1. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT Optimizing Compilers CISC 673 Spring 2011 Yet More Data flow analysis John Cavazos University of Delaware

2. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 2 Overview Additional Data Flow Problem Constant Propagation Generalization of Data Flow

3. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 3 Constant Propagation Along every path to point p, variable v has same “known” value

4. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 4 Constant Propagation Specialize computation at p based on v’s value

5. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 5 Constant Propagation Lattice Lattice is a poset with (V, ≤) V meet (or join) operator ≤ ordering operator … -2 -1 0 1 2... The Lattice of Constant Prop Hasse diagram

6. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 6 Two Special Elements in Lattice ⊤ (called top) Not a Constant ⊥ (called bottom) Unknown

7. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 7 Some Lattice Theory Generalized Meet Operator ⊔ Used at “merge” points

8. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 8 Some Lattice Theory (cont’d) Meet Operator ( ⊔ ) for Avail Expressions Avail Expression (w/ sets):  Avail Expression (w/ Bit Vectors): ⋀ Produces Least Upper Bound (LUB) X ⊔ Y = least element Z s.t. X ≤ Z and Y ≤ Z

9. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 9 Iterative Data Flow Analysis Initialize non-entry nodes to identify element Identity element for meet function Remember last time: X ⋀ 1 = X

10. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 10 Iterative Data Flow Analysis (cont’d) If node function is monotone: Each re-evaluation of node moves up the lattice, if it moves at all If height of lattice is finite, must terminate

11. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 11 Top ⊤ versus Bottom ⊥ ⊤ means definitely NOT a constant Inputs are ⊤ ⊥ means Unknown or Undefined Could be constant but we do not know!

12. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 12 More Lattice Theory: Ordering Op ≤ ordering operator ⊥ ≤ anything ∀ x ⊥ ⊔ x = x anything ≤ ⊤ ∀ x x ⊔ ⊤ = ⊤ ∀ x x ⊔ x = x … -2 -1 0 1 2... The Lattice of Constant Prop

13. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 13 Relate to Constant Prop?! i and j are integer values i ⊔ i = i i ⊔ j = ⊤ if i ≠ j

14. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 14 What is X in Block 3? 1.2. 3.

15. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 15 What is X in Block 3? 1.2. 3. x is ⊤

16. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 16 Constant Propagation Problem Variable v has constant value c at point p, iff ∀ paths from entry to p, the most recently assigned value for v is c

17. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 17 Constant Prop Example Meet Operator = ⊔ Identity= ? X ⊔ ? = X

18. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 18 Constant Prop Example Meet Operator = ⊔ Identify= ⊥ X ⊔ ⊥ = X Initial Condition?

19. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 19 Constant Prop Example Meet Operator = ⊔ Identify= ⊥ X ⊔ ⊥ = X Initial Condition Boundary Condition? ⊥⊥⊥ XYZ

20. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 20 Constant Prop Example Meet Operator = ⊔ Identify= ⊥ X ⊔ ⊥ = X Initial Condition Boundary Condition ⊥⊥⊥ 1. 2.3. 4. XYZ

21. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 21 Constant Prop Example ⊥⊥⊥ 1. 2.3. 4. out 1 = ??? ⊥⊥ ⊥ XYZ

22. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 22 Constant Prop Example ⊥⊥⊥ 1. 2.3. 4. out 1 = 1 ⊥⊥ out 2 = ??? ⊥⊥ ⊥ 1 ⊥⊥ XYZ

23. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 23 Constant Prop Example ⊥⊥⊥ 1. 2.3. 4. out 1 = 1 ⊥⊥ out 2 = 023 out 3 = ??? ⊥⊥ ⊥ 1 ⊥⊥ 023 XYZ

24. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 24 Constant Prop Example ⊥⊥⊥ 1. 2.3. 4. out 1 = 1 ⊥⊥ out 2 = 023 out 3 = 12 ⊥ out 4 = ??? ⊥⊥ ⊥ 1 ⊥⊥ 023 XYZ 12 ⊥

25. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 25 Constant Prop Example ⊥⊥⊥ 1. 2.3. 4. out 1 = 1 ⊥⊥ out 2 = 023 out 3 = 12 ⊥ out 4 = ⊤ 23 ⊥⊥ ⊥ 1 ⊥⊥ 023 XYZ 12 ⊥ ⊤ 23 Does this make sense?

26. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 26 Next Time Static-Single Assignment (SSA Form) Read Efficiently computing static single assignment form and the control dependence graph, Cytron et al. http://portal.acm.org/citation.cfm?id=115320

27. U NIVERSITY OF D ELAWARE C OMPUTER & I NFORMATION S CIENCES D EPARTMENT 27 Next Time Intelligent Compilation