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 and def-use chains.

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

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

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

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

• ${\rm GEN}[d : y \leftarrow f(x_1,\cdots,x_n)] = \{d\}$
• ${\rm KILL}[d : y \leftarrow f(x_1,\cdots,x_n)] = {\rm DEFS}[y] - \{d\}$

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

Further reading

• Aho, Alfred V.; Sethi, Ravi; & Ullman, Jeffrey D. (1986). Compilers: Principles, Techniques, and Tools. Addison Wesley. ISBN 0-201-10088-6.
• Appel, Andrew W. (1999). Modern Compiler Implementation in ML. Cambridge University Press. ISBN 0-521-58274-1.
• Cooper, Keith D.; & Torczon, Linda. (2005). Engineering a Compiler. Morgan Kaufmann. ISBN 1-55860-698-X.
• Muchnick, Steven S. (1997). Advanced Compiler Design and Implementation. Morgan Kaufmann. ISBN 1-55860-320-4.
• Nielson F., H.R. Nielson;, C. Hankin (2005). Principles of Program Analysis. Springer. ISBN 3-540-65410-0.

See also

• Static single assignment form