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)

// InitializeforallCFGnodesninN,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){chooseanodeninChanged;// remove it from the changed setChanged=Changed-{n};// init IN[n] to be emptyIN[n]=emptyset;// calculate IN[n] from predecessors' OUT[p]forallnodespinpredecessors(n)IN[n]=IN[n]UnionOUT[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 setforallnodessinsuccessors(n)Changed=ChangedU{s};}}

See also

• Dead-code elimination
• Loop-invariant code motion
• Reachable uses
• Static single assignment form

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.