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

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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} 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]}
* 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])}

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

For a generic instruction, we define the  G E N {\displaystyle {\rm {GEN}}} and  K I L L {\displaystyle {\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\\}} , 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\\}} , 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]} is the set of all definitions that assign to the variable  y {\displaystyle y} . Here  d {\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

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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 };

## See also

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

## Further reading

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* 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.

* v
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* e

Compiler optimizations

Basic block|

* Peephole optimization
* Local value numbering

Loop|

* Automatic parallelization
* Automatic vectorization
* Induction variable
* Loop fusion
* Loop-invariant code motion
* Loop inversion
* Loop interchange
* Loop nest optimization
* Loop splitting
* Loop unrolling
* Loop unswitching
* Software pipelining
* Strength reduction

Data-flow
analysis|

* Available expression
* Common subexpression elimination
* Constant folding
* Dead store elimination
* Induction variable recognition and elimination
* Live-variable analysis
* Upwards exposed uses
* Use-define chain
* Reaching definitions

SSA-based|

* Global value numbering
* Sparse conditional constant propagation

Code generation|

* Instruction scheduling
* Instruction selection
* Register allocation
* Rematerialization

Functional|

* Deforestation
* Tail-call elimination

Global|

* Interprocedural optimization

Other|

* Bounds-checking elimination
* Compile-time function execution
* Dead-code elimination
* Expression templates
* Inline expansion
* Jump threading
* Partial evaluation
* Profile-guided optimization

Static analysis|

* Alias analysis
* Array-access analysis
* Control-flow analysis
* Data-flow analysis
* Dependence analysis
* Escape analysis
* Pointer analysis
* Shape analysis
* Value range analysis

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