More CFGStatic Single Assignment

ECE 5775High-Level Digital Design Automation

Fall 2018

▸ HW 1 due next Monday (9/17)

▸ Lab 2 will be released tonight (due 9/24)

▸ Instructor OH cancelled this afternoon

1

Announcements

▸ More on control flow analysis – Dominance frontier

▸ Dataflow analysis – static single assignment (SSA)– SSA Definition– PHI node (F-node) placement– Code optimizations with SSA

2

Outline

▸ What becomes a leader statement of a basic block? (three cases)

▸ Does a node strictly (or properly) dominate itself?

▸ Does a predecessor of a node B always dominate B?

▸ Suppose A that dominates all of B’s predecessors. Does A always dominate B?

3

Revisiting Control Flow Analysis

Review: Finding Loops

▸ Loop identification algorithm– Find an edge B®H where H dominates B;

This edge is called a back-edge

– Find all nodes that (1) are dominated by H and (2) can reach B through nodes dominated by H; add them to the loop • H and B are naturally included

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5

Finding Loops (1)

1

2

3

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1)4

7

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Finding Loops (1)

1

2

3

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph4

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7

Finding Loops (2)

1

2

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph(10,7)

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4

7

Intuition of Dominance Relation

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Imagine a source of lightat the entry node, and thatthe edges are optical fibers

To find which nodes are dominated by a given node, place an opaque barrier at that node and observe which nodes become dark

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2

5 6

8

9 10

3

4

7

entry

9

Finding Loops (2)

1

2

3

4

7

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph(10,7)

10

Finding Loops (2)

1

2

3

4

7

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph(10,7) {7,8,10}

(7,4)

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Finding Loops (3)

1

2

3

4

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph(10,7) {7,8,10}

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(7,4)

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Finding Loops (3)

1

2

3

4

7

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph(10,7) {7,8,10}

13

Finding Loops (3)

1

2

3

4

7

5 6

8

9 10

Find all back edges in this graphand the natural loop associated with each back edge

(9,1) Entire graph(10,7) {7,8,10}(7,4) {4,5,6,7,8,10}

Static Single Assignment

▸ Static single assignment (SSA) form is a restricted IR where– Each variable definition has a unique name– Each variable use refers to a single definition

▸ SSA simplifies data flow analysis & many compiler optimizations – Eliminates artificial dependences (on scalars)

• Write-after-write• Write-after-read

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SSA within a Basic Block

▸ Assign each variable definition a unique name▸ Update the uses accordingly

x = read()x = x * 5x = x + 1y = x * 9

x0 = read()x1 = x0 * 5x2 = x1 + 1y = x2 * 9

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×

+

×

5

x1

x0

x2

1

9

yOriginal code SSA form

Corresponding data flow graph

SSA with Control Flow

▸ Consider a situation where two control-flow paths merge– e.g., due to an if-then-else statement or a loop

x = read()if (x > 0)

y = 5else

y = 10x = y

y = 5 y = 10

x = y

x = read()if (x > 0)

y0 = 5 y1 = 10

x1 = y

x0 = read()if (x0 > 0)

should this be y0 or y1?

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Introducing ϕ-Node

▸ Inserts special join functions (called ϕ-nodes or PHI nodes) at points where different control flow paths converge

y0 = 5 y1 = 10

y2 = ϕ(y0, y1)x1 = y2

if (x0 > 0) Note: ϕ is not an executable function!To generate executable code from this form, appropriate copy statements need to be generated in the predecessors (in other words, reversing the SSA process for code generation)

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SSA in a Loop

▸ Insert ϕ-nodes in the loop header block

x = 0 i = 1 while (i<10) {

x = x+ii = i+1

}

if (i < 10)

Outsidex = x + 1i = i + 1

x = 0i = 1

x1 = ϕ(x0, x2)i1 = ϕ(i0, i2)if (i1 < 10)

Outsidex2 = x1 + 1i2 = i1 + 1

x0 = 0i0 = 1

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ϕ-Node Placement

▸ When and where to add ϕ-nodes?– If two control paths A à C and B à C

converge at a node C, and both A and B contain assignments to variable X, then ϕ-node for X must be placed at C • We often call C a join node or

convergence point• Can be generalized to more than two

converging control paths

▸ Objective: Minimize the number of ϕ-nodes– Need to compute dominance frontier

sets

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x1 = ϕ(x0, x2)i1 = ϕ(i0, i2)if (i1 < 10)

Outsidex2 = x1 + 1i2 = i1 + 1

x0 = 0i0 = 1

Dominance Frontier

▸ A basic block F is in the dominance frontier set (DF)of basic block B if and only if– B does NOT strictly dominate F– B dominates some predecessor(s) of FIf above two conditions hold, F Î DF(B)

▸ Intuitively, the basic blocks in the dominance frontier set of B are almost strictly dominated by B

▸ Useful for efficiently computing the SSA form

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Dominance Frontier and SSA

B0

B1

B2 B3

B4

B6

B7

B5

Area dominated by B3

▸ B7 is in the dominance frontier set of B3– B3 does not dominate B7,

but dominates one of its predecessors (i.e., B6)

▸ For each variable definition in B3, a ϕnode is needed in B7– B7 is the destination of

some edge(s) leaving an area dominated by B3

DF(B)–1776671

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Example: Dominance Frontiers

B0

B1

B2 B3

B4

B6

B7

B5

CFG

B01234567

Dominance frontiers of B

▸ F is in the dominance frontier set (DF) of B iff– B does NOT strictly dominate F– B dominates some predecessor(s) of FIf above two conditions hold, F Î DF(B)

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Iterative ϕ-Node InsertionB DF0 -1 12 73 74 65 66 77 1

b =c =

c =

b =

a =b =c =B0

B1

B2 B3

B4

B6

B7

B5

a = ϕ(…)b = ϕ(…)c = ϕ(…)

c = ϕ(…)

a = ϕ(…)b = ϕ(…)c = ϕ(…)

a =

• a is defined in 0, 3 => need ϕ in 7, then a defined in 7=> need ϕ in 1

• b is defined in 0, 2, 6=> need ϕ in 7then b defined in 7 => need ϕ in 1

• c is defined in 0,2,5=> need ϕ in 6,7then c defined in 7=> need ϕ in 1

▸ SSA form simplifies data flow analysis and many code transformations– Primarily due to explicit & simplified (sparse) def-use

chains

▸ Here we show two simple examples – Dead code elimination– Loop induction variable detection

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SSA Applications

Dead Code in CDFG

▸ Dead code is either– Unreachable code– Definitions never used

▸ Dead statements?

x = a + by = c + d

x = a – b

z = z + 1

y = c – d z = x + y

f(x, y)

B1

B2

B4

B5

B6

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Dead Code Elimination with SSA

x1 = a + by1 = c + d

x2 = a – b

z2 = z1 + 1

x3 = ϕ(x1, x2)y2 = c – d

z1 = x3 + y2

f(x3, y2)

B1

B2

B3

B4

B5

x1 = a + b

x2 = a – b

x3 = ϕ(x1, x2)y2 = c – d

f(x3, y2)

Iteratively remove unused definitions: remove y1, z2 (and B4) à then remove z1

▸ An induction variable is a variable that

– Gets increased or decreased by a fixed amount (loop invariant) on every iteration of a loop (basic induction variable)• i = i + c

– or is an affine function of another induction variable (mutual induction variable)• j = a * i + b

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Loop Induction Variables

▸ Find basic loop induction variable(s)1. Inspect back edges in the loop

2. Each back edge points to a ϕ node in the loop header, which may indicate a basic induction variable

3. ϕ is a function of an initialized variable and a definition in the form of “i + c” (i.e., increment operation)

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Identifying Basic Loop Induction Variable

i1 = ϕ(0, i2)if (i1 < 10)

Outsidei2 = i1 + 1

…

▸ Importance of compilers– Essential component of SoC software development flow– Essential component of high-level synthesis

▸ A good intermediate representation (IR) enables efficient and effective analysis and optimization– Dominance relation helps effective CFG analysis– SSA form facilitates efficient IR-level optimization

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Summary

▸ These slides contain/adapt materials developed by– Prof. Scott Mahlke (UMich)

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Acknowledgements