Advanced Program Representations: Understanding Static Single Assignment Form (SSA)

1. CS 201Compiler Construction Lecture 9 Static Single Assignment Form

2. Program Representations Why develop Advanced Program Representations? • To develop faster algorithms • To develop more powerful algorithms Superior representation for Data Flow Static Single Assignment Form (SSA Form) superior to def-use chains Superior representation for Control Flow Control Dependence Graph superior to control flow graph

3. SSA-Form A program in SSA-form satisfies the following two properties: • A use of a variable is reached by exactly one definition of that variable. • The program is augmented with ϕ–nodes that distinguish values of variables transmitted on distinct incoming control flow edges.

4. Example K11 L11 Repeat K2ϕ(K1,K5) L2ϕ(L1,L6) if (P) then if (Q) then L32 else L43 L5ϕ(L3,L4) K3K2+1 else K4K2+2 K5ϕ(K3,K4) L6ϕ(L2,L5) Until (T) K  1 L  1 Repeat if (P) then if (Q) then L2 else L3 KK+1 else KK+2 Until (T)

5. SSA-Form Observations: • SSA-form has def-use information textually embedded in it. Given a use, we know where the definition comes from. • SSA-form is more compact representation of def-use chains. • Def-use chains: #defs x #uses – O(n2) • SSA-form: 2 x #defs or #uses – O(n)

6. Constructing SSA-Form Step 1: Introduce functions at certain points in the program -- v  ϕ (v,v,….) where of operands equals number of control predecessors and ith operand corresponds to ith predecessor.

7. Contd.. Step 2: Each variable v is given several new names v1, v2, …. Such that every name appears exactly once on the left hand side of an assignment.

8. Step 1: Introducing ϕ-functions Node Z needs a ϕ-function for variable V if Z is the first node common to two non-null paths that originate at two different nodes each containing: an assignment to V; or a ϕ-function for V.

9. Step 1 Contd.. Definition: X strictly dominates Y ≅ X dominates Y & X != Y. Definition: Immediate dominator of a node is its closest strict dominator. Notation: X = idom(Y). Definition: Dominance Frontier DF(X) = {Y: there exists P εpred(Y) such that X dominates P & X does not strictly dominate Y} ϕ-functions are placed at nodes in DF nodes of nodes with assignments.

10. Step 1 Contd.. Y ε DF(X) Strict Domination Does Not Strictly Domination Dominate

11. Step 1 Contd.. Observation: if Y εDF(X) then there may or may not be a direct edge from X to Y.

12. Step 1 Contd.. Computing Dominance Frontier: for each Y εsucc(X) do if idom(Y) != X then DF(X) = DF(X) U {Y} for each Z εChidren(X) in the dominator tree do for each Y εDF(Z) do if idom(Y) != X then DF(X) = DF(X) U {Y} Compute bottom-up order According to dominator tree

13. Step 1 Contd.. Dominator Tree

14. Step 1 Contd.. Dominance Frontier of a Set of Nodes S DF(S) = DF(X) Iterated Dominance Frontier DF+(S): DF1 = DF(S) DFi+1 = DF(S U DFi) S – set of nodes which assign to variable V DF+(S) – set of nodes including those where ϕ-functions must be placed.

15. Step 2: Rename the Variables Step 2: For each variable v rename its left hand side occurrences as v1, v2, …. Perform reaching definition analysis to identify names to use in the right hand side occurrences of v.