Static single assignment form

In compiler design, static single assignment form (often abbreviated as SSA form or simply SSA) is a property of an intermediate representation (IR), which requires that each variable be assigned exactly once, and every variable be defined before it is used. Existing variables in the original IR are split into "versions", new variables typically indicated by the original name with a subscript in textbooks, so that every definition gets its own version. In SSA form, use-def chains are explicit and each contains a single element.
SSA was proposed by Barry K. Rosen, Mark N. Wegman, and F. Kenneth Zadeck in 1988. Ron Cytron, Jeanne Ferrante and the previous three researchers at IBM developed an algorithm that can compute the SSA form efficiently.
One can expect to find SSA in a compiler for Fortran, C or C++, whereas in functional language compilers, such as those for Scheme and ML, continuation-passing style (CPS) is generally used. SSA is formally equivalent to a well-behaved subset of CPS excluding non-local control flow, which does not occur when CPS is used as intermediate representation. So optimizations and transformations formulated in terms of one immediately apply to the other.
Benefits.
The primary usefulness of SSA comes from how it simultaneously simplifies and improves the results of a variety of compiler optimizations, by simplifying the properties of variables. For example, consider this piece of code:
 y := 1
 y := 2
 x := y
Humans can see that the first assignment is not necessary, and that the value of codice_1 being used in the third line comes from the second assignment of codice_1. A program would have to perform reaching definition analysis to determine this. But if the program is in SSA form, both of these are immediate:
 y1 := 1
 y2 := 2
 x1 := y2
Compiler optimization algorithms which are either enabled or strongly enhanced by the use of SSA include:
Converting to SSA.
Converting ordinary code into SSA form is primarily a matter of replacing the target of each assignment with a new variable, and replacing each use of a variable with the "version" of the variable reaching that point. For example, consider the following control-flow graph:
Changing the name on the left hand side of "x formula_1 x - 3" and changing the following uses of x to that new name would leave the program unaltered. This can be exploited in SSA by creating two new variables: x1 and x2, each of which is assigned only once. Likewise, giving distinguishing subscripts to all the other variables yields:
It is clear which definition each use is referring to, except for one case: both uses of y in the bottom block could be referring to either y1 or y2, depending on which path the control flow took.
To resolve this, a special statement is inserted in the last block, called a "Φ (Phi) function". This statement will generate a new definition of "y" called y3 by "choosing" either y1 or y2, depending on the control flow in the past.
Now, the last block can simply use y3, and the correct value will be obtained either way. A Φ function for "x" is not needed: only one version of x, namely x2 is reaching this place, so there is no problem (in other words, Φ(x1,x2)=x2).
Given an arbitrary control-flow graph, it can be difficult to tell where to insert Φ functions, and for which variables. This general question has an efficient solution that can be computed using a concept called "dominance frontiers" (see below).
Φ functions are not implemented as machine operations on most machines. A compiler can implement a Φ function by inserting "move" operations at the end of every predecessor block. In the example above, the compiler might insert a move from y1 to y3 at the end of the middle-left block and a move from y2 to y3 at the end of the middle-right block. These move operations might not end up in the final code based on the compiler's register allocation procedure. However, this approach may not work when simultaneous operations are speculatively producing inputs to a Φ function, as can happen on wide-issue machines. Typically, a wide-issue machine has a selection instruction used in such situations by the compiler to implement the Φ function.
According to Kenny Zadeck, Φ functions were originally known as "phony" functions while SSA was being developed at IBM Research in the 1980s. The formal name of a Φ function was only adopted when the work was first published in an academic paper.
Computing minimal SSA using dominance frontiers.
First, we need the concept of a "dominator": we say that a node A "strictly dominates" a different node B in the control-flow graph if it is impossible to reach B without passing through A first. This is useful, because if we ever reach B we know that any code in A has run. We say that A "dominates" B (B is "dominated by" A) if either A strictly dominates B or A = B.
Now we can define the "dominance frontier": a node B is in the dominance frontier of a node A if A does "not" strictly dominate B, but does dominate some immediate predecessor of B, or if node A is an immediate predecessor of B, and, since any node dominates itself, node A dominates itself, and as a result node A is in the dominance frontier of node B. From A's point of view, these are the nodes at which other control paths, which don't go through A, make their earliest appearance.
Dominance frontiers capture the precise places at which we need Φ functions: if the node A defines a certain variable, then that definition and that definition alone (or redefinitions) will reach every node A dominates. Only when we leave these nodes and enter the dominance frontier must we account for other flows bringing in other definitions of the same variable. Moreover, no other Φ functions are needed in the control-flow graph to deal with A's definitions, and we can do with no less.
There is an efficient algorithm for finding dominance frontiers of each node. This algorithm was originally described in Cytron "et al." 1991. Also useful is chapter 19 of the book "Modern compiler implementation in Java" by Andrew Appel (Cambridge University Press, 2002). See the paper for more details.
Keith D. Cooper, Timothy J. Harvey, and Ken Kennedy of Rice University describe an algorithm in their paper titled "A Simple, Fast Dominance Algorithm". The algorithm uses well-engineered data structures to improve performance. It is expressed simply as follows:
 for each node b
 for each node b
 if the number of immediate predecessors of b ≥ 2
 for each p in immediate predecessors of b
 runner := p
 while runner ≠ idom(b)
 runner := idom(runner)
Note: in the code above, an immediate predecessor of node n is any node from which control is transferred to node n, and idom(b) is the node that immediately dominates node b (a singleton set).
Variations that reduce the number of Φ functions.
"Minimal" SSA inserts the minimal number of Φ functions required to ensure that each name is assigned a value exactly once and that each reference (use) of a name in the original program can still refer to a unique name. (The latter requirement is needed to ensure that the compiler can write down a name for each operand in each operation.)
However, some of these Φ functions could be "dead". For this reason, minimal SSA does not necessarily produce the fewest Φ functions that are needed by a specific procedure. For some types of analysis, these Φ functions are superfluous and can cause the analysis to run less efficiently.
Pruned SSA.
Pruned SSA form is based on a simple observation: Φ functions are only needed for variables that are "live" after the Φ function. (Here, "live" means that the value is used along some path that begins at the Φ function in question.) If a variable is not live, the result of the Φ function cannot be used and the assignment by the Φ function is dead.
Construction of pruned SSA form uses live variable information in the Φ function insertion phase to decide whether a given Φ function is needed. If the original variable name isn't live at the Φ function insertion point, the Φ function isn't inserted.
Another possibility is to treat pruning as a dead code elimination problem. Then, a Φ function is live only if any use in the input program will be rewritten to it, or if it will be used as an argument in another Φ function. When entering SSA form, each use is rewritten to the nearest definition that dominates it. A Φ function will then be considered live as long as it is the nearest definition that dominates at least one use, or at least one argument of a live Φ.
Semi-pruned SSA.
Semi-pruned SSA form is an attempt to reduce the number of Φ functions without incurring the relatively high cost of computing live variable information. It is based on the following observation: if a variable is never live upon entry into a basic block, it never needs a Φ function. During SSA construction, Φ functions for any "block-local" variables are omitted.
Computing the set of block-local variables is a simpler and faster procedure than full live variable analysis, making semi-pruned SSA form more efficient to compute than pruned SSA form. On the other hand, semi-pruned SSA form will contain more Φ functions.
Converting out of SSA form.
SSA form is not normally used for direct execution (although it is possible to interpret SSA), and it is frequently used "on top of" another IR with which it remains in direct correspondence. This can be accomplished by "constructing" SSA as a set of functions which map between parts of the existing IR (basic blocks, instructions, operands, "etc.") and its SSA counterpart. When the SSA form is no longer needed, these mapping functions may be discarded, leaving only the now-optimized IR.
Performing optimizations on SSA form usually leads to entangled SSA-Webs, meaning there are Φ instructions whose operands do not all have the same root operand. In such cases color-out algorithms are used to come out of SSA. Naive algorithms introduce a copy along each predecessor path which caused a source of different root symbol to be put in Φ than the destination of Φ. There are multiple algorithms for coming out of SSA with fewer copies, most use interference graphs or some approximation of it to do copy coalescing.
Extensions.
Extensions to SSA form can be divided into two categories.
"Renaming scheme" extensions alter the renaming criterion. Recall that SSA form renames each variable when it is assigned a value. Alternative schemes include static single use form (which renames each variable at each statement when it is used) and static single information form (which renames each variable when it is assigned a value, and at the post-dominance frontier).
"Feature-specific" extensions retain the single assignment property for variables, but incorporate new semantics to model additional features. Some feature-specific extensions model high-level programming language features like arrays, objects and aliased pointers. Other feature-specific extensions model low-level architectural features like speculation and predication.
Compilers using SSA form.
SSA form is a relatively recent development in the compiler community. As such, many older compilers only use SSA form for some part of the compilation or optimization process, but most do not rely on it. Examples of compilers that rely heavily on SSA form include: