1. Global optimizations

2. Local analysis framework
Define an analysis of a basic block as a quadruple (D, V, F, I) where
D is a direction (forwards or backwards)
V is a set of values the program can have at any point
F is a family of transfer functions defining the meaning of any expression as a function f : V  V
I is the initial information at the top (or bottom) of a basic block

3. Available Expressions
Direction: Forward
Values: Sets of expressions assigned to variables
Transfer functions: Given a set of variable assignments V and statement a = b + c:
Remove from V any expression containing a as a subexpression
Add to V the expression a = b + c
Formally: Vout = (Vin \ {e | e contains a})  {a = b + c}
Initial value: Empty set of expressions

4. Available expressions analysis
IN[s]
OUT[s]
entry: {}
Initial value
a = b;
{a=b}
c = b;
{a=b, c=b}
d = a + b;
{a=b, c=b, d=a+b}
e = a + b;
{a=b, c=b, d=a+b, e=a+b}
d = b;
{a=b, c=b, d=b, e=a+b}
f = a + b;
{a=b, c=b, d=b, e=a+b, f=a+b}

5. Optimizing from available expressions
Common sub-expression elimination
If {… t = y op z … } x = y op z
Can transform statement into x = t
Copy propagation
If {… y = t … } x = y op z
Can transform statement into x = t op z
Note: same for x=y and x=op y

6. Common sub-expression elimination
entry: {}
a = b;
{a=b}
c = b;
{a=b, c=b}
d = a + b;
{a=b, c=b, d=a+b}
e = d;
e = a + b;
{a=b, c=b, d=a+b, e=a+b}
d = b;
{a=b, c=b, d=b, e=a+b}
f = e;
f = a + b;
{a=b, c=b, d=b, e=a+b, f=a+b}

7. Liveness Analysis Direction: Backward Values: Sets of variables
Transfer functions: Given a set of variable assignments V and statement a = b + c:
Remove a from V (any previous value of a is now dead)
Add b and c to V (any previous value of b or c is now live)
Formally: Vin = (Vout \ {a})  {b,c}
Initial value: Depends on semantics of language
E.g., function arguments and return values (pushes)
Result of local analysis of other blocks as part of a global analysis

8. Liveness analysis s IN[s] OUT[s] { b } a = b; { a, b } c = a; { a, b }
d = a + b;
{ a, b, d }
e = d;
{ a, b, e }
d = a;
{ b, d, e }
f = e;
Initial value
{ b, d }
exit:

9. Optimizing from liveness analysis
Dead code elimination
If x = y op z {v1,…,vk}
And x  {v1,…,vk}
We can eliminate x = y op z
Note: same for x=y and x=op y

10. Dead code elimination { b } a = b; c = a; d = a + b; e = d; { a, b}
f = e;
{ b, d }
exit:

11. Zero value Analysis Direction: Forward
Values: Sets of valuations (values assigned to variables)
Transfer functions:
Given a set of variable assignments V and statement a = b op c: update valuation of a at V according to values of b, c and operator op.
Formally: Vout = (Vin \ {a = value’})  {a = value}
Initial value: Depends on semantics of language
E.g., function arguments and return values (pushes)
Result of local analysis of other blocks as part of a global analysis

12. Zero value Analysis s IN[s] OUT[s] entry: {b= 𝟎 + } a = b;
Initial value
a = b;
{b= 𝟎 + , a= 𝟎 + }
c = a + b;
{b= 𝟎 + , a= 𝟎 + , c= 𝟎 + }
d = a / c;
{b= 𝟎 + , a= 𝟎 + , c= 𝟎 + , d= 𝟎 + }
c = b - b;
{b= 𝟎 + , a= 𝟎 + , c=0, d= 𝟎 + }
a = d - b;
{b= 𝟎 + , a=?, c=0, d= 𝟎 + }
f = d / c;
{b= 𝟎 + , a=?, c=0, d= 𝟎 + , f=?}

13. Optimizing from zero value analysis
Safe division
If x = y / z {v1,…,vk}
And z  {v1,…,z=0,…,vk} or {v1,…,z=?,…,vk}
We can update x = y safe div z

14. Safe division entry: {b= 𝟎 + } a = b; {b= 𝟎 + , a= 𝟎 + } c = a + b;
d = a / c;
{b= 𝟎 + , a= 𝟎 + , c= 𝟎 + , d= 𝟎 + }
c = b - b;
{b= 𝟎 + , a= 𝟎 + , c=0, d= 𝟎 + }
a = d - b;
{b= 𝟎 + , a=?, c=0, d= 𝟎 + }
f = d safe div c;
f = d / c;
{b= 𝟎 + , a=?, c=0, d= 𝟎 + , f=?}

15. Global Optimizations

16. Global dead code elimination
Local dead code elimination needed to know what variables were live on exit from a basic block
This information can only be computed as part of a global analysis
How do we modify our liveness analysis to handle a CFG?

17. CFGs without loops b = c + d; e = c + d; Entry x = c + d; a = b + c;
y = a + b;
x = a + b; y = c + d;
Exit

18. CFGs without loops ? {a, c, d} b = c + d; e = c + d; Entry
Which variables may be live on some execution path?
Entry
{a, b, c, d} ?
x = c + d; a = b + c;
{b, c, d}
{a, b, c, d}
y = a + b;
{a, b, c, d}
{a, b, c, d}
x = a + b; y = c + d;
{a, b, c, d}
{x, y}
{x, y}
Exit

19. CFGs without loops b = c + d; e = c + d; {c, d} Entry {b, c, d}
Need to combine currently-computed value with new value
Entry
{b, c, d}
x = c + d; a = b + c;
{b, c, d}
{a, b, c, d}
y = a + b;
{a, b, c, d}
{a, b, c, d}
x = a + b; y = c + d;
{a, b, c, d}
{x, y}
{x, y}
Exit

20. CFGs without loops {a, c, d} b = c + d; e = c + d; Entry {a, b, c, d}
x = c + d; a = b + c;
{a, b, c, d}
y = a + b;
{a, b, c, d}
{a, b, c, d}
x = a + b; y = c + d;
{a, b, c, d}
{x, y}
{x, y}
Exit

21. CFGs without loops b = c + d; Entry a = b + c; x = a + b; y = c + d;
Exit

22. CFGs without loops b = c + d; Entry a = b + c; x = a + b; y = c + d;
Exit

23. CFGs with loops b = c + d; c = c + d; IfZ ... Entry
a = b + c; d = a + c;
c = a + b;
a = a + b; d = b + c; ?
Exit
{a}

24. CFGs with loops - initialization{}
b = c + d; c = c + d;
Entry
a = b + c; d = a + c; {} {}
c = a + b;
a = a + b; d = b + c; {}
Exit
{a}

25. CFGs with loops - iteration{}
b = c + d; c = c + d;
Entry {}
a = b + c; d = a + c; {}
c = a + b;
a = a + b; d = b + c; {}
{a}
Exit
{a}

26. CFGs with loops - iteration{}
b = c + d; c = c + d;
Entry {}
a = b + c; d = a + c; {}
c = a + b;
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

27. CFGs with loops - iteration{}
b = c + d; c = c + d;
Entry {}
a = b + c; d = a + c; {}
c = a + b;
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

28. CFGs with loops - iteration{}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c; {}
c = a + b;
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

29. CFGs with loops - iteration{}
b = c + d; c = c + d;
Entry
{b, c}
{b, c}
a = b + c; d = a + c; {}
c = a + b;
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

30. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c} {}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

31. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

32. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a}
Exit
{a}

33. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

34. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

35. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

36. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

37. CFGs with loops - iteration
{c, d}
b = c + d; c = c + d;
Entry
{a, b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

38. CFGs with loops - iteration
{a, c, d}
b = c + d; c = c + d;
Entry
{a, b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

39. CFGs with loops - iteration
{a, c, d}
b = c + d; c = c + d;
Entry
{a, b, c}
a = b + c; d = a + c;
{b, c}
{a, b}
c = a + b;
{a, b, c}
{a, b, c}
a = a + b; d = b + c;
{a, b, c}
{a, c, d}
Exit
{a}

40. Join semilattices
A join semilattice is a ordering defined on a set of elements
Any two elements have some join that is the smallest element larger than both elements
There is a unique bottom element, which is smaller than all other elements
Intuitively:
The join of two elements represents combining information from two elements by an overapproximation
The bottom element represents “no information yet” or “the least conservative possible answer”

41. Formal definitions A join semilattice is a pair (V, ), where
V is a domain of elements
 is a join operator that is
commutative: x  y = y  x
associative: (x  y)  z = x  (y  z)
idempotent: x  x = x
If x  y = z, we say that z is the join or (least upper bound) of x and y
Every join semilattice has a bottom element denoted  such that   x = x for all x

42. A general framework
A global analysis is a tuple (D, V, , F, I), where
D is a direction (forward or backward)
The order to visit statements within a basic block, not the order in which to visit the basic blocks
V is a set of values
 is a join operator over those values
F is a set of transfer functions f : V  V
I is an initial value
The only difference from local analysis is the introduction of the join operator

43. A semilattice for zero value analysis
One possible semilattice for this analysis is shown here (for each variable): ?
𝟎 −
𝟎 +
Undefined

44. Global constant propagation
entry Undefined
x = 6; Undefined
y = x; Undefined
z = y; Undefined
x=Undefined y=Undefined
z=Undefined
w=Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

45. Global constant propagation
x = 6; Undefined
entry Undefined
y = x; Undefined
z = y; Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

46. Global constant propagation
Undefined
x = 6; Undefined
entry Undefined
y = x; Undefined
z = y; Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

47. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
y = x; Undefined
z = y; Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

48. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
y = x; Undefined
z = y; Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

49. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; Undefined
z = y; Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

50. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

51. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
w = x; Undefined
y= 𝟎 +  y=Undefined gives what?
z = y / x; Undefined
x = -5; Undefined
exit

52. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; Undefined
z = y / x; Undefined
x = -5; Undefined
exit

53. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
z = y / x; Undefined
x = -5; Undefined
exit

54. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
z = y / x; Undefined
x = -5; Undefined
exit

55. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; Undefined
x = -5; Undefined
exit

56. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x = -5; Undefined
exit

57. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x = -5; Undefined
exit

58. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; Undefined
exit

59. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

60. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

61. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; Undefined
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

62. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; x = 𝟎 +
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

63. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; x = 𝟎 +
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
x= 𝟎 +  x= 𝟎 − gives what?
x=y=w= 𝟎 +
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

64. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; x = 𝟎 +
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
y=w= 𝟎 + ,x=?
z = y / x; x=y=w=z= 𝟎 +
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

65. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; x = 𝟎 +
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
y=w= 𝟎 + ,x=?
z = y / x; y=w= 𝟎 + ,x=z=?
x=y=w=z= 𝟎 + x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

66. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; x = 𝟎 +
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
y=w= 𝟎 + ,x=?
z = y / x; y=w= 𝟎 + ,x=z=?
y=w= 𝟎 + ,x=z=? x = -5; x= 𝟎 − , y=w=z= 𝟎 +
exit

67. Global constant propagation
Undefined
x = 6; x = 𝟎 +
entry Undefined
x= 𝟎 + y = x; x= 𝟎 + ,y= 𝟎 +
x = 𝟎 + z = y; x = 𝟎 +
x= 𝟎 + ,y= 𝟎 +
w = x; x=y=w= 𝟎 +
Global analysis reached fixpoint
y=w= 𝟎 + ,x=?
z = y / x; y=w= 𝟎 + ,x=z=?
y=w= 𝟎 + ,x=z=? x = -5; y=w= 𝟎 + ,x=z=?
exit