1. Worklist algorithm Initialize all d i to the empty set Store all nodes onto a worklist while worklist is not empty: –remove node n from worklist –apply flow function for node n –update the appropriate d i, and add nodes whose inputs have changed back onto worklist

2. Worklist algorithm let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ; for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length-1 do if (m(n.outgoing_edges[i])  info_out[i]) m(n.outgoing_edges[i]) := info_out[i]; worklist.add(n.outgoing_edges[i].dst);

3. Termination Why is termination important? Can we stop the algorithm in the middle and just say we’re done... No: we need to run it to completion, otherwise the results are not safe...

4. Termination Assuming we’re doing reaching defs, let’s try to guarantee that the worklist loop terminates, regardless of what the flow function F does while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length-1 do if (m(n.outgoing_edges[i])  info_out[i]) m(n.outgoing_edges[i]) := info_out[i]; worklist.add(n.outgoing_edges[i].dst);

5. Termination Assuming we’re doing reaching defs, let’s try to guarantee that the worklist loop terminates, regardless of what the flow function F does while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length-1 do let new_info := m(n.outgoing_edges[i]) [ info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

6. Structure of the domain We’re using the structure of the domain outside of the flow functions In general, it’s useful to have a framework that formalizes this structure We will use lattices

7. Background material

8. Relations A relation over a set S is a set R µ S £ S –We write a R b for (a,b) 2 R A relation R is: –reflexive iff 8 a 2 S. a R a –transitive iff 8 a 2 S, b 2 S, c 2 S. a R b Æ b R c ) a R c –symmetric iff 8 a, b 2 S. a R b ) b R a –anti-symmetric iff 8 a, b, 2 S. a R b ) : (b R a)

9. Relations A relation over a set S is a set R µ S £ S –We write a R b for (a,b) 2 R A relation R is: –reflexive iff 8 a 2 S. a R a –transitive iff 8 a 2 S, b 2 S, c 2 S. a R b Æ b R c ) a R c –symmetric iff 8 a, b 2 S. a R b ) b R a –anti-symmetric iff 8 a, b, 2 S. a R b ) : (b R a) 8 a, b, 2 S. a R b Æ b R a ) a = b

10. Partial orders An equivalence class is a relation that is: A partial order is a relation that is:

11. Partial orders An equivalence class is a relation that is: –reflexive, transitive, symmetric A partial order is a relation that is: –reflexive, transitive, anti-symmetric A partially ordered set (a poset) is a pair (S, · ) of a set S and a partial order · over the set Examples of posets: (2 S, µ ), (Z, · ), (Z, divides)

12. Lub and glb Given a poset (S, · ), and two elements a 2 S and b 2 S, then the: –least upper bound (lub) is an element c such that a · c, b · c, and 8 d 2 S. (a · d Æ b · d) ) c · d –greatest lower bound (glb) is an element c such that c · a, c · b, and 8 d 2 S. (d · a Æ d · b) ) d · c

13. Lub and glb Given a poset (S, · ), and two elements a 2 S and b 2 S, then the: –least upper bound (lub) is an element c such that a · c, b · c, and 8 d 2 S. (a · d Æ b · d) ) c · d –greatest lower bound (glb) is an element c such that c · a, c · b, and 8 d 2 S. (d · a Æ d · b) ) d · c lub and glb don’t always exists:

14. Lub and glb Given a poset (S, · ), and two elements a 2 S and b 2 S, then the: –least upper bound (lub) is an element c such that a · c, b · c, and 8 d 2 S. (a · d Æ b · d) ) c · d –greatest lower bound (glb) is an element c such that c · a, c · b, and 8 d 2 S. (d · a Æ d · b) ) d · c lub and glb don’t always exists:

15. Lattices A lattice is a tuple (S, v, ?, >, t, u ) such that: –(S, v ) is a poset – 8 a 2 S. ? v a – 8 a 2 S. a v > –Every two elements from S have a lub and a glb – t is the least upper bound operator, called a join – u is the greatest lower bound operator, called a meet

16. Examples of lattices Powerset lattice

17. Examples of lattices Powerset lattice

18. Examples of lattices Booleans expressions

19. Examples of lattices Booleans expressions

20. Examples of lattices Booleans expressions

21. Examples of lattices Booleans expressions

22. End of background material

23. Back to our example let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ; for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) [ info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

24. Back to our example We formalize our domain with a powerset lattice What should be top and what should be bottom?

25. Back to our example We formalize our domain with a powerset lattice What should be top and what should be bottom? Does it matter? –It matters because, as we’ve seen, there is a notion of approximation, and this notion shows up in the lattice

26. Direction of lattice Unfortunately: –dataflow analysis community has picked one direction –abstract interpretation community has picked the other We will work with the abstract interpretation direction Bottom is the most precise (optimistic) answer, Top the most imprecise (conservative)

27. Direction of lattice Always safe to go up in the lattice Can always set the result to > Hard to go down in the lattice So... Bottom will be the empty set in reaching defs

28. Worklist algorithm using lattices let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ? for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) t info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

29. Termination of this algorithm? For reaching definitions, it terminates... Why? –lattice is finite Can we loosen this requirement? –Yes, we only require the lattice to have a finite height Height of a lattice: length of the longest ascending or descending chain Height of lattice (2 S, µ ) =

30. Termination of this algorithm? For reaching definitions, it terminates... Why? –lattice is finite Can we loosen this requirement? –Yes, we only require the lattice to have a finite height Height of a lattice: length of the longest ascending or descending chain Height of lattice (2 S, µ ) = | S |

31. Termination Still, it’s annoying to have to perform a join in the worklist algorithm It would be nice to get rid of it, if there is a property of the flow functions that would allow us to do so while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) t info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

32. Even more formal To reason more formally about termination and precision, we re-express our worklist algorithm mathematically We will use fixed points to formalize our algorithm

33. Fixed points Recall, we are computing m, a map from edges to dataflow information Define a global flow function F as follows: F takes a map m as a parameter and returns a new map m’, in which individual local flow functions have been applied

34. Fixed points We want to find a fixed point of F, that is to say a map m such that m = F(m) Approach to doing this? Define ?, which is ? lifted to be a map: ? = e. ? Compute F( ? ), then F(F( ? )), then F(F(F( ? ))),... until the result doesn’t change anymore

35. Fixed points Formally: We would like the sequence F i ( ? ) for i = 0, 1, 2... to be increasing, so we can get rid of the outer join Require that F be monotonic: – 8 a, b. a v b ) F(a) v F(b)

36. Fixed points

38. Back to termination So if F is monotonic, we have what we want: finite height ) termination, without the outer join Also, if the local flow functions are monotonic, then global flow function F is monotonic

39. Another benefit of monotonicity Suppose Marsians came to earth, and miraculously give you a fixed point of F, call it fp. Then:

40. Another benefit of monotonicity Suppose Marsians came to earth, and miraculously give you a fixed point of F, call it fp. Then:

41. Another benefit of monotonicity We are computing the least fixed point...

42. Recap Let’s do a recap of what we’ve seen so far Started with worklist algorithm for reaching definitions

43. Worklist algorithm for reaching defns let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ; for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) [ info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

44. Generalized algorithm using lattices let m: map from edge to computed value at edge let worklist: work list of nodes for each edge e in CFG do m(e) := ? for each node n do worklist.add(n) while (worklist.empty.not) do let n := worklist.remove_any; let info_in := m(n.incoming_edges); let info_out := F(n, info_in); for i := 0.. info_out.length do let new_info := m(n.outgoing_edges[i]) t info_out[i]; if (m(n.outgoing_edges[i])  new_info]) m(n.outgoing_edges[i]) := new_info; worklist.add(n.outgoing_edges[i].dst);

45. Next step: removed outer join Wanted to remove the outer join, while still providing termination guarantee To do this, we re-expressed our algorithm more formally We first defined a “global” flow function F, and then expressed our algorithm as a fixed point computation

46. Guarantees If F is monotonic, don’t need outer join If F is monotonic and height of lattice is finite: iterative algorithm terminates If F is monotonic, the fixed point we find is the least fixed point. Any questions so far?

47. What about if we start at top? What if we start with > : F( > ), F(F( > )), F(F(F( > )))

48. What about if we start at top? What if we start with > : F( > ), F(F( > )), F(F(F( > ))) We get the greatest fixed point Why do we prefer the least fixed point? –More precise

49. Graphically x y 10

50. Graphically x y 10

51. Graphically x y 10

52. Graphically, another way

53. Another example: constant prop Set D = x := N in out F x := N (in) = x := y op z in out F x := y op z (in) =

54. Another example: constant prop Set D = 2 { x ! N | x 2 Vars Æ N 2 Z } x := N in out F x := N (in) = in – { x ! * } [ { x ! N } x := y op z in out F x := y op z (in) = in – { x ! * } [ { x ! N | ( y ! N 1 ) 2 in Æ ( z ! N 2 ) 2 in Æ N = N 1 op N 2 }

55. Another example: constant prop *x := y in out F *x := y (in) = x := *y in out F x := *y (in) =

56. Another example: constant prop *x := y in out F *x := y (in) = in – { z ! * | z 2 may-point(x) } [ { z ! N | z 2 must-point-to(x) Æ y ! N 2 in } [ { z ! N | (y ! N) 2 in Æ (z ! N) 2 in } x := *y in out F x := *y (in) = in – { x ! * } [ { x ! N | 8 z 2 may-point-to(x). (z ! N) 2 in }

57. Another example: constant prop x := f(...) in out F x := f(...) (in) = *x := *y + *z in out F *x := *y + *z (in) =

58. Another example: constant prop x := f(...) in out F x := f(...) (in) = ; *x := *y + *z in out F *x := *y + *z (in) = F a := *y;b := *z;c := a + b; *x := c (in)

59. Another example: constant prop s: if (...) in out[0]out[1] merge out in[0]in[1]