Reaching definition

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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 [math]\displaystyle{ S }[/math] in reaching definitions are:

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

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

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

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

where [math]\displaystyle{ {\rm DEFS}[y] }[/math] is the set of all definitions that assign to the variable [math]\displaystyle{ y }[/math]. Here [math]\displaystyle{ d }[/math] 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 };

See also

Further reading