The document discusses the process of converting a program into static single assignment (SSA) form. It involves inserting phi-functions at join points to rename variables and ensure each variable has a single definition. Phi-functions are inserted at nodes in the dominance frontier of a definition, which contains predecessors that are not strictly dominated. The dominance frontier and dominator tree properties are used to determine where to place phi-functions to construct the SSA form.

1. SSA

2. SSA Properties
● Each definition in the procedure creates a unique name.
● Each use refers to a single definition.

3. Construct SSA Form
● Insert phi-functions.
● Rename variables to conform the properties.

4. Where to Insert phi-functions
● A definition in node n forces a phi-function at join points that lie just outside
the region of the CFG that n dominates.
○ n dominate a predecessor of m
○ n does not strictly dominate m
■ m belongs DF(n)
● DF(n) contains the first node reachable from n that n does not dominate.
● DF: Dominance Frontier

5. Dominance
● In a flow graph with entry node e, node n dominates node m if and only if n
lies on every path from e to m.
● Annotate each node m in the CFG with a set DOM(m) that contains the
names of all nodes that dominate m

6. e
a
b c
d f
g
h
i
DOM(d) = {e, a, c, d}

7. Dominance Frontier e
a
b c
d f
g
h
i
DF(c) = {h}

8. Immediate Dominator
● The node in strictly dominator
that is closest to n is called n’s
immediate dominator IDOM(n)
e
a
b c
d f
g
h
i
IDOM(g) = {c}

9. Dominator Tree
● Node
○ Every node of the flow graph
● Edge
○ If n is in IDOM(m), then the
dominator tree has an edge
from n to m.
e
a
b c
d f
g
h
i

10. Dominator Tree Properties
● Given a node n in the dominator tree
○ IDOM(n) is just its parent in the tree
○ DOM(n): the nodes that lie on the path from the root of the dominator tree to n.

11. Placing phi-functions
● A definition of x in block b forces a phi-function at every node in DF(b)
● Global names
○ The set of names that are live across multiple blocks
● Insert phi-functions for global names and ignore any name that is not in the
set
○ semipruned SSA form

12. Global Names
● In each block, it looks for names with upward exposed uses, UEVAR.
● Union of all the UEVAR sets gives global names.
Globals = {}
Initialize all the Blocks sets to {}
for each block b
varkill = {}
for each operation i in b in order
assume operation i is x = y op z
if y is not in varkill
Globals = Globals + {y}
if z is not in varkill
Globals = Globals + {z}
varkill = varkill + {x}
block(x) = block(x) + {b}

13. Computing Dominance Frontiers
for all nodes n in the CFG
DF(n) = {}
for all nodes n in the CFG
if n has multiple predecessors
for each predecessor p of n
runner = p
while runner != IDOM(n)
DF(runner) = DF(runner) + {n}
runner = IDOM(runner)

14. Insert phi-functions
for each name x in Globals
worklist = blocks(x)
for each block b in worklist
for each block d in DF(b)
if d has no phi-function for x
insert phi-function for x in d
worklist = worklist + {d}