Reaching definition (original) (raw)

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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.

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 S {\displaystyle S} $S$ in reaching definitions are:

R E A C H i n [ S ] = ⋃ p ∈ p r e d [ S ] R E A C H o u t [ p ] {\displaystyle {\rm {REACH}}_{\rm {in}}[S]=\bigcup _{p\in pred[S]}{\rm {REACH}}_{\rm {out}}[p]} ${\rm {REACH}}_{\rm {in}}[S]=\bigcup _{p\in pred[S]}{\rm {REACH}}_{\rm {out}}[p]$
R E A C H o u t [ S ] = G E N [ S ] ∪ ( R E A C H i n [ S ] − K I L L [ S ] ) {\displaystyle {\rm {REACH}}_{\rm {out}}[S]={\rm {GEN}}[S]\cup ({\rm {REACH}}_{\rm {in}}[S]-{\rm {KILL}}[S])} ${\rm {REACH}}_{\rm {out}}[S]={\rm {GEN}}[S]\cup ({\rm {REACH}}_{\rm {in}}[S]-{\rm {KILL}}[S])$

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

For a generic instruction, we define the G E N {\displaystyle {\rm {GEN}}} ${\rm {GEN}}$ and K I L L {\displaystyle {\rm {KILL}}} ${\rm {KILL}}$ sets as follows:

G E N [ d : y ← f ( x 1 , ⋯ , x n ) ] = { d } {\displaystyle {\rm {GEN}}[d:y\leftarrow f(x_{1},\cdots ,x_{n})]=\{d\}} ${\rm {GEN}}[d:y\leftarrow f(x_{1},\cdots ,x_{n})]=\{d\}$ , a set of locally available definitions in a basic block
K I L L [ d : y ← f ( x 1 , ⋯ , x n ) ] = D E F S [ y ] − { d } {\displaystyle {\rm {KILL}}[d:y\leftarrow f(x_{1},\cdots ,x_{n})]={\rm {DEFS}}[y]-\{d\}} ${\rm {KILL}}[d:y\leftarrow f(x_{1},\cdots ,x_{n})]={\rm {DEFS}}[y]-\{d\}$ , a set of definitions (not locally available, but in the rest of the program) killed by definitions in the basic block.

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

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 };
}

}

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