Teichmüller space

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:

• ${\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{C}\mathrm{H}}_{\mathrm{i}\mathrm{n}}[S]=\bigcup _{p\in pred[S]}{\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{C}\mathrm{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}[p]$
• ${\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{C}\mathrm{H}}_{\mathrm{o}\mathrm{u}\mathrm{t}}[S]=\mathrm{G}\mathrm{E}\mathrm{N}[S]\cup ({\mathrm{R}\mathrm{E}\mathrm{A}\mathrm{C}\mathrm{H}}_{\mathrm{i}\mathrm{n}}[S]-\mathrm{K}\mathrm{I}\mathrm{L}\mathrm{L}[S])$

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 $\mathrm{G}\mathrm{E}\mathrm{N}$ and $\mathrm{K}\mathrm{I}\mathrm{L}\mathrm{L}$ sets as follows:

• $\mathrm{G}\mathrm{E}\mathrm{N}[d:y\leftarrow f(x_1,\cdots ,x_n)]=\{d\}$
• $\mathrm{K}\mathrm{I}\mathrm{L}\mathrm{L}[d:y\leftarrow f(x_1,\cdots ,x_n)]=\mathrm{D}\mathrm{E}\mathrm{F}\mathrm{S}[y]-\{d\}$

where $\mathrm{D}\mathrm{E}\mathrm{F}\mathrm{S}[y]$ 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

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

See also

• Static single assignment form