galois connection 1: P(States) L1 1: L1 P(States)

– Transfer functionsop1:L1 L1

Analysis 2

– Lattice (L2, 2, 2, 2, 2,2)

– Galois connection2: P(States) L2 1: L2 P(States)

– Transfer functionsop2:L2 L2

Combined Analysis

– L = (L1 L2, ) where (l1, l2) (u1, u2) if l1 1 u1 and l2 2 u2

– Galois connection

– Transfer functions

Course Summary Techniques Studied

– Operational Semantics

– Dataflow Analysis and Monotone Frameworks (Imperative Programs)

– Control Flow Analysis and Set Constraints (Functional Programs)

Techniques Sketched– Abstract interpretation

– Interprocedural Analysis

– Type and effect systems

Not Covered– Efficient algorithms

– Applications in compilers

– Logic programming

Course Summary

Able to understand advanced static analysis techniques

Find faults in existing algorithms Be able to develop new algorithms Gain a better understanding of programming

languages– Functional Vs. Imperative

– Operational Semantics

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