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Optimizing CompilersCISC 673

Spring 2011

Yet More Data flow analysis

John CavazosUniversity of Delaware

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Overview

Additional Data Flow Problem Constant Propagation Generalization of Data Flow

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Constant Propagation

Along every path to point p, variable v has same “known” value

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Constant Propagation

Specialize computation at p based on v’s value

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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

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Two Special Elements in Lattice

⊤ (called top) Not a Constant

⊥ (called bottom) Unknown

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Some Lattice Theory

Generalized Meet Operator ⊔ Used at “merge” points

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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

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Iterative Data Flow Analysis

Initialize non-entry nodes to identify element Identity element for meet function

Remember last time:

X ⋀ 1 = X

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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

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Top ⊤ versus Bottom ⊥

⊤ means definitely NOT a constant Inputs are ⊤

⊥ means Unknown or Undefined Could be constant but we do not

know!

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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

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Relate to Constant Prop?!

i and j are integer values i ⊔ i = i i ⊔ j = ⊤ if i ≠ j

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What is X in Block 3?

1. 2.

3.

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What is X in Block 3?

1. 2.

3.

x is ⊤

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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

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Constant Prop Example

Meet Operator = ⊔ Identity= ? X ⊔ ? = X

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Constant Prop Example

Meet Operator = ⊔ Identify= ⊥ X ⊔ ⊥ = X Initial Condition?

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Constant Prop Example

Meet Operator = ⊔ Identify= ⊥ X ⊔ ⊥ = X Initial Condition Boundary Condition?

⊥⊥⊥

⊥⊥⊥⊥⊥⊥

⊥⊥⊥

XYZ

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Constant Prop Example

Meet Operator = ⊔ Identify= ⊥ X ⊔ ⊥ = X Initial Condition Boundary Condition

⊥⊥⊥

⊥⊥⊥⊥⊥⊥

⊥⊥⊥

⊥⊥⊥

1.

2. 3.

4.

XYZ

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Constant Prop Example

⊥⊥⊥

⊥⊥⊥⊥⊥⊥

⊥⊥⊥

1.

2. 3.

4.

out1= ???⊥⊥⊥XYZ

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Constant Prop Example

⊥⊥⊥⊥⊥⊥

⊥⊥⊥

1.

2. 3.

4.

out1= 1⊥⊥out2= ???

⊥⊥⊥

1⊥⊥

XYZ

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Constant Prop Example

⊥⊥⊥

⊥⊥⊥

1.

2. 3.

4.

out1= 1⊥⊥out2= 023out3= ???

⊥⊥⊥

1⊥⊥

023

XYZ

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Constant Prop Example

⊥⊥⊥

1.

2. 3.

4.

out1= 1⊥⊥out2= 023out3= 12⊥out4= ???

⊥⊥⊥

1⊥⊥

023

XYZ

12⊥

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Constant Prop Example

1.

2. 3.

4.

out1= 1⊥⊥out2= 023out3= 12⊥

out4= ⊤23

⊥⊥⊥

1⊥⊥

023

XYZ

12⊥

⊤23

Does this make sense?

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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

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Next Time

Intelligent Compilation