Reaching definition

In compiler theory, a reaching definition for a given instruction is another instruction, the target variable of which may reach the given instruction without an intervening assignment. For example, in the following code:

d1 : y := 3
d2 : x := y

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

d1 : y := 3
d2 : y := 4
d3 : x := y

d1 is no longer a reaching definition at d3, because d2 kills its reach.

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 GEN and 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 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. 

See also

• Static single assignment form