1

CS 201Compiler Construction

Lecture 9Static Single Assignment Form

Program Representations

Why develop Advanced Program Representations?

-To develop faster algorithms-To develop more powerful algorithms

Superior representation for Data FlowStatic Single Assignment Form (SSA Form)

superior to def-use chains

Superior representation for Control FlowControl Dependence Graph

superior to control flow graph2

SSA-Form

A program in SSA-form satisfies the following two properties:

1. A use of a variable is reached by exactly one definition of that variable.

1. The program is augmented with ϕ–nodes that distinguish values of variables transmitted on distinct incoming control flow edges.

3

Example

K 1L 1Repeat

if (P) thenif (Q) then

L2else L3KK+1

else KK+2Until (T)

4

K11L11Repeat

K2ϕ(K1,K5)L2ϕ(L1,L6)if (P) then

if (Q) thenL32

else L43L5ϕ(L3,L4)K3K2+1

else K4K2+2K5ϕ(K3,K4)L6ϕ(L2,L5)

Until (T)

SSA-Form

Observations:

1. SSA-form has def-use information textually embedded in it. Given a use, we know where the definition comes from.

1. 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)

5

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.

6

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.

7

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.

8

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.

9

Step 1 Contd..

Strict Domination Does Not StrictlyDomination Dominate

10

Y ε DF(X)

Step 1 Contd..

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

11

Step 1 Contd..

12

Computing Dominance Frontier:for each Y εsucc(X) do

if idom(Y) != X thenDF(X) = DF(X) U {Y}

for each Z εChidren(X) in the dominator tree dofor each Y εDF(Z) do

if idom(Y) != X then DF(X) = DF(X) U {Y}

Compute bottom-up order According to dominator tree

Step 1 Contd..

13

Dominator Tree

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.

14

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.

15

16

Sample ProblemsStatic Single Assignment

17

SSA Form

18

Applications of SSA Form

Global Value Numbering

A technique for determining when two computations in a program are equivalent can be used for redundancy removal.

Constant Propagation -- by computing values of two computations they can be shown to be equivalent.

Common Subexpression Elimination -- lexically identical expressions can be shown to be equivalent.

Value Numbering -- lexically different expressions can be shown to be equivalent without computing their values.

19

Examples

20

Examples

21

Value Numbering Algorithm

22

Value Numbering Algorithm

23