Code
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
B0["0: i = 0
1: s = 0"]
B1["2: x = m
3: s = s + x
4: i = i +4
5: if i < n go to B0"]
B0 --> B1
B1 --> B1
A variable in a program can have multiple definitions. In Bril definitions are instructions which compute values. Up till now we have been thinking about analysis which look at variables (names) but a different way to look at this is based on values, If we think of instructions calculating values, and uses being uses of values we can picture a graph called the data flow graph showing how values move through a program
in SSA we change our IR so that every variable has exactly one definition in the program (each variable is assigned only once). The name SSA means statically there is only a single assignment per variable.
In addition to a language form, SSA is also a philosophy! It can fundamentally change the way you think about programs. In the SSA philosophy:
In LLVM, for example, instructions do not refer to argument variables by name—an argument is a pointer to defining instruction.
Static means in the text, not in the execution.
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
B0["0: i = 0
1: s = 0"]
B1["2: x = m
3: s = s + x
4: i = i +4
5: if i < n go to B0"]
B0 --> B1
B1 --> B1
variable i has two static assignments 0 and 4, so this program is not in SSA
Variable s has two static assignments, x has one static assignment but x has lots of dynamic assignments (when the program executes)
We call a program without branches a piece of straight line code.
@main {
a: int = const 4;
b: int = const 2;
a: int = add a b;
b: int = add a b;
print b;
}Its easy to see how to convert straight line code into ssa
@main {
a.1: int = const 4;
b.1: int = const 2;
a.2: int = add a.1 b.1;
b.2: int = add a.2 b.1;
print b.2;
}for each variable a:
Count[a] = 0
Stack[a] = [0]
rename_basic_block(B):
for each instruction S in block B:
for each use of a argument x in S:
i = top(Stack[x])
replace the use of x with x_i
for each variable a that S defines (a dest)
count[a] = Count[a] + 1
i = Count[a]
push i onto Stack[a]
replace definition of a with a_iWe don’t need the stack here but we will need it later.
Of course, things will get a little more complicated when there is control flow. And because real machines are not SSA, using separate variables (i.e., memory locations and registers) for everything is bound to be inefficient.
The idea in SSA is to convert general programs into SSA form, do all our optimization there, and then convert back to a standard mutating form before we generate backend code.
Just renaming assignments will quickly run into problems. Consider this program:
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
B0[".b0
a: int = const 47;
br cond .left .right;"]
left["a: int = add a a;
jmp .exit;"]
right["a: int = mul a a;
jmp .exit;"]
exit["print a;"]
B0 --> left
B0 --> right
left --> exit
right --> exit
Which “version” of a should we use in the print statement?
To match the expressiveness of unrestricted programs, SSA adds a new kind of instruction: a phi-node.
phi-nodes are flow-sensitive copy instructions: they get a value from one of several variables, depending on which incoming CFG edge was most recently taken to get to them.
In Bril, a phi-node appears as a phi instruction:
a.4: int = phi .left a.2 .right a.3;
The phi instruction chooses between any number of variables, and it picks between them based on labels. If the program most recently executed a basic block with the given label, then the phi instruction takes its value from the corresponding variable.
You can write the above program in SSA like this:
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
B0[".b0
a: int = const 47;
br cond .left .right;"]
left["a: int = add a a;
jmp .exit;"]
right["a: int = mul a a;
jmp .exit;"]
exit["print a;"]
B0 --> left
B0 --> right
left --> exit
right --> exit
@main(cond: bool) {
.entry:
a.1: int = const 47;
br cond .left .right;
.left:
a.2: int = add a.1 a.1;
jmp .exit;
.right:
a.3: int = mul a.1 a.1;
jmp .exit;
.exit:
a.4: int = phi .left a.2 .right a.3;
print a.4;
}Bril has an SSA extension It adds support for a phi instruction. Beyond that, SSA form is just a restriction on the normal expressiveness of Bril—if you solemnly promise never to assign statically to the same variable twice, you are writing “SSA Bril.”
The reference interpreter has built-in support for phi, so you can execute your SSA-form Bril programs without fuss.
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph LR
X["Block X
a =
b =
if s > b"]
Y["Block Y
b = a"]
Z["Block Z
ret b"]
X --> Y
Y--> Z
X --> Z
Where do we need phi-functions?
Which variables
phi At the merge (join) node
variable b
conditions: phi-function for variable b at node z
this is iterative since when we add a phi, we are creating a new defintion, which may add new phi-functions
When we find nodes X,Y,Z that match these steps and z does not contain a phi function for b, insert a phi
While really expensive this will work
using dash for path
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
a["x:x="]
b["y:x="]
c[join]
d
a-.-> c
b-.-> c
c-.->d
We could have complex flow - including loops on the paths
while dom is changing
for vertex in cfg
dom[vertex] =while dom is changing
for vertex in cfg
dom[vertex] = {vertex} + ...if b has multiple preds, and a dominates all of them, a dom b
while dom is changing
for vertex in cfg
dom[vertex] = {vertex} + Intersection( dom(p) for p a pred of vertex)The method for this has two steps
To convert to SSA, we want to insert phi-nodes whenever there are distinct paths containing distinct definitions of a variable. We don’t need phi-nodes in places that are dominated by a definition of the variable. So what’s a way to know when control reachable from a definition is not dominated by that definition?
The dominance frontier!
| block | A | B | C | D | E |
| frontier | empty | F | E | E | F |
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
A--> B
B--> A
B--> C
why: A dom B, but B does not dom A. so A is in the dom frontier of A
We do it in two steps.
let b be a block with a def of a variable v, if b has multiple defs of v, use the last one
What is the first block following v that can be reached by a different def of v
in blocks dominated by b, b’s def must have been executed, (other defs of v in a dominated block may overwrite it)
we need to place a phi function for b at the start of all blocks in the dom frontier of b.
after we add phi functions to S where S = df(b) we have more defs, so we need to add phi’s in the dom frontier of all the blocks in S
::: {.columns}
::: {.column}
graph TD
V["v=init"]
Z{"z:v = v+1"}
V --> Z
Z -->Z
:::
::: {.column}
graph TD
V["v1=init"]
Z{"z:v2 = phi(v1,v3)\nv3 = v2+1"}
V --> Z
Z -->Z
:::
:::
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
B1[v1=1]
B2[v2=2]
B3["v3=phi(v1,v2)"]
B4[v4=3]
B5["v5=phi(v3,v4)"]
B1--> B3
B2--> B3
B3--> B5
B4--> B5
for each block b in the cfg
for each var v defined in b
add block to the set defs(v) ## blocks that contain an assignment to v
W = Defs[v]
while W is not empty
remove a node n from w
for block in DF[n]: # Dominance frontier.
Add a phi-node to block,
unless we have done so already.
Add block to W (because it now writes to v),
unless it's already in there.%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
N1["1:x = 1"]
N2["2:"]
N3["3:x= 2"]
N4["4:"]
N5["5:x=3"]
N6[6:x=4]
N7[7:]
N1--> N2
N1--> N3
N2--> N4
N3--> N4
N4--> N5
N5--> N4
N5--> N6
N6--> N7
N6--> N5
add phi’s to blocks 4 and 5
# allocate a stack and a counter for each variable
for each V a variable
c[v] = 0
s[v] = empty stack
search(entry)
search(n):
for each instr i in n:
if instr is not a phi
replace every variable in the rhs of instr by vi where i = top(s[v])
if instr has a dest v
i = C(v)
replace v by new vi, push i onto s[v]
increment c[v]
for each y a successor of n
j = which pred (y,n)
for each phi function pinstr in Y replace the jth opernmad of pinstr by vi where
i = top(s(v)
for each Y a child of n in the dominator tree
call search(Y)
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5--> L9
L7 --> L9
L9 --> L3%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5--> L9
L7 --> L9
L9 --> L3
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 -.-> L9
L9 --> L3%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 -.-> L9
L9 --> L3
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 -.-> L9
L9 --> L3%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 -.-> L9
L9 --> L3
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 --> L9
L9 --> L3%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9[L9: goto l3]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 --> L9
L9 --> L3
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["j = phi(j,j)
k = phi(k,k)
L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9["j = phi(j,j)
k = phi(k,k)
L9: goto l3"]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 --> L9
L9 --> L3%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
L0["L0: i = 1
L1: j = 1
L2: k = 0"]
L3["j = phi(j,j)
k = phi(k,k)
L3: if j <20 go to l4 else l10"]
L4["l4: if j < 20 goto l7 else l5"]
L5["l5: j = i
l6: k = k +1"]
L7["l7: j = k
l8: k = k +2"]
L9["j = phi(j,j)
k = phi(k,k)
L9: goto l3"]
L10["l10: ret j"]
L0--> L3
L3--> L4
L3 --> L10
L4 --> L5
L4 --> L7
L5 --> L9
L7 --> L9
L9 --> L3
Could we have a phi-function in a node with only one predecessor?
could we have a phi-function wit more then two arguments?
This algorithm computes what is called minimal SSA form which is not so mimimal since it can leave dead assignments
doing dead code elimination pruned ssa form
Compilers that use the SSA form usually contain a step, before the generation of actual assembly code, in which phi functions are replaced by ordinary instructions. Normally these instructions are simple copies.
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
A0["io =
j0 =
k0 ="]
A1["i1 =
j1 =
k1 = "]
A2["i2 = phi(i0, i1)
j2 = phi(j0, j1)
k2 = phi(k0, k1)
...
= i2
= j2
= k2"]
A0 --> A2
A1--> A2
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TD
B0["io =
j0 =
k0 ="]
B1["i1 =
j1 =
k1 = "]
B2["
...
= i2
= j2
= k2"]
B0 --"i2 = i0
j2 = j0
k2 = k0"--> B2
B1 --"i2 = i1
j2 = j1
k2 = k1"--> B2
we cannot put instructions on edges, but we can add to prev block
%%{init: {"flowchart": {"htmlLabels": false}} }%%
graph TB
A0["L1:
a0 =
b0 =
if A0 > b0"]
A1["b1 = a0"]
A2["l2:
b2 = phi(b1,b0)"]
A0 --> A1
A1 --> A2
A0 --> A2
b2 = b0?
The placement of the copy b2 = b0 is not simple, because the edge that links L2 to L5 is critical. A critical edge connects a block with multiple successors to a block with multiple predecessors. This should remind you of adding a preheader to a loop
We can solve this problem by doing critical edge splitting. This CFG transformation consists in adding an empty basic block (empty, except by – perhaps – a goto statement) between each pair of blocks connected by a critical edge.
Our previous analyses always used a (variable, program point), but in ssa these are the same
while there is some variable v with no uses and the statement that defines v has no other side effects, delete the statement that defines v from the program.
we need a counter for each variable (or each instruction)
walk the program once increment the counter each time the variable is used
while there exists v, such that counter[v] = 0 remove the instruction that defined v, e.g., “v = E for each variable x used in E decrement counter[x]
we define a partial order on constats, any > all constants > undefined and define the intersection of two states as the common parent
with each variable we have an abstract state (like a value number)
v = const c ==> v state is const
v = id q ==> v state is the state of q
v = v0 op v1 ==> if both are constants v = c0 op c1
==> if one is any, v's state is any
v = phi(v0,..vn) ==> v's state is the intersection of the states of v0,..,vnbecause the program is in ssa form we can do the nodes in dominator tree order, then before processing any instruction that is not a phi, we will have processed all the arguments
B0: x0 = input
a0 = 1
c0 = a0 +10
if a0 < c0 go to b1
B1: a1 phi(a1,a2 )
b0 = x0 * a1
print b0
a2 = a1 +1
go to b1B0: x0 = input
a0 = 1
c0 = a0 +10
if a0 < c0 go to b1
B1: a1 phi(a0,a2 )
b0 = x0 * a1
print b0
a2 = a1 +1
go to b1B0:
x0 - any
a0 - 1
c0 - 11 (folding the constant)
a0 < c0 skip
B1:
a1 - 1 (only one input defined)
b0 - any
a2 - 2
update the uses of a2 - the phi
a1 - any
update the uses of a1
no change Eventually, we need to convert out of SSA form to generate efficient code for real machines that don’t have phi-nodes and do have finite space for variable storage.
The basic algorithm is pretty straightforward. If you have a phi-node:
v = phi .l1 x .l2 y;Then there must be assignments to x and y (recursively) preceding this statement in the CFG.
The paths from x to the phi-containing block and from y to the same block must “converge” at that block. So insert code into the phi-containing block’s immediate predecessors along each of those two paths: one that does v = id x and one that does v = id y. Then you can delete the phi instruction.
This basic approach can introduce some redundant copying. (Take a look at the code it generates after you implement it!) Non-SSA copy propagation optimization can work well as a post-processing step. For a more extensive take on how to translate out of SSA efficiently, see “Revisiting Out-of-SSA Translation for Correctness, Code Quality, and Efficiency” by Boissinot et al.
its possible that an optimization can give overlapping phi-functions
b0
x1 = 1
y1 = 2
B1
x2 = phi(x1,x3)
y2 = phi(y1, y3)
z = x2
x3 = y2
y3= z
if() go to b1b0
x1 = 1
y1 = 2
B1
x2 = phi(x1, y2)
y2 = phi(y1, x2)
if() go to b1if we add copies x2 = y3 y2 = x2 (uses the wrong value of x2)
phi nodes execute all at once - not one at a time
Some SSA slides from Todd Mowry at CMU