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Software Verification With Liquid Types

Software Verification With Liquid Types. Ranjit Jhala , UC San Diego (with Pat Rondon , Ming Kawaguchi). Software Verification. i nt min_index ( int [] a){ r = 0; n = a.length ; for (i = 0; i < n; i++){ if (a[i] < a[r]) r = i; } return r; }.

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Software Verification With Liquid Types

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  1. Software Verification With Liquid Types RanjitJhala, UC San Diego (with Pat Rondon, Ming Kawaguchi)

  2. Software Verification intmin_index(int [] a){ r = 0; n = a.length; for (i = 0; i < n; i++){ if (a[i] < a[r]) r = i; } return r; }

  3. Example: Memory Safety intmin_index(int [] a){ r = 0; n = a.length; for (i = 0; i < n; i++){ if (a[i] < a[r]) r = i; } return r; } • Access In Array Bounds ?

  4. Example: Memory Safety intmin_index(int [] a){ r = 0; n = a.length; for (i = 0; i < n; i++){ if (a[i] < a[r]) r = i; } return r; } assert (0<=r && r<n) • How to prove assert ?

  5. Example: Memory Safety intmin_index(int [] a){ r = 0; n = a.length; //@invariant: 0·iÆ0·rÆr<n for (i = 0; i < n; i++){ if (a[i] < a[r]) r = i; } return r; }

  6. Invariants intmin_index(int [] a){ r = 0; n = a.length; //@invariant: 0·iÆ0·rÆr<n for (i = 0; i < n; i++){ if (a[i] < a[r]) r = i; } return r; } • By Man [Floyd-Hoare] • Or Machine[Cousot-Cousot]

  7. But, what of … Closures Collections Polymorphism Data Structures …?

  8. But, what of … H-O Logics? Quantifiers? Induction?

  9. Automation

  10. H-O Logics? Quantifiers? Induction? A Solution: Types

  11. Part I • Types and Invariants • Part II • Closures, Collections, Generics • Part III • Induction for Data Structures

  12. Part I • Types and Invariants

  13. Part I • Refinement Types [Zenger ’97, Owre-Rushby-Shankar ’98, Xi-Pfenning ’98]

  14. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 min_index:a:intarray->int

  15. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 Refinement min_index:{a:intarray|0≺ a.len}->{v:int|0≼v≺a.len} Type

  16. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 “Requires” min_index:{a:intarray|0≺ a.len}->{v:int|0≼v≺a.len}

  17. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 “Ensures” min_index:{a:intarray|0≺ a.len}->{v:int|0≼v≺a.len}

  18. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 loop: {r:int|0≼ r≺ a.len}->{i:int|0≼i}->{v:int|0≼ v≺ a.len}

  19. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 loop: {r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len}

  20. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 loop: {r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} “Requires”

  21. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 loop: {r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} “Ensures”

  22. Part I • Refinement Types • 1. Type-checking • 2. Refine-checking • 3. Refine-Inference

  23. loop : r:int -> i:int -> int let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is int • loop : … • r : int • i: int ⊢ r “subtype-of” int

  24. loop : r:int -> i:int -> int let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is int • loop : … • r : int • i: int ⊢ r <: int

  25. loop : r:int -> i:int -> int let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 Assume Prove index is int • loop : … • r : int • i: int ⊢ i <: int

  26. loop : r:int -> i:int -> int let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 3 Assume Prove input is int • loop : … • i : int • r : int • r': int • i': int ⊢ r' <: int

  27. loop : r:int -> i:int -> int let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 3 4 Assume Prove output is int • loop : … • i : int • r : int • r': int • i': int ⊢ loopr'i'<: int

  28. loop : r:int -> i:int -> int let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 3 4 5 Assume Prove input is int • loop : … • i : int • r : int • r': int • i': int ⊢ 0 <: int

  29. Part I • Refinement Types • 1. Type-checking • 2. Refine-checking • 3. Refine-Inference

  30. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 min_index {a:0≺a.len}->{v:0≼v≺ a.len} loop {r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len}

  31. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 3 4 5 Assume Prove output is ensured • r :0≼ r≺ a.len • i :0≼ i • [i≽ a.len]

  32. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is ensured • r :0≼ r≺ a.len • i :0≼ i • [i≽ a.len] ⊢ {v=r}<:{0≼ v≺ a.len}

  33. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is ensured • r :0≼ r≺ a.len • i :0≼ i • [i≽ a.len] ⊢ {v=r}<:{0≼ v≺ a.len}

  34. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is ensured 0≼ r≺ a.len ⋀0≼ i ⋀i≽ a.len ⊢ {v=r}<:{0≼ v≺ a.len}

  35. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is ensured 0≼ r≺ a.len ⋀0≼ i ⋀i≽ a.len ⋀ {v=r}<:{0≼ v≺ a.len}

  36. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 Assume Prove output is ensured 0≼ r≺ a.len ⋀0≼ i ⋀i≽ a.len ⇒ ⋀ v=r0≼ v≺ a.len

  37. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 0≼ r≺ a.len ⋀0≼ i ⋀i≽ a.len Is Valid? Yes. ⇒ ⋀ v=r0≼ v≺ a.len

  38. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 4 3 5 Assume Prove index in bounds • r :0≼ r≺ a.len • i :0≼ i • [i≺ a.len] ⊢ {v=i}<:{0≼ v≺ a.len} • [i≺ a.len]

  39. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 2 0≼ r≺ a.len ⋀0≼ i ⋀i≺ a.len Is Valid? Yes. ⇒ ⋀ v=i0≼ v≺ a.len

  40. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 4 3 5 Prove input as required ⊢ {v=r'}<:{0≼ v≺ a.len}

  41. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 3 Assume Prove input as required • r :0≼ r≺ a.len • i :0≼ i • [i≺ a.len] • i':i'=i+1 ⊢ {v=r'}<:{0≼ v≺ a.len} r':r'=i∨r'=r

  42. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 3 • 0≼ r≺ a.len • ⋀ 0≼ i • ⋀i≺ a.len • ⋀ r'=i∨r'=r • ⋀ i'=i+1 Is Valid? Yes! ⇒ ⋀ v=r'0≼ v≺ a.len

  43. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 4 3 5 Prove output is ensured ⊢ {0≼ v≺a.len}<:{0≼ v≺a.len} Trivial.

  44. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 1 2 4 3 5 Assume Prove input as required ⊢ {v=0}<:{0≼ v≺a.len} • a :0≺ a.len

  45. loop:{r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len} let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 5 Is Valid? Indeed. ⇒ ⋀ v=0 0≼ v≺a.len • 0≺ a.len

  46. Part I • Refinement Types • 1. Type-checking • 2. Refine-checking • 3. Refine-Inference

  47. let min_indexa = let rec loopr i = if i>= a.lengththen r else let r' = a[i] < a[r] ? i : r let i' = i + 1 loopr' i' loop 0 0 min_index {a:0≺a.len}->{v:0≼v≺ a.len} loop {r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len}

  48. Writing Types is Hard Work! min_index {a:0≺a.len}->{v:0≼v≺ a.len} loop {r:0≼ r≺a.len}->{i:0≼i}->{v:0≼v≺ a.len}

  49. Automatic Refinement Inference • Writing Types is Hard Work! loop:???

  50. Automatic Refinement Inference • 1. Templates • 2. Constraints • 3. Fixpoint[MC/AI] loop:???

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