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Shape Analysis for Low-level Code

Hongseok Yang(Seoul National University)

(Joint work with Cristiano Calcagno, Dino Distefano and Peter O’Hearn)

Dream

Automatically verify the memory safety of systems code, such as device derivers and memory managers.

Challenges: 1. Pointer arithmetic.2. Scalability.3. Concurrency.

Our Analyzer Handles programs for dynamic memory

management. Experimental results (Pentium

3.2GHz,4GB)Found a hidden assumption of the K&R memory manager. These are “fixed” versions.

Proved memory safety and even partial correctness.

Sample Analysis Result

Program: ans = malloc_bestfit_acyclic(n);Precondition: n¸2 Æ mls(freep,0)

Postcondition: (ans=0 Æ n¸2 Æ mls(freep,0)) Ç(n¸2 Æ nd(ans,q’,n) * mls(freep,0)) Ç(n¸2 Æ nd(ans,q’,n) * mls(freep,q’) * mls(q’,0))

Hidden Assumption in K&R Malloc/Free

0 220

Global Vars Stack Heap

Hidden Assumption in K&R Malloc/Free

0 220

Global VarsStack Heap

Multiword Lists

24

515 3 18 3 nil 2

lp 15 18

24

Link Field Size Field

Coalescing

24 515 3 18 3 nil 25 15 18 24

p = lp; while (p!=0) { local q = *p; if (p + *(p+1) == q) { *(p+1) = *(p+1) + *(q+1); *p = *q; } else { p = q; } }

p

Coalescing

24 515 3 18 3 nil 2

15 18 24


5p

Coalescing

24 515 3 18 3 nil 2

15 18 24


5p q

Coalescing

24 515 3 18 8 nil 2

15 18 24


5p q

Coalescing

24 515 3 24 8 nil 2

15 18 24


5p q

Coalescing

15 3 24 8 nil 2

15 24


5p

Coalescing

15 3 24 8 nil 2

15 24


5p=0

Nodeful High-level View

Nodeful High-level View

Nodeless Low-level View

Complex numerical relationships are used only for reconstructing a high-

level view.

Separation Logic blk(p+2,p+5)

nd(p,q,5) =def (pq) * (p+15) * blk(p+2,p+5)

mls(p,q)

p+2 p+5

p+5

5q

p

3 4 2

qp

Symbolic Heaps

9x’,y’. (P1 Æ P2 Æ … Æ Pn) Æ (H1 * H2 * … * Hm)

whereP ::= E=F | E·F | E!=F | …H ::= EF | blk(E,F) | mls(E,F) | nd(E,F,G) |…

Abstract Domain

P(CanSymH)>,µ

Pfin(SymH)>,µ

P(Emb) P(Abs)

y=x+z Æ x y*x+1 z*blk(x+2,0)*mls(y,0)

nd(x,y,z) * mls(y,0)

{Q1, Q2, … ,Qn}

{T1,T2,…,Tn}

Our Analysis

while(B) { C;

}

{T1,T2,…,Tn}

{ T’1,T’2,…,T’m}

{Q1, Q2, … ,Qn}

Nodeful View:

P(CanSymH)>

Nodeless View:

Pfin(SymH)>

{Q’1, Q’2, … ,Q’m}

Emb; Rearrangement

Abstraction

Sym. Execution

Our Analysis

while(B) { C;

}

{T1,T2,…,Tn}

{ T’1,T’2,…,T’m}

{Q1, Q2, … ,Qn}

Nodeful View:

P(CanSymH)>

Nodeless View:

Pfin(SymH)>

{Q’1, Q’2, … ,Q’m}

Analysis

«C¬ : Pfin(SymH)> ! Pfin(SymH)>

«A¬d = P(SymExec(A) o Rearrange(A))d «while b C¬d = FixComp(P(Abs) o F)

where F : P(CanSymHeaps) ! P(CanSymHeaps) F(d’) = P(Abs)(d [ «C¬d’)

Analysis


«A¬d = (P(SymExec(A)) o lift(Rearrange(A)))d «while b C¬d = FixComp(P(Abs) o F)

where F : P(CanSymHeaps) ! P(CanSymHeaps) F(d’) = P(Abs)(d [ «C¬d’)

SymExec(A) :

Proof Rules in Sep. Log.

Rearrange(A) :

Unrolling of mls and nd

Analysis


«A¬d = (P(SymExec(A)) o lift(Rearrange(A)))d «while b C¬d = FixComp(F)

where F : P(CanSymH)> ! P(CanSymH)>

F(d’) = P(Abs)(d [ («C¬o P(Emb))d’)

Emb: CanSymH !SymH Abs : SymH ! CanSymH

Information Loss

Widened Differential Fixpoint Algorithm

Abstraction Function Abs

Abs : SymH ! CanSymH

1. Package all nodes.2. Drop numerical relationships.3. Combine two connected multiword lists.(5 · x+x Æ p+3=z’) Æ(p q’ * p+1 3 * blk(p+2,z’) * mls(q’,0))



1. Package all nodes.2. Drop numerical relationships.3. Combine two connected multiword lists.(5 · x+x Æ p+3=z’) Æ(nd(p,q’,3) * mls(q’,0))




(5 · x+x Æ p+3=z’) Æ(nd(p,q’,3) * mls(q’,0) * r 4)




(5 · x+x Æ p+3=z’) Æ(nd(p,q’,3) * mls(q’,0) * true)



1. Package all nodes.2. Drop numerical relationships.3. Combine two connected multiword lists. (nd(p,q’,3) * mls(q’,0))



1. Package all nodes.2. Drop numerical relationships.3. Combine two connected multiword lists. mls(p,0)



1. Package all nodes.2. Drop numerical relationships.3. Combine two connected multiword lists.Precondition: true

… (xx’,s) * blk(x+2,x+s) Ã … nd(x,x’,s)

x’ s

x x+2 x+s

x’ s

x x+s



1. Package all nodes.2. Drop numerical relationships.3. Combine two connected multiword lists.Precondition: s = s’+i

… (xx’,s) * blk(x+2,x+i) * nd(x+i,y’,s’) Ã … nd(x,x’,s)

y’ s’x’ s

x x+2 x+i x+i+s’

x’ s

x x+s

Coalescing while (p!=0){local q=p*;

if (p + *(p+1) == q) {

*(p+1) = *(p+1) + *(q+1);

*p = *q; } else {

p = *p;

} }

mls(lp,p) * mls(p,0)…

p!=0 Æ p+s’=q Æ mls(lp,p)*p q,s’ * blk(p+2,p+s’) * mls(q,0)

p!=0Æp+s’=qÆmls(lp,p)* pq,s’+t’ * blk(p+2,p+s’) *qr’,t’*blk(q+2,q+t’)*mls(r’,0)

p!=0Æp+s’=qÆmls(lp,p)*pr’,s’+t’*blk(p+2,p+s’)*qr’,t’*blk(q+2,q+t’)*mls(r’,0)


p!=0 Æ mls(lp,p) * p q,s’ * blk(p+2,p+s’) * mls(q,0)


if (p + *(p+1) == q) {

*(p+1) = *(p+1) + *(q+1);

*p = *q; } else {

p = *p;

} }





p!=0Æp+s’=q’Æmls(lp,p)*pr’,s’+t’*blk(p+2,p+s’)*q’r’,t’*blk(q’+2,q’+t’)*mls(r’,0)



if (p + *(p+1) == q) {

*(p+1) = *(p+1) + *(q+1);

*p = *q; } else {

p = *p;

} }





p!=0Æp+s’=q’Æmls(lp,p)*pr’,s’+t’*blk(p+2,p+s’)*nd(q’,r’,t’) *mls(r’,0)



if (p + *(p+1) == q) {

*(p+1) = *(p+1) + *(q+1);

*p = *q; } else {

p = *p;

} }





p!=0Æp+s’=q’Æmls(lp,p)*nd(p,r’,s’+t’)* *mls(r’,0)



if (p + *(p+1) == q) {

*(p+1) = *(p+1) + *(q+1);

*p = *q; } else {

p = *p;

} }





mls(lp,p)*nd(p,r’,s’+t’)* *mls(r’,0)



if (p + *(p+1) == q) {

*(p+1) = *(p+1) + *(q+1);

*p = *q; } else {

p = *p;

} }





mls(lp,p)*mls(p,0)


Theorem Prover for “Q1 ` Q2”

without prover with prover

malloc_K&R about 20 hours 502.23 secs

free_K&R 23.844 secs 9.69 secs

Put Prover inside Hoare Powerdomain?

Q1 ` Q2, Q3 ` Q4

{Q1, Q2, Q3, Q4}

x0 = {}

x1 = F(x0) = {Q1, Q2, Q4}

x2 = F(x1) = {Q1, Q2, Q3, Q4}

P(CanSymH), µ vs. PH(CanSymH), v

{Q2, Q3} v

But, works only when ` is transitive.

Put Prover inside Hoare Powerdomain?

Q1 ` Q2, Q2 ` Q3, Q3 ` Q1

x0 = {}

x1 = F(x0) = {Q1, Q2}

x2 = F(x1) = {Q2, Q3}

x3 = F(x2) = {Q3, Q1}

x4 = F(x3) = {Q1, Q2}

P(CanSymH), µ vs. PH(CanSymH), v

But, works only when ` is transitive.

Put Prover inside Widening!

r : P(CanSymH) £ P(CanSymH) ! P(CanSymH)

x0r x1 =def x0 [ { Q 2 x1 | 8Q’ 2 x0. Q ` Q’ }

x0 = {}

x1 = x0 r F(x0)

x2 = x1 r F(x1)

xn+1 = xn r F(xn)

…x0 µ x1 µ x2 µ x3 …

Add Differencing

F : P(CanSymH) ! P(CanSymH)

x0 = {}

x1 = x0rF({}) = {Q1}

x2 = x1rF({Q1}) = {Q1,Q2}

x3 = x2rF({Q1,Q2}) = {Q1,Q2,Q3}

x4 = x3rF({Q1,Q2,Q3}) = {Q1,Q2,Q3}xn+1 = xnrF(yn), yn+1 = xn+1-xn

Nonstandard Fixpoint Algorithm:

• NOT y µ (x r y).

• NOT F(wdfix F) µ wdfix F.

NOT (F(wdfix F)) µ (wdfix F)

Soundness

Analysis results can be compiled into separation-logic proofs.

Widened Differential Fixpoint Algo.

«while (*) C¬d0 = ??

x0 = d0

x1 = x0r F(x0) y1 = x1 – x0

x2 = x1r F(y1) y2 = x2 – x1

x3 = x2r F(y2) = x2(x3) µ (d0) [ (y1)

[ (y2)x3 = d0r F(d0) r F(y1) r F(y2)(x3) (d0) [ (F(d0)) [ (F(y1))

[ (F(y2))

Widened Differential Fixpoint Algo.

{d0} C {F(d0)} {y1} C {F(y1)} {y2} C {F(y2)}

{d0} C {x3} {y1} C {x3} {y2} C {x3}

{d0 Ç y1 Ç y2} C {x3}

{x3} C {x3}

{x3} while (*) C {x3}

{d0} while (*) C {x3}

Disjunction Rule

Consequence:

(x3) (d0) [ (F(d0)) [ (F(y1)) [ (F(y2))

Consequence:

(x3) µ (d0) [ (y1)

[ (y2)