Teichmüller space: Difference between revisions

Revision as of 03:17, 4 November 2013

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:

In other words, the set of reaching definitions going into $S$ are all of the reaching definitions from $S$ 's predecessors, $p r e d [S]$ . $p r e d [S]$ 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 $G E N$ and $K I L L$ sets as follows:

where $D E F S [y]$ 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.

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+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:
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+ d1 : y := 3
+ d2 : x := y
+<code>d1</code> is a reaching definition at <code>d2</code>. In the following, example, however:
+ d1 : y := 3
+ d2 : y := 4
+ d3 : x := y
+<code>d1</code> is no longer a reaching definition at <code>d3</code>, because <code>d2</code> 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 chain]]s and [[def-use chain]]s.
+The data-flow equations used for a given basic block <math>S</math> in reaching definitions are:
+* <math>{\rm REACH}_{\rm in}[S] = \bigcup_{p \in pred[S]} {\rm REACH}_{\rm out}[p]</math>
+* <math>{\rm REACH}_{\rm out}[S] = {\rm GEN}[S] \cup ({\rm REACH}_{\rm in}[S] - {\rm KILL}[S])</math>
+In other words, the set of reaching definitions going into <math>S</math> are all of the reaching definitions from <math>S</math>'s predecessors, <math>pred[S]</math>.  <math>pred[S]</math> consists of all of the basic blocks that come before <math>S</math> in the [[control flow graph]]. The reaching definitions coming out of <math>S</math> are all reaching definitions of its predecessors minus those reaching definitions whose variable is killed by <math>S</math> plus any new definitions generated within <math>S</math>.
+For a generic instruction, we define the <math>{\rm GEN}</math> and <math>{\rm KILL}</math> sets as follows:
+* <math>{\rm GEN}[d : y \leftarrow f(x_1,\cdots,x_n)] = \{d\}</math>
+* <math>{\rm KILL}[d : y \leftarrow f(x_1,\cdots,x_n)] = {\rm DEFS}[y] - \{d\}</math>
+where <math>{\rm DEFS}[y]</math> is the set of all definitions that assign to the variable <math>y</math>. Here <math>d</math> is a unique label attached to the assigning instruction; thus, the domain of values in reaching definitions are these instruction labels.
+== Further reading ==
+*{{cite book
+|author=Aho, Alfred V.; Sethi, Ravi; & Ullman, Jeffrey D.
+|title=[[Compilers: Principles, Techniques, and Tools]]
+|publisher=Addison Wesley
+|year=1986
+|isbn=0-201-10088-6}}
+*{{cite book
+|author=Appel, Andrew W.
+|title=Modern Compiler Implementation in ML
+|publisher=Cambridge University Press
+|year=1999
+|isbn=0-521-58274-1}}
+*{{cite book
+|author=Cooper, Keith D.; & Torczon, Linda.
+|title=Engineering a Compiler
+|publisher=Morgan Kaufmann
+|year=2005
+|isbn=1-55860-698-X}}
+*{{cite book
+|author=Muchnick, Steven S.
+|title=Advanced Compiler Design and Implementation
+|publisher=Morgan Kaufmann
+|year=1997
+|isbn=1-55860-320-4}}
+== See also ==
+*[[Static single assignment form]]
+[[Category:Program analysis]]
+[[Category:Data-flow analysis]]

Teichmüller space: Difference between revisions

Revision as of 03:17, 4 November 2013

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