Two Approaches to Interprocedural Data Flow Analysis

Transcription

1 Two Approaches to Interprocedural Data Flow Analysis Micha Sharir Amir Pnueli Part one: The Functional Approach Klaas Boesche

2 Intraprocedural analysis procedure main read a, b procedure p if a=0 F a := a - 1 T a*b available? print t stop return Klaas Boesche 2

3 Interprocedural challenges Recursion Infeasible paths main p Function variables & Virtual functions main p if a=0 F a := a - 1 T x := call x p q q return Infinite paths Efficiency vs. Precision Filter invalid paths Precision and Efficiency No static call graph Klaas Boesche 3

4 Outline Notation and Review Functional Approach Interprocedural MOP Pragmatic Considerations Klaas Boesche 4

5 Notations main read a, b Control Flow Graphs r main no parameters p if a=0 F a := a - 1 T r p print t stop e main return e p Klaas Boesche 5

6 Data Flow Frameworks (L, F) is a data flow framework: L is a meet-semilattice = greatest lower bound = smallest element (no information) = largest element ("undefined") bounded - No infinite descending chain Analysis direction Klaas Boesche 6

7 Data Flow Frameworks (L, F) is a data flow framework: F is a monotone space of transfer functions 1. Closed under composition and meet f g x = f x g(x) 2. Contains id L x = x and f x = F is distributive iff f, x, y: f x f(y) = f(x y) Restrict F to graph G = (N, E): Smallest S F such that f m,n m, n E S and 1. and 2. hold Klaas Boesche 7

8 Intraprocedural example Available expression framework for the single expression a * b: L =, 1, F = f, f 1, id L, f : a * b not available 1: a * b available f x =, f 1 x = Klaas Boesche 8

9 Intraprocedural example main p read a, b r main if a=0 F id L T r p f 1 a := a - 1 n p 1 c main f f c p print t n main f m p f 1 stop 1 e main f 1 return id L e p Klaas Boesche 9

10 Intraprocedural equations The data flow equations x r = x n = f m,n x m m,n E n N *r+ approximate the meet-over-all paths (MOP) solution y n = *f p ( ) p pat G r, n + n N where f p=(n1,,n k ) = f (nk 1,n k ) f (n1,n 2 ) Klaas Boesche 10

11 Intraprocedural solutions F is distributive The maximum fixed point solution x n = y n F is monotone x n y n Klaas Boesche 11

12 Outline Notation and Review Functional Approach Interprocedural MOP Pragmatic Considerations Klaas Boesche 12

13 Interprocedural Graphs Two representations: 1. G = p N p, p E p 2. G = p N p, E s E s 0 E s 1 E 1 E p = E p 0 E p 1 E = E 0 E 1 p E p 0 E p 0 r p E 0 = E p 0 E p 0 p Intraprocedural edges E p 1 Intraprocedural edges with interprocedural control flow E s 0 E 1 return e p E 1 Interprocedural edges Klaas Boesche 13

14 Example G main p read a, b r main if a=0 F T r p a := a E main print t E p 1 stop e main return e p Klaas Boesche 14

15 Example G main p read a, b r main if a=0 F T r p a := a - 1 print t stop e main return e p Klaas Boesche 15

16 Interprocedurally Valid paths IVP 0 (r p, n) p r p call q m q r q m return e q n Klaas Boesche 16

17 Interprocedurally Valid paths main IVP(r main, n) r main q 1 IVP 0 (r main, n) p 2 2 c 1 r p2 3 q 2 IVP 0 (r p2, n) c 2 p j r pj q j IVP 0 (r pj, n) n Klaas Boesche 17

18 Path notations p 1, p 2 pat G r q, n p 1 E 1 Sequence of call & return edges in p 1 p = p 1 p 2 Concatenation of p 1, p Klaas Boesche 18

19 Interprocedurally Valid paths p pat G r q, n is in IVP 0 r q, n p E 1 is complete defined as: call q' m q' p 1 r q 1. p E 1 = ε m return e q 2. p E 1 = p 1 p 2 and p 1, p 2 are complete 3. p E 1 = m, r q p 1 e q, m and p 1 is complete Klaas Boesche 19

20 Interprocedurally Valid paths q IVP r main, n q = q 1 c 1, r p2 q 2 c j 1, r pj q j i < j: q i IVP 0 r pi, c i and q j IVP 0 r pj, n Also called Path Decomposition Klaas Boesche 20

21 Examples G c main, r p, c p, r p, e p, n read a, b r main if a=0 F T r p a := a - 1 c main c p print t n stop e main return e p Klaas Boesche 21

22 Examples G c main, r p, c p, r p, c p, r p, e p, n, e p, n read a, b r main if a=0 F T r p a := a - 1 c main c p print t n stop e main return e p Klaas Boesche 22

23 Functional approach p r p φ rp,m φ rp,n φ rp,l call q m r q φ rq,e q l return e q f l,n n Information x at r p is transformed to φ rp,n x at n Klaas Boesche 23

24 Functional Approach equations φ rp,r p = id L φ rp,n = m,n φ rp,m n N p *r p + m,n E p m,n = f m,n m, n E0 p φ rq,e q m, n E 1 p, m calls q Initialize the equations with Recursion implicitly encoded in equations φ rp,r p = id L φ rp,n = f n N p *r p + Compute the maximal fixed point Klaas Boesche 24

25 Example main p id L read a, b r m if a=0 F id L T r p f 1 a := a - 1 n p f 1 φ rm,r m c m f c p φ rp,e p φ rm,c m print t n m m p f 1 f 1 id L f 1 φ rm,n m stop e m return e p Klaas Boesche 25

26 Example main p read a, b r m if a=0 F id L T r p id L f 1 a := a - 1 n p id L φ rp,r p c m f c p f φ rp,n p print t n m m p φ rp,e p φ rp,c p f 1 f 1 id L stop e m return e p id L φ rp,r p f 1 φ rp,m p Klaas Boesche 26

27 Example φ rm,r m = id L φ rm,c m = f 1 φ rm,r m φ rm,e m = f 1 φ rm,n m φ rp,r p = id L φ rp,c p = f φ rp,n p φ rp,m p = φ rp,e p φ rp,c p φ rm,n m = φ rp,e p φ rm,c m φ rp,n p = id L φ rp,r p φ rp,e p = id L φ rp,r p f 1 φ rp,m p Function Initial value Iteration φ rm,r m id L id L id L id L φ rm,c m f f 1 f 1 f 1 φ rm,n m f f f 1 f 1 φ rm,e m f f 1 f 1 f 1 φ rp,r p id L id L id L id L φ rp,n p f id L id L id L φ rp,c p f f f f φ rp,m p f f f f φ rp,e p f id L id L id L Klaas Boesche 27

28 Solution x rmain = x rp = φ rq,c x rq c calls p in q Compute the maximal fixed point iteratively x n = φ rp,n x rp Computes the solution for all other nodes Klaas Boesche 28

29 Example (continued) x rp = φ rm,c m x rm φ rp,c p x rp main = f 1 x rm f (x rp ) p read a, b r m Functions φ rm,r m = id L if a=0 F T r p φ rm,c m = f 1 a := a - 1 n p 1 c m φ rm,n m = f 1 1 print t n m φ rm,e m = f 1 φ rp,r p = id L φ rp,n p = id L φ rp,c p = f c p m p φ rp,m p = f 1 stop e m φ rp,e p = id L return e p Klaas Boesche 29

30 Outline Notation and Review Functional Approach Interprocedural MOP Pragmatic Considerations Klaas Boesche 30

31 Interprocedural MOP solution ψ n = f q q IVP r main, n y n = ψ n main r main ψ n f q1 f q2 n Klaas Boesche 31

32 IVP 0 Lemma φ rp,n = f q q IVP 0 r p, n p r p call q φ rp,n f q q IVP 0 r p, n return n Klaas Boesche 32

33 IVP 0 Lemma: Proof φ rp,n f q q IVP 0 r p, n By induction on φi rp,n, the i-th approximation For i = 0: 0 If n = r p then φ rp,n = id L = f ε f q q IVP 0 r p, r p If n r p then φ0 rp,n = f f F i IH: φ rp,n f q q IVP 0 r p, n For i + 1: see blackboard Klaas Boesche 33

34 IVP 0 Lemma: Proof f q q IVP 0 r p, n φ rp,n Show: q IVP 0 r p, n : f q φ rp,n by induction over the length k of q For k = 0: Then f q = φ rp,r p = id L IH: q IVP 0 r p, n, q k: f q φ rp,n For k + 1: see blackboard Klaas Boesche 34

35 MOP Rewritten ψ n = f q q IVP r main, n χ n = φ rpj,n φ rpj 1,c j 1 φ rmain,c 1 c i calls p i+1 Then ψ n = χ n with IVP 0 Lemma and Path decomposition Thus y n = ψ n = χ n Klaas Boesche 35

36 Equivalence to MOP F distributive y n = x n Proof: Show that x rp = y rp then with x n = φ rp,n x rp = φ rp,n y rp, y n = χ n = φ rpj,n φ rpj 1,c j 1 φ rmain,c 1 c i calls p i+1 = φ rp,n χ rp = φ rp,n y rp it follows that x n = y n n N Klaas Boesche 36

37 Equivalence to MOP New "Intraprocedural" Data Flow Problem: G c = N c, E c E c = E c 0 E c 1 s p r s r p E c 0 E c 1 E c 1 E c 0 g m,n = φ m,n, m, n E c 0 id L, m, n E c Klaas Boesche 37

38 Equivalence to MOP s p r s r p E c 0 E c 1 E c 1 E c 0 g m,n = φ m,n, m, n E c 0 id L, m, n E c 1 interprocedural intraprocedural x rmain = x rp = φ rq,c x rq c calls p in q x r = x n = g m,n x m m,n E x n = φ rp,n x rp Klaas Boesche 38

39 Equivalence to MOP x rmain = interprocedural x rp = φ rq,c x rq c calls p in q x rmain = intraprocedural x rp = id L x m m,r p E c 1 x n = φ rp,n x rp = x m m,r p E c 1 = φ rq,c x rq c calls p in q x c = φ m,c x m m,c E c 0 = φ rp,c x rp Klaas Boesche 39

40 Equivalence to MOP s p r s r p E c 0 E c 1 E c 1 E c 0 g m,n = φ m,n, m, n E c 0 id L, m, n E c 1 interprocedural y n = f q q IVP r main, n intraprocedural y n = *f p ( ) p pat G r, n + Follows from y n = φ rpj,n φ rmain,c 1 c i calls p i+1 and by construction of G c Klaas Boesche 40

41 Equivalence to MOP F is distributive The maximum fixed point solution x n of the functional approach = the interprocedural MOP solution y n F is monotone x n y n Klaas Boesche 41

42 Outline Notation and Review Functional Approach Interprocedural MOP Pragmatic Considerations Klaas Boesche 42

43 Pragmatic problems Representation of φ: Symbolic not always possible Explicit representation maybe not finite if L infinite: main: A := 0 call P print A p: if cond then A := A + 1 call P A := A 1 endif return φ rp,e p A, 0 φ rp,e p A, 1 φ rp,e p A, Klaas Boesche 43

44 Pragmatic problems Approach with symbolic representation may not converge when L infinite and F unbounded: k 0: Need k iterations to show φ rp,e p A, k = main: A := 0 call P print A p: if cond then A := A (A<0)?-1:1 call P A := A 1 endif return Klaas Boesche 44

45 Practical result If L, F distributive and L finite then the iterative solution of φ rp,r p = id L φ rp,n = m,n φ rp,m m,n E p converges and x rmain = x rp = φ rq,c x rq c calls p in q x n = φ rp,n x rp results in the interprocedural MOP solution Klaas Boesche 45

46 Algorithm Assume L finite and no symbolic representation available Compute φ rp,n only for necessary values W r main, ; PHI r main, = wile W remove some n, x from W update PHIs and W x n = a L PHI n, a Klaas Boesche 46

47 Algorithm: updating PHIs and W propagate x, z, m : PHI m, x PHI m, x z; if PHI canged: W W m, x Klaas Boesche 47

48 Algorithm: updating PHIs and W x q Case 1 y PHI(n, x) r q n p r p z PHI(e p, y) m return e p if z undefined: propagate y, y, r p else: propagate x, z, m Klaas Boesche 48

49 Algorithm: updating PHIs and W u x = PHI(c p, u) p Case 2 r p x q c p r q y PHI(n, x) m p return n for eac call c p to q and u L wit x = PHI c p, u propagate u, y, m p Klaas Boesche 49

50 Algorithm: updating PHIs and W x Case 3 q y PHI(n, x) r q n m 1 m k for eac n, m E q 0 : propagate x, f n,m y, m Klaas Boesche 50

51 Example W * r m, main read a, b f 1 print t f 1 stop r m c m n m e m c m, r p, 1 p if a=0 F a := a - 1 return PHI r m, Klaas Boesche 51 id L f f 1 T e p, 1 r p c p id L n p m p e p + = propagate, f 1, c m PHI c m, = 1 propagate 1,1, r p PHI r p, 1 = 1 propagate 1, id L (1), n p propagate 1, id L (1), e p PHI n p, 1 = 1 PHI e p, 1 = 1 propagate, 1, n m PHI n m, = 1

52 Summary: Define valid paths IVP 0 (r p, n) p r p main IVP(r main, n) r main call q m m q return r q e q q 1 IVP 0 (r main, n) callp 2 c 1 p j r pj n q j IVP 0 (r pj, n) n Klaas Boesche 52

53 Summary: Functional Approach p r p φ rp,m φ rp,n φ rp,l call q m r q φ rq,e q l return e q f l,n n Information x at r p is transformed to φ rp,n x at n Klaas Boesche 53

54 Summary: Functional Result F is distributive The maximum fixed point solution x n of the functional approach = the interprocedural MOP solution y n Klaas Boesche 54

55 Summary: Practical Aspects Represent φ rp,n symbolically or by explicit relation Explicit approach may require much space L infinite Explicit approach may diverge L infinite, F unbounded Symbolic approach may diverge Klaas Boesche 55