1 CS 201 Compiler Construction Lecture 9 Static Single Assignment Form

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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.

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Example

K 1L 1Repeat

if (P) thenif (Q) then

L2else L3KK+1

else KK+2Until (T)

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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)

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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.

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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.

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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.

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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.

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Step 1 Contd..

Strict Domination Does Not StrictlyDomination Dominate

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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.

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Step 1 Contd..

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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..

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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.

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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.

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Sample ProblemsStatic Single Assignment

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SSA Form

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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.

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Examples

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Examples

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Value Numbering Algorithm

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Value Numbering Algorithm

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