1. 1 CS 201 Compiler Construction Data Flow Framework

2. 2 The various problems considered have things in common: –Transfer functions –Confluence Operator –Direction of Propagation These problems can be treated in a unified way  data flow framework is an algebraic structure used to encode and solve data flow problems.

3. 3 Monotone Data Flow Framework Components of the framework: 1. Information Set: L 2. Effect of joining paths: ∧ (meet operator) 3. Effect of basic blocks: f n (monotone transfer func.) 4. Iterative Solution: can be shown to terminate (L, ∧ ) is a semilattice st ∨ a,b,c εL 1. a ∧ a = a (idempotent) 2. a ∧ b = b ∧ a (commutative) 3. a ∧ (b ∧ c) = (a ∧ b) ∧ c (assocative) Bottom Element st ∨ a ε L, a ∧ = Top Element Τ st ∨ a ε L, a ∧ Τ = a Top & Bottom elements are unique.

4. Contd.. Relation ≤ is a partial order on L a ≤ b ≅ a ∧ b = a Similarly relation > can be defined. A semilattice is bounded iff ∨ a εL there exists a constant c a st length of chain beginning at a is at most c a. 4 Max c a

5. 5 Monotonic Functions Effect of each basic block is modeled by a transfer function f: L  L. Function f must be monotonic. A total function f: L  L is monotonic iff ∨ a,b ε L f(a ∧ b) ≤ f(a) ∧ f(b) Distributive function: f(a ∧ b) = f(a) ∧ f(b) For monotonic functions: a ≤ b => f(a) ≤ f(b)

6. 6 Contd.. For monotonic functions: a ≤ b => f(a) ≤ f(b) Proof: f(a ∧ b) ≤ f(a) ∧ f(b) Defn. of Monotonicity f(a ∧ b) ∧ f(a) ∧ f(b) = f(a ∧ b) Defn. of ≤:(a ∧ b=a) f(a) ∧ f(a) ∧ f(b)= f(a) Given a ≤ b: a ∧ b = a f(a) ∧ f(b) = f(a) f(a) ∧ f(a) = f(a) idempotence f(a) ≤ f(b) Defn. of ≤

7. 7 Fixpoint A fixpoint of a monotonic function f: L  L is a value a ε L such that f(a) = a Τ > f (Τ) > f ( f (Τ) ) > f ( f ( f (Τ) ) ) …….. There exists t such that f ( f t (Τ) ) = f t (Τ) f t (Τ) is the greatest fixpoint of f.

8. 8 Monotone Function Space A monotone function space for a semilattice is a set F of monotonic functions which: 1. Contains the identity function (id) -- basic blocks may not modify information 2. Is closed under function composition -- to model the effects of paths 3. For each a ε L, there exists fεF st f( ) = a -- to model gen functions A distributive function space is a monotone function space in which all functions are distributive.

9. 9 A Monotone Data Flow System A monotone data flow system is a tuple 1. (L, ∧ ) is a bounded semilattice with Τ & 2. F is the monotone function space 3. G = (N, E, s) is the program flow graph 4. FM: N  F is a total function that associates a function from F with each basic block.

10. 10 Meet Over All Paths Solution Meet over all paths solution (MOP) of a data flow system – MOP: N  L MOP(s) = NULL (NULL is the element in L which represents “no information”) F f π is composition of functions from nodes along path π excluding node n. n 1  n 2  n 3  ….  n k-1  n k  fn k o fn k-1 o….o fn 2 o fn 1

11. 11 MOP Solution Finding MOP solution is undecidable, i.e. there does not exist a general algorithm that computes MOP solution for all monotone data flow systems. Let X: N  L denote a total function that associates nodes with lattice elements. X is conservative or safe iff ∨ n εN, X(n) ≤ MOP(n) Iterative algorithm computes conservative approximation of MOP. For distributive data flow systems, it computes solution that is identical to MOP solution.

12. 12 Iterative Algorithm

13. 13 Reaching Definitions

14. 14 Contd…,

15. 15 Dominators

16. 16 Constant Propagation f (X)={(a,2),(b,3),(c,5)}f (Y)={(a,3),(b,2),(c,5)} f (X) ∧ f (Y) = {(a,not-const), (b, not-const), (c,5)} X ∧ Y = {(a,not-const),(b,not-const),(c,undef)} f (X ∧ Y) = {(a,not-const),(b,not-const),(c,not-const)} f (X ∧ Y) ≤ f(X) ∧ f(Y)

17. 17 Sample Problems Data Flow Framework

18. 18 Data Flow Framework For each of the given problems provide the following: Lattice values Meet operator Top and bottom elements The partial order relation, including its pictorial representation The transfer function Data flow equations.

19. 19 1.Reachable uses -- for each definition identify the set of uses reachable by the definition. This information is used for computing def-use chains. 2. Reaching uses -- given a definition of variable x, identify the set of uses of x that are encountered prior to reaching the definition and there is no other definitions of x that intervene the use and the definition. This information is used for computing antidependences.

20. 20 3. Classify Variable Values -- classify the value of each program variable at each program point into one of the following categories: (a) the value is a unique constant -- you must also identify this constant value; (b) the value is one-of-many constants – you do not have to compute the identities of these constants as part of your solution; and (c) the value is not-a-constant, that is, it is neither a unique constant nor a one-of-many constants. This is a generalization of constant propagation.

21. 21 4. Postdominators -- postdominator set of a node is the set of nodes that are encountered along all paths from the node to the end node of the control flow graph. This information is used for computing control dependence.