# Reaching definition

In compiler theory, a reaching definition for a given instruction is an earlier instruction whose target variable can reach (be assigned to) the given one without an intervening assignment. For example, in the following code:

d1 : y := 3
d2 : x := y


d1 is a reaching definition for d2. In the following, example, however:

d1 : y := 3
d2 : y := 4
d3 : x := y


d1 is no longer a reaching definition for d3, because d2 kills its reach: the value defined in d1 is no longer available and cannot reach d3.

## As analysis

The similarly named reaching definitions is a data-flow analysis which statically determines which definitions may reach a given point in the code. Because of its simplicity, it is often used as the canonical example of a data-flow analysis in textbooks. The data-flow confluence operator used is set union, and the analysis is forward flow. Reaching definitions are used to compute use-def chains.

The data-flow equations used for a given basic block $\displaystyle{ S }$ in reaching definitions are:

• $\displaystyle{ {\rm REACH}_{\rm in}[S] = \bigcup_{p \in pred[S]} {\rm REACH}_{\rm out}[p] }$
• $\displaystyle{ {\rm REACH}_{\rm out}[S] = {\rm GEN}[S] \cup ({\rm REACH}_{\rm in}[S] - {\rm KILL}[S]) }$

In other words, the set of reaching definitions going into $\displaystyle{ S }$ are all of the reaching definitions from $\displaystyle{ S }$'s predecessors, $\displaystyle{ pred[S] }$. $\displaystyle{ pred[S] }$ consists of all of the basic blocks that come before $\displaystyle{ S }$ in the control-flow graph. The reaching definitions coming out of $\displaystyle{ S }$ are all reaching definitions of its predecessors minus those reaching definitions whose variable is killed by $\displaystyle{ S }$ plus any new definitions generated within $\displaystyle{ S }$.

For a generic instruction, we define the $\displaystyle{ {\rm GEN} }$ and $\displaystyle{ {\rm KILL} }$ sets as follows:

• $\displaystyle{ {\rm GEN}[d : y \leftarrow f(x_1,\cdots,x_n)] = \{d\} }$ , a set of locally available definitions in a basic block
• $\displaystyle{ {\rm KILL}[d : y \leftarrow f(x_1,\cdots,x_n)] = {\rm DEFS}[y] - \{d\} }$, a set of definitions (not locally available, but in the rest of the program) killed by definitions in the basic block.

where $\displaystyle{ {\rm DEFS}[y] }$ is the set of all definitions that assign to the variable $\displaystyle{ y }$. Here $\displaystyle{ d }$ is a unique label attached to the assigning instruction; thus, the domain of values in reaching definitions are these instruction labels.

## Worklist algorithm

Reaching definition is usually calculated using an iterative worklist algorithm.

Input: control-flow graph CFG = (Nodes, Edges, Entry, Exit)

// Initialize
for all CFG nodes n in N,
OUT[n] = emptyset; // can optimize by OUT[n] = GEN[n];

// put all nodes into the changed set
// N is all nodes in graph,
Changed = N;

// Iterate
while (Changed != emptyset)
{
choose a node n in Changed;
// remove it from the changed set
Changed = Changed -{ n };

// init IN[n] to be empty
IN[n] = emptyset;

// calculate IN[n] from predecessors' OUT[p]
for all nodes p in predecessors(n)
IN[n] = IN[n] Union OUT[p];

oldout = OUT[n]; // save old OUT[n]

// update OUT[n] using transfer function f_n ()
OUT[n] = GEN[n] Union (IN[n] -KILL[n]);

// any change to OUT[n] compared to previous value?
if (OUT[n] changed) // compare oldout vs. OUT[n]
{
// if yes, put all successors of n into the changed set
for all nodes s in successors(n)
Changed = Changed U { s };
}
}