1. Pentagons: A Weakly Relational Abstract Domain for the Efficient Validation of Array Accesses Francesco Logozzo, Manuel Fahndrich Microsoft Research, Redmond

2. The Background 2  Efficient static checking of.NET assemblies  Foxtrot  Foxtrot: a language agnostic contract language  Clousot  Clousot: a language agnostic static analyzer  Based on abstract interpretation  Checks contracts, array bounds, memory accesses, nullness, …

3. Demo 3 Wrong? Ok: not null

4. Demo 4 Ok: index in bounds Ok: not null Ok: index in bounds

5. The paper in a nutshell Program executions  Is 0 ≤ y < x ?  Testing: try some points  What for the others?  Model checking: try all the points  What if we have ∞ points?  Abstract interpretation: approximation Intervals No  in O(n) Octagons Yes! in Θ (n 3 ) Polyhedra Yes! in O(2 n ) Pentagons Yes! In O(n) 5

6. Pentagons? 6  A lightweight numerical domain  Keep relations in the form a ≤ x ≤ b && x < y  a, b numerical constants  x, y variables  Enough to validate > 83% of the accesses of mscorlib.dll  Mscorlib.dll is the main library in.NET  Fast: Analyze it in a couple of minutes

7. Abstract domain 7  An abstract domain is a complete lattice endowed with  Widening operator  To ensure the convergence of the analysis  Ex. The increasing chain [0,1] ⊑ [0,2] ⊑ [0,3] ⊑ [0, 4] ⊑... Is extrapolated by widening to [0, +∞]  Transfer functions  To capture the abstract semantics of statements x → [4,5]  Ex. x := y + 3([y → [1, 2]) = [y → [1,2], x → [4,5]]

8. Interval domain 8  Elements:  { [a, b] | a ∈ Z ∪ { -∞ }, b ∈ Z ∪ { +∞ } }  Order  [a,b] ⊑ [c,d] iff c ≤ a and b ≤ d  Join  [a,b] ⊔ [c,d] =[min(a,c), max(b,d)]  Meet  [a,b] ⊓ [c,d] = [max(a,c), min(b,d)]  Widening: Keep the stable bounds  Transfer functions: ordinary interval arithmetic

9. LT Domain 9  Elements  ℘ ({ X < Y | X and Y are variables })  Efficient representation with Hashtables  Order  A ⊑ B iff B ⊆ A  Join  A ⊔ B = A \cap B  Meet  A ⊓ B = A ∪ B  Widening: just the join as the lattice has finite height  Transfer functions: y := x + 1 (A) = (A-{y}) ∪ { x < y }

10. Pentagons 10  Reduced  Reduced Cartesian product of Intervals and LT  Reduced?  Not just pairs: information flows from one element to the other  Ex. 2  (x → [1, 4], y → [3, 3], { x (x → [1, 2], y → [3, 3], { x < y })  May introduce cubic slowdown  Reduction is applied  In precise points of the analysis  Lazily at join points

11. The (Naif) Join of Pentagons 11  Left_P = (left_intv, left_lt), Right_P = (right_intv, right_lt) 1. Close Left_P and Right_P 2. Apply the join pairwisely  Closure (intv, lt) iterates until saturation this rule: if x → [a,b], y → [c,d] ∈ intv. If b< c then lt = lt ∪ { x < y }  Problem: It introduces a quadratic slowdown

12. The smarter join on Pentagons 12  Idea: 1. Apply the pairwise join 2. If a symbolic constraint x < y is dropped, check if the other branch implies it 3. If it does, then keep the constraint  Formal details in the paper  Results:  For mscorlib we moved from > 1h to a couple of minutes  No access is lost!

13. Experiment: Array bounds analysis 13  Assemblies as shipped  No pre-processing  No pre-selection  Intra-procedular analysis only  Contracts will improve the precision

14. Conclusions 14  A lightweight abstract domain  Used for array bounds validation  Efficient, and scalable  Implemented in Clousot  To be used  as a first pass to drop most of the proof obligations  In combination with other domains