1. Spring 2014Program Analysis and Verification Lecture 12: Abstract Interpretation IV Roman Manevich Ben-Gurion University

2. Syllabus

3. Previously • Solving monotone systems via chaotic iteration • Relating abstract semantics and concrete semantics • Galois connection • Sound abstract transformers • Partial reduction • Implementing transformers

4. Vanilla algorithm Non-incremental. Most variables don’t change. Problem Definition: • Lattice of properties L of finite height (ACC) • For each statement define a monotone transformer Preparation: • Parse program into AST • Convert AST into CFG • Generate system of equations from CFG Analysis: • Initialize each analysis variable with  • Update all analysis variables of each equation until reaching a fixed point

5. Chaotic iteration fori:=1 to n do X[i] := WL = {1,…,n}while WL  do j := pop WL // choose index non-deterministically N := F[i](X) if N  X[i] then X[i] := Nadd all the indexes that directly depend on i to WL (X[j] depends on X[i] if F[j] contains X[i])return X • Input: • A cpoL = (D, , , ) satisfying ACC • Ln = LL … L • A monotone function f : DnDn • A system of equations { X[i] | f(X) | 1  i  n} • Output: lfp(f) • A worklist-based algorithm

6. Relating the concrete domainwith the abstract domain

7. Galois Connection • Given two complete latticesC = (DC, C, C, C, C, C) – concrete domainA = (DA, A, A, A, A, A) – abstract domain • A Galois Connection (GC) is quadruple (C, , , A)that relates C and A via the monotone functions • The abstraction function  : DC DA • The concretization function  : DA DC • for every concrete element cDCand abstract element aDA((a)) a and c ((c)) • Alternatively (c)  aiffc (a)

8. Galois Connection: c ((c)) C A The most precise (least) element in A representing c  ((c)) 3  (c) 2 c  1

9. Galois Connection: ((a)) a What a represents in C(its meaning) C A  a (a) 1 2  ((a))  3

10. Relating abstract transformers to concrete transformers

11. Sound abstract transformer • Given two latticesC = (DC, C, C, C, C, C)A = (DA, A, A, A, A, A)and GCC,A=(C, , , A) with • A concrete transformer f : DC DCan abstract transformer f# : DA DA • We say that f#is a sound transformer (w.r.t. f) if • c: f(c)=c’ (f#(c))  (c’) • For every a and a’ such that (f((a)))A f#(a)

12. Transformer soundness condition 1 c: f(c)=c’ (f#(c))  (c’) C A f#  5 f 4 1 2 3

13. Transformer soundness condition 2 a: f#(a)=a’ f((a))  (a’) C A 4  f 5 1 2 f# 3

14. Best (induced) transformer f#(a)=(f((a))) C A f# 4 f 3 1 2 Problem:  incomputable directly

15. Best abstract transformer [CC’77] • Best in terms of precision • Most precise abstract transformer • May be too expensive to compute • Constructively defined asf# =  f   • Induced by the GC • Not directly computable because first step is concretization • We often compromise for a “good enough” transformer • Useful tool: partial concretization

16. Global (fixed point) Soundness theorems

17. Soundness theorem 1 aDA: f((a))  (f#(a))  aDA: fn((a))  (f#n(a))  aDA: lfp(fn)((a))  (lfp(f#n)(a)) lfp(f)  lfp(f#)  C A lpf(f)  lpf(f#)  fn  f#n  f3 f#3 … … f2  f#2  f  f#   

18. Soundness theorem 2 c DC : (f(c))  f#((c))  c DC : (fn(c))  f#n((c))  c DC : (lfp(f)(c))  lfp(f#)((c)) lfp(f)  lfp(f#)  C A lpf(f#)  lpf(f)  f#n  fn  … … f#3 f3 f#2  f2  f#  f   

19. A recipe for a sound static analysis Define an “appropriate” operational semantics Define “collecting” structural operational semantics Establish a Galois connection between collecting states and abstract states Local correctness: show that the abstract interpretation of every atomic statement is soundw.r.t. the collecting semantics Global correctness: conclude that the analysis is sound

20. Today Question: given existing abstract interpreters GC + transformers) can we reuse them to construct a new abstract interpreter? Composing abstract domains (and GCs) Reduced product Implementing composition of analyses

21. Composing lattices

22. Three example analyses • Abstract states are conjunctions of constraints • Variable Equalities • VE-factoids = { x=y | x, y Var}  falseVE = (2VE-factoids, , , , false, ) • Constant Propagation • CP-factoids = { x=c | x Var, c  Z}  falseCP = (2CP-factoids, , , , false, ) • Available Expressions • AE-factoids = { x=y+z | x Var, y,z  VarZ}  falseAE = (2AE-factoids, , , , false, )

23. Lattice combinators reminder • Cartesian Product • L1 = (D1, 1, 1, 1, 1, 1)L2 = (D2, 2, 2, 2, 2, 2) • Cart(L1, L2) = (D1D2, cart, cart, cart, cart, cart) • Disjunctive completion • L = (D, , , , , ) • Disj(L) = (2D, , , , , ) • Relational Product • Rel(L1, L2) = Disj(Cart(L1, L2))

24. Cartesian product of complete lattices • For two complete lattices L1 = (D1, 1, 1, 1, 1, 1) L2 = (D2, 2, 2, 2, 2, 2) • Define the posetLcart = (D1D2, cart, cart, cart, cart, cart)as follows: • (x1, x2) cart (y1, y2) iffx1 1 y1 andx2 2 y2 • Lemma: L is a complete lattice • Define the Cartesian constructor Lcart = Cart(L1, L2)

25. Composing Galois connections

26. Goal Given two Galois connections, can we combine them to form a new Galois connection?

27. Product connection: abstraction CPVE lattice collecting lattice {[a9, c 9]}  {a=9, c=9}{c=a} {}

28. Product connection: concretization CPVE lattice collecting lattice   {a=9}{c=a} {[a9, c 0] [a9, c 1] [a9, c 9] ... } {[a9, c 9] [a1, c 1] [a2, c 2] ... } {}

29. Product connection: concretization CPVE lattice collecting lattice  {a=9}{c=a} {[a9, c 9]} {}

30. Cartesian product of GCs • GCC,A=(C, C,A, A,C, A)GCC,B=(C, C,B, B,C, B) • Cartesian Product GCC,AB = (C, C,AB, AB,C, AB) • C,AB(X)= ? • AB,C(Y)= ?

31. Cartesian product of GCs • GCC,A=(C, C,A, A,C, A)GCC,B=(C, C,B, B,C, B) • Cartesian Product GCC,AB = (C, C,AB, AB,C, AB) • C,AB(X) = C,A(X), C,B(X) • AB,C(Y1,Y2)= A,C(Y1)  B,C(Y2)

32. Product vs. reduced product

33. Product vs. reduced product CPVE lattice collecting lattice   {a=9}{c=a} {c=9}{c=a}  {[a9, c 9]}  {a=9, c=9}{c=a} {}

34. Reduced product For two complete lattices L1 = (D1, 1, 1, 1, 1, 1) L2 = (D2, 2, 2, 2, 2, 2) Define the reduced posetD1D2 = {(d1,d2)D1D2 | (d1,d2) = (d1,d2) } L1L2 = (D1D2, cart, cart, cart, cart, cart)

35. Cartesian product transformers • GCC,A=(C, C,A, A,C, A) FA[st] : A  AGCC,B=(C, C,B, B,C, B) FB[st] : B  B • Cartesian Product GCC,AB = (C, C,AB, AB,C, AB) • C,AB(X) = (C,A(X), C,B(X)) • AB,C(Y)= A,C(X)  B,C(X) • How should we define FAB[st] : AB AB ?

36. Cartesian product transformers • GCC,A=(C, C,A, A,C, A) FA[st] : A  AGCC,B=(C, C,B, B,C, B) FB[st] : B  B • Cartesian Product GCC,AB = (C, C,AB, AB,C, AB) • C,AB(X) = (C,A(X), C,B(X)) • AB,C(Y)= A,C(X)  B,C(X) • How should we define FAB[st] : AB AB ? • Idea: FAB[st](a, b) = (FA[st] a, FB[st] b) • Are component-wise transformers precise?

37. Cartesian product analysis example CP analysis VE analysis a := 9;b := 9;c := a; {a=9}{a=9, b=9}{a=9, b=9, c=9} a := 9;b := 9;c := a; {}{}{c=a} • Abstract interpreter 1: Constant Propagation • Abstract interpreter 2: Variable Equalities • Let’s compare • Running them separately and combining results • Running the analysis with their Cartesian product

38. Cartesian product analysis example CP analysis + VE analysis a := 9;b := 9;c := a; {a=9}{a=9, b=9}{a=9, b=9, c=9, c=a} • Abstract interpreter 1: Constant Propagation • Abstract interpreter 2: Variable Equalities • Let’s compare • Running them separately and combining results • Running the analysis with their Cartesian product

39. Cartesian product analysis example CPVE analysis Missing a := 9;b := 9;c := a; {a=9}{a=9, b=9}{a=9, b=9, c=9, c=a} {a=b, b=c} • Abstract interpreter 1: Constant Propagation • Abstract interpreter 2: Variable Equalities • Let’s compare • Running them separately and combining results • Running the analysis with their Cartesian product

40. Transformers for Cartesian product • Naïve (component-wise) transformers do not utilize information from both components • Same as running analyses separately and then combining results • Can we treat transformers from each analysis as black box and obtain best transformer for their combination?

41. Can we combine transformer modularly? No generic method for any abstract interpretations

42. Implementing approximate reduced product

43. Reducing values for CPVE X = set of CP constraints of the form x=c(e.g., a=9) Y = set of VE constraints of the form x=y ReduceCPVE(X, Y) = (X’, Y’) such that(X’, Y’)  (X’, Y’) Ideas?

44. Reducing values for CPVE • X = set of CP constraints of the form x=c(e.g., a=9) • Y = set of VE constraints of the form x=y • ReduceCPVE(X, Y) = (X’, Y’) such that (X’, Y’)  (X’, Y’) • ReduceRight: • if a=b  X and a=c  Y then add b=c to Y • ReduceLeft: • If a=c and b=c  Y then add a=b to X • Keep applying ReduceLeft and ReduceRight and reductions on each domain separately until reaching a fixed-point

45. All the code you need to obtaina basic CPVE implementation

46. Implementing reduction for CPVE

47. Transformers for Cartesian product • Do we get the best transformer by applying component-wise transformer followed by reduction? • Unfortunately, no (what’s the intuition?) • Can we do better? • Logical Product [Gulwani and Tiwari, PLDI 2006]

49. A modular approachfor approximatingreduced product

50. Logical product-- • Assume A=(D,…) is an abstract domain that supports two operations: for xD • inferEqualities(x) = { a=b | (x) a=b }returns a set of equalities between variables that are satisfied in all states given by x • refineFromEqualities(x, {a=b}) = ysuch that • (x)=(y) • y  x