Bernhard Gleiss, Gilles Barthe, Renate Eilers, Pamina...

Verifying Relational Properties using Trace Logic

Bernhard Gleiss, Gilles Barthe, Renate Eilers,Pamina Georgiou, Laura Kovacs, Matteo Maffei

October 25, 2019

Motivating example

1 func main() {

2 const Int[] a;

34 Int sum = 0;

56 for (Int i=0; i < a.length; ++i)

7 {

8 sum += a[i];

9 }

10 }

v w

w v

k

a(t1) :

a(t2) :

⇒sum(end , t1)

==

sum(end , t2)

Motivating example - Human Proof

FirstIteration

LastIteration

iteration where i = k

iteration where i = k + 2

Induction' Induction'Comm.+

Focus

I (Software) programs containing loops and arrays

I Proving Relational Safety Properties

I Proving Correctness (instead of finding Counterexamples)

Remaining Talk - Outline

I Part 1: Language and Semantics - Trace Logic

I Part 2: Verification Approach - Vampire and Trace Lemmas

I Part 3: Extension - Relational Properties

Part 1: Language and Semantics - Trace Logic

Trace Logic

I full first-order logic over UFDTLIA

I explicit notion of time: able to refer to each timepoint of theexecution uniquely, while preserving control flow structure

I can formulate induction directly in the language

I can denote parts of a loop and reason about those partsseparately

Timepoints

1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

l4 l8(0) l6(s(0)) l6(n6) l6(it)

Program Variables

1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

sum(l8(0)) i(l6(n6)) a(l2, pos)

a(pos)

Program Variables

1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

sum(l8(0)) i(l6(n6)) a(pos)

a(l2, pos)

Semantics in Trace Logic

1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

i(l6(0)) ' 0

Semantics in Trace Logic

1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

∀itN.(it < n6 → i(l6(s(it)))' i(l8(it))+1

)

Part 2: Verification Approach - Vampire and Trace Lemmas

Workflow - Rapid

P JPKSemantics FO-clauses

PropertyFO-clauses

JPK �UFDTLIA Property

Workflow - Rapid


PropertyFO-clauses

VAMPIRE

Trace-LemmasFO-clauses

Trace Lemmas

I provide necessary inductive reasoning

I valid formulas, derivable from instances of the inductionaxiom scheme

I can’t be automatically generated by state-of-the-art tools

I manually identified set of useful Trace Lemmas

Trace Lemmas - Example 1

”For an arbitrary interval: if the value of v stays the same in eachstep, then the value of v at the end is the same as the value of v atthe beginning”

∀itNL ,∀itNR .(∀itN.

(itL ≤ it < itR → v(l6(it))' v(l6(s(it)))

)→

∀itN.v(l6(itL))' v(l6(itR)))

Trace Lemmas - Example 2

Intermediate Value Theorem: ”If i ≤ v at the beginning and i > vat the end and if i is incremented by 1 in each iteration, then thereexists an iteration, where i = v .”

∀v I.((

i(l6(0)) ≤ v ∧i(l6(n6)) > v ∧∀itN.

(it < n→ i(l6(s(it)))' i(l6(it)) + 1

))

→ ∃it ′N.i(l6(it ′))' v

)

Trace Lemmas

I can be instantiated to parts of the loop

I can feature existential quantification over iterations

I can feature quantifier alternations

I can not be synthesized automatically by state-of-the-arttechniques

Workflow - Rapid


PropertyFO-clauses

VAMPIRE

Trace-LemmasFO-clauses

- synthesize split-timepoints- reason about loop-parts sep.- perform interleaved

Part 3: Extension to Relational Properties

Extension - Timepoints and Program Variables

1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

n6 sum(l8(0)) i(l6(n8))


1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

n6(t1) sum(l8(0), t1) i(l6(n8), t2)


1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

i(l6(0)) ' 0


1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

∀trT.i(l6(0), tr) ' 0


1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

∀itN.(it < n6 → i(l6(s(it)))' i(l8(it))+1

)


1 func main() {

2 const Int[] a;

34 Int sum = 0;


7 {

8 sum += a[i];

9 }

10 }

∀trT.∀itN.(it < n6(tr)→ i(l6(s(it)), tr)' i(l8(it), tr)+1

)

Extension - Trace Lemmas

sum has the same value in both traces in iteration it:

Eqsum(it) := sum(l6(it), t1)' sum(l6(it), t2).

Bounded induction with induction hypothesis Eqsum(it):

∀itLN, itRN.(

(Eqsum(itL) ∧∀itN.((itL ≤ it < itR ∧ Eqsum(it))→ Eqsum(s(it)))

)→ Eqsum(itR)

)

Benchmarks

I 27 challenging benchmarks from security applications

I 2-safety properties: non-interference and sensitivity

I 60 second timeout

BenchmarksVampire

CVC4 Z3S S+A F F+A

1-hw-equal-arrays X X - X X X2-hw-last-position-swapped - X - - X X3-hw-swap-and-two-arrays - X - - - -4-hw-swap-in-array-lemma - X - - - -

4-hw-swap-in-array-full - X - - - -1-ni-assign-to-high X X X X X X

2-ni-branch-on-high-twice X X X X X X3-ni-high-guard-equal-branches X X X X X X

4-ni-branch-on-high-twice-prop2 X X - - X X5-ni-temp-impl-flow - - X X X X

6-ni-branch-assign-equal-val - - X X X X7-ni-explicit-flow X X X X X X

8-ni-explicit-flow-while X X - X X X9-ni-equal-output X - - - - X

10-ni-rsa-exponentiation X X X X X -1-sens-equal-sums X X X X X X

2-sens-equal-sums-two-arrays X X X X - -3-sens-abs-diff-up-to-k - - - - X X

4-sens-abs-diff-up-to-k-two-arrays - - - - - -5-sens-two-arrays-equal-k X X X X - -6-sens-diff-up-to-explicit-k X X X X - -

7-sens-diff-up-to-explicit-k-sum - - X X - -8-sens-explicit-swap - - X X - -

9-sens-explicit-swap-prop2 - X X - -10-sens-equal-k X X X X - -

11-sens-equal-k-twice X X X X - -12-sens-diff-up-to-forall-k - - X X X -

Total Vampire 15 18 17 19Unique Vampire 1 4 0 0

Total 25 14 13

Conclusion

I Trace Logic Language and Semantics

I Verification Approach - Vampire and Trace Lemmas

I Application to Relational Verification

Extra slides

Background Theory

I Full first-order logic with equality and uninterpreted functions

I Iterations - Datatype (0, s, p, <) (no arithmetic!)

I Timepoints - Uninterpreted Sort

I Values of program variables - Integers

Motivating example - Property in Trace Logic

Swappeda(k) :=

0 ≤ k < k + 1 < a.length

∧ ∀posI.((pos 6' k ∧ pos 6' k+1)→a(pos, t1)' a(pos, t2))

∧ a(k , t1)' a(k+1, t2)

∧ a(k , t2)' a(k+1, t1)

∀kI.(Swappeda(k)→ sum(end , t1)' sum(end , t2)

)

Rapid Tool

Available at:

https://github.com/gleiss/rapid

Theory reasoning - Blowup

new: 40588. less(-4,0) — less(-6,4)new: 40589. less(-4,-1) — less(-4,4)new: 40590. less(-4,-1) — less(-3,4)new: 40591. less(-4,-1) — less(-2,4)new: 40592. less(-4,-1) — 0 = 4 — less(0,4)new: 40593. less(-4,-1) — less(4,0)new: 40594. less(-4,2) — less(-2,4)new: 40595. less(-4,2) — less(-1,4)new: 40596. less(-4,2) — less(0,4)new: 40597. less(-4,2) — less(4,3)new: 40598. less(-4,2) — less(1,4)new: 40599. less(-4,-2) — less(-4,4)new: 40600. less(-4,-2) — less(-3,4)new: 40601. less(-4,-3) — less(-4,4)new: 40602. less(-4,3) — less(-3,4)new: 40603. less(-4,3) — less(-1,4)new: 40604. less(-4,3) — less(-2,4)new: 40605. less(-4,3) — less(0,4)new: 40606. less(-4,3) — less(1,4)new: 40607. less(-4,3) — less(2,4)new: 40608. less(-4,4) — less(-3,4)new: 40609. less(-4,4) — less(-2,4)new: 40610. less(-4,4) — less(2,4)new: 40611. less(-4,4) — less(1,4)new: 40612. less(-4,4) — less(0,4)new: 40613. less(-4,4) — less(-1,4)new: 40614. less(-4,5) — less(-1,4)new: 40615. less(-4,5) — less(1,4)new: 40616. less(-4,5) — less(0,4)

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