3 Address Code to SSA to Assembly Explanation (by Jason Hiebel)

notes.md

Notes

It should be noted that:

1. Basic blocks in SSA can be identified at 3 points:

• Start of the program
• Labels
• Branches

2. Basic blocks are used in optimisation.
3. PHI instructions are only required for SSA, they are needed so that the correct variable to use can be determined. This choice determinator is generally required in loops or when an existing identifier's value is mutated in the source program, such as in a for loop.
4. SSA is great for optimisation of code, our example shows dead code elimination.
5. To convert SSA back to 3 address form, one just needs to remove the dead code, remove the PHI instructions, and drop the counter subscripts.
6. Generally the compiler flow is 3 Address Form -> SSA -> Optimised 3 Address Form.
7. SSA is generally useful for compiled languages, although it may not be used in interpreted languages due to time/space constraints.
8. Languages that cannot mutate already assigned variables such as Haskell and Erlang is relatively easier to map to SSA form.
9. SSA is useful for constant propagation, dead code elimination, and redundancy elimination (and maybe inlining constant values?).

three_address_code_to_ssa_to_assembly.txt

# Three Address Codes

A three address code is an assembly like language where each instruction takes at most
three arguments, for example:
  ADD $1 <- y, 5
is an instruction which takes variables y and constant 5, adds them, and stores the 
result in to a variable $1. From this point on, I'm going to denote variables with a $ 
followed by an identifier e.g. $1, $x23, etc.

A lot of our binary operations follow the same pattern: OP result <- arg1 arg2.
But what about other types of instructions, such as loading and storing to memory /
arrays? Well, for those we have load and store instructions:
  LOAD result <- base offset    // result = base[offset]
  STORE base offset <- result   // base[offset] = result

These instructions still follow our convention of three addresses. Sometimes we have
instructions which use fewer than three addresses, such as a MOVE instruction:
  MOVE result <- source

To use as a running example, lets write a small little program:
  int A[] = { 1, 2, 3, 4, 5 };
  int avg = 0;
  int min = A[0];
  for(int x = 0; x < length(A); x += 1) {
    avg = avg + A[x];
    min = MIN(min, A[x]);
  }
  avg = avg / length(A)
  return avg;
  

and then translate it in to a three address code:
  .data A { 1, 2, 3, 4, 5 }
    MOVE $avg <- 0
    LOAD $min <- A 0

    MOVE $x   <- 0           // loop initialization
  start:
    LESS_THAN $t0  <- $x  5  // check loop condition
    IF_FALSE   end <- $t0

    LOAD $t1  <-  A   $x     // loop body
    ADD  $avg <- $avg $t1
    MIN  $min <- $min $t1

    ADD  $x   <- $x    1     // loop effect
    JUMP start
  end:
    DIV  $avg <- $avg 5
    RET  $avg


A few comments before we move on. Fixed tables are encoded directly in to the program,
and so its natural to have some way of describing that to our three address code. That
is where the .data directive comes in to play. It doesn't need to be limited to three
addresses because its not an instruction - its more just telling us to dump these values
in to memory somewhere. Because A is known at compile time, we can treat length(A) as
a constant in our compiler.

Instruction names are not usually so verbose, but for our sake here I used the full names.
I also used a different form of three address code to emphasize that it really is a lower 
level language that you are generating.



# Static Single Assignment

In order to talk about SSA, we first need to talk about basic blocks. A basic block is
a chunk of code which has a single entry and a single exit. If we look at the above
example, we have 4 basic blocks:
  -------- (1)
    MOVE $avg <- 0
    LOAD $min <- A 0

    MOVE $x   <- 0           // loop initialization
  -------- (2)
  start:
    LESS_THAN $t0  <- $x  5  // check loop condition
    IF_FALSE   end <- $t0
  -------- (3)
    LOAD $t1  <-  A   $x     // loop body
    ADD  $avg <- $avg $t1
    MIN  $min <- $min $t1

    ADD  $x   <- $x    1     // loop effect
    JUMP start
  -------- (4)
  end:
    DIV  $avg <- $avg 5
    RET  $avg

Knowing that each block has a single entry and exit, we can now reason about each basic
block individually in a powerful way.

SSA is a modification to our three address code which states that there can only be exactly
one assignment to each variable. This is done by indexing each assignment, so the first
assignment to $x would be $x_1, the second would be $x_2, and so on. Making the necessary
modifications:
  -------- (1)
    MOVE $avg_1 <- 0
    LOAD $min_1 <- A 0

    MOVE $x_1   <- 0           // loop initialization
  -------- (2)
  start:
    LESS_THAN $t0_1 <- $x   5  // check loop condition
    IF_FALSE   end  <- $t0_1
  -------- (3)
    LOAD $t1_1  <-  A   $x     // loop body
    ADD  $avg_2 <- $avg $t1_1
    MIN  $min_2 <- $min $t1_1

    ADD  $x_2   <- $x    1     // loop effect
    JUMP start
  -------- (4)
  end:
    DIV  $avg_3 <- $avg 5
    RET  $avg_3

Now you'll note that this example is incomplete. There are still unsubscripted references to
$x, $avg, and $min. This is because there are now multiple possible choices for each of these. 
In order to resolve these, we introduce PHI nodes.

A PHI node indicates that we need to make a decision about which subscripted version to use.
It is not an instruction; there is no (serious, non-theoretical) architecture out there which
has a PHI instruction. It is simply a tool we will use in our intermediate representation and
we will remove it when it comes time to output assembly code.

Adding PHI nodes to our example:
  -------- (1)
    MOVE $avg_1 <- 0
    LOAD $min_1 <- A 0

    MOVE $x_1   <- 0               // loop initialization
  -------- (2)
  start:
    PHI       $x_3  <- $x_1  $x_2
    LESS_THAN $t0_1 <- $x_3   5    // check loop condition
    IF_FALSE   end  <- $t0_1
  -------- (3)
    PHI  $avg_4 <- $avg_1 $avg_2
    PHI  $min_3 <- $min_1 $min_2
    LOAD $t1_1  <-  A     $x_3     // loop body
    ADD  $avg_2 <- $avg_4 $t1_1
    MIN  $min_2 <- $min_3 $t1

    ADD  $x_2   <- $x_3    1       // loop effect
    JUMP start
  -------- (4)
  end:
    PHI  $avg_5 <- $avg_1 $avg_2
    PHI  $min_3 <- $min_1 $min_2
    DIV  $avg_3 <- $avg_5  5
    RET  $avg_3

So why do we care? The two most obvious answers are constant propegation and dead code elimination.
Constant propegation isn't shown here, but dead code elimination is. Lets start at the end of our
program, and keep a list of things that are requested. 
 * Starting with our return, we request $avg_3.
 * Since $avg_3 is assigned exactly once, we add the $avg_3 arguments to our list as well, which are
   $avg_5 and 5.
 * The constant 5 doesn't have any arguments. The $avg_5 arguments are $avg_1 and $avg_2
 * ...
Assume we repeat this procedure until we have marked all of the reachable assignments. I also assume
that any jumps are always marked (to preserve the flow of the program). The result is:
  -------- (1)
  *  MOVE $avg_1 <- 0
     LOAD $min_1 <- A 0

  *  MOVE $x_1   <- 0               // loop initialization
  -------- (2)
  start:
  *  PHI       $x_3  <- $x_1  $x_2
  *  LESS_THAN $t0_1 <- $x_3   5    // check loop condition
  *  IF_FALSE   end  <- $t0_1
  -------- (3)
  *  PHI  $avg_4 <- $avg_1 $avg_2
     PHI  $min_3 <- $min_1 $min_2
  *  LOAD $t1_1  <-  A     $x_3     // loop body
  *  ADD  $avg_2 <- $avg_4 $t1_1
     MIN  $min_2 <- $min_3 $t1

  *  ADD  $x_2   <- $x_3    1       // loop effect
  *  JUMP start
  -------- (4)
  end:
  *  PHI  $avg_5 <- $avg_1 $avg_2
     PHI  $min_3 <- $min_1 $min_2
  *  DIV  $avg_3 <- $avg_5  5
  *  RET  $avg_3

Since all the $min assigments are never requested, they all amount to dead code and can be removed.
Could this be done without SSA? Sure. But SSA is a strong assumption which simplifies a huge class of
different optimizations.



# Assembly

Now, I won't explain how to turn SSA in to assembly right now, but I did want to show what our example
program would look like if it was written in MIPS assembly:
  .text
  .globl main
    main:
      la   $t0, A          // $t0 = A
      li   $t1, 0          // $t1 = avg
      lw   $t2, 0($t0)     // $t2 = min

      li   $t3, 0          // $t3 = x
    start:
      slti $t4, $t3, 5     // x < length(A)
      beq  $t4, $zero, end

      lw   $t5, 0($t0)     // $t5 = A[x]
      add  $t1, $t1, $t5   // avg = avg + A[x]
      min  $t2, $t2, $t5   // min = MIN(min, A[x])

      addi $t0, $t0, 4     // x = x + 1
      addi $t3, $t3, 1
      j    start
    end:
      li   $t6, 5
      div  $a0, $t1, $t6   // avg = avg / length(A)
      li   $v0, 1          // specify "print int" syscall
      syscall              // print $a0
  
  .data
    A: .word 1 .word 2 .word 3 .word 4 .word 5

And there we have it. Almost. I glossed over the fact that MIPS doesn't actually have a 'MIN'
instruction, but I didn't want to make the examples even more complex.

To convert SSA back to 3 address form, one just needs to remove the dead code, remove the PHI instructions, and drop the counter subscripts.

This is not necessarily true. Check Cooper & Torczon (Engineering a compiler, 2nd Edition) in chapter 9.3.5.

Are you able to provide an example?

@CMCDragonkai @hosewiejacke
It strikes me that this would be an example of where converting from SSA back to machine code is more complex than this explanation eludes to:

i = 0;
y = 0;
while(i < 10) {
	y = i;
	i++;
}
z = y;

SSA:
i0 = 0
y0 = 0
loop:
	i1 = phi(i0, i2)
	if i1 >= 10 branch to end
	y1 = i1
	i2 = i1 + 1
goto loop
end:
z0 = y1

y0 is never used and y1 is redundant, they can be eliminated:
i0 = 0
loop:
	i1 = phi(i0, i2)
	if i1 >= 10 branch to end
	i2 = i1 + 1
goto loop
end:
z0 = i1

Resolving the phi node by removing it and dropping sub-scripts clearly leads to an incorrect assignment to z:
i = 0
loop:
	if i >= 10 branch to end
	i = i + 1
goto loop
end:
z = i

Note: this example comes from a lecture by Hugh Leather.