Live-variable analysis: Difference between revisions

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In [[compiler theorycompilers]], '''live variable analysis''' (or simply '''liveness analysis''') is a classic [[data-flow analysis]] performed by [[compiler]]s to calculate for each program point the [[Variable (programming)|variables]] that mayare be''live'' potentiallyat readeach beforepoint theirin next write,the thatprogram. is,A thevariable variables that areis ''live'' at some point if it holds a value that may be needed in the exitfuture, fromor eachequivalently programif pointits value may be read before the next time the variable is written to.

== Example ==

Stated simply: a variable is '''live''' if it holds a value that may be needed in the future.

Consider the following program:

L1: b := 3;

<!-- Is this paragraph relevant? -->

L2: c := 5;

Recently {{As of|2006|lc=on}}, various program analyses such as live variable analysis have been solved using [[Datalog]]. The Datalog specifications for such analyses are generally an order of magnitude shorter than their imperative counterparts (e.g. [[iterative analysis]]), and are at least as efficient.<ref>{{cite book | author=Whaley | title=Using Datalog with Binary Decision Diagrams for Program Analysis | year=2004 | url=http://people.csail.mit.edu/mcarbin/papers/aplas05.pdf}}</ref>

L3: a := f(b +* c);

The set of live variables atbetween linelines 2 and L33 is {<code>b</code>, <code>c</code>} because both are used in the addition,multiplication andon therebyline 3. But the callset toof live variables after line 1 is only {<code>fb</code>}, andsince assignment tovariable <code>ac</code>. is Butupdated thelater, seton ofline live2. variablesThe atvalue lineof L1variable <code>a</code> is not used in this code.

== Example ==

{| class="floatright"

Note that the assignment to <code>a</code> may be eliminated as <code>a</code> is not used later, but there is insufficient information to justify removing all of line 3 as <code>f</code> may have [[Side effect (computer science)|side effects]] (printing <code>b * c</code>, perhaps).

|<code>

L1: b := 3;

L2: c := 5;

L3: a := f(b + c);

goto L1;

</code>

|}

The set of live variables at line L3 is {<code>b</code>, <code>c</code>} because both are used in the addition, and thereby the call to <code>f</code> and assignment to <code>a</code>. But the set of live variables at line L1 is

only {<code>b</code>} since variable <code>c</code> is updated in L2. The value of variable <code>a</code> is never used. Note that <code>f</code> may be stateful, so the never-live assignment to <code>a</code> can be eliminated, but there is insufficient information to rule on the entirety of <code>L3</code>.

== Expression in terms of dataflow equations ==

Liveness analysis is a "backwards may"{{Elucidate}} analysis. The analysis is done in a [[Data-flow_analysis#Backward_Analysis|backwards]] order, and the dataflow [[confluence operator]] is [[set union]]. In other words, if applying liveness analysis to a function with a particular number of logical branches within it, the analysis is performed starting from the end of the function working towards the beginning (hence "backwards"), and a variable is considered live if any of the branches moving forward within the function might potentially (hence "may") need the variable's current value. This is in contrast to a "backwards must" analysis which would instead enforce this condition on all branches moving forward.

The dataflow equations used for a given basic block ''<math>s''</math> and exiting block ''f''<math>\mathit{final}</math> in live variable analysis are the following:

:<math>

{\mbox{GEN}}[s] </math>: The set of variables that are used in s before any assignment in the same basic block.

:<math>

{\mbox{KILL}}[s] </math>: The set of variables that are assigned a value in s (in many books{{Clarify}} that discuss compiler design, KILL (s) is also defined as the set of variables assigned a value in s ''before any use'', but this doesn'tdoes not change the solution of the dataflow equation):

:<math>

{\mbox{LIVE}}_{\mathrm{in}}[s] = {\mbox{GEN}}[s] \cup ({\mbox{LIVE}}_{\mathrm{out}}[s] - {\mbox{KILL}}[s])

</math>

:<math>

{\mbox{LIVE}}_{\mathrm{out}}[\mathit{final}] = {\emptyset}

</math>

:<math>

{\mbox{LIVE}}_{\mathrm{out}}[s] = \bigcup_{p \in \mathrm{succ}[Ss]} {\mbox{LIVE}}_{\mathrm{in}}[p]

</math>

Line 49 ⟶ 43:

The in-state of a block is the set of variables that are live at the start of the block. Its out-state is the set of variables that are live at the end of it. The out-state is the union of the in-states of the block's successors. The transfer function of a statement is applied by making the variables that are written dead, then making the variables that are read live.

== Second example ==

{|

|

// in: {}; predecessor blocks: none

b1: a = 3;

b = 5;

Line 62 ⟶ 55:

// out: {a,b,d} //union of all (in) successors of b1 => b2: {a,b}, and b3:{b,d}

// in: {a,b}; predecessor blocks: b1

b2: c = a + b;

d = 2;

// out: {b,d}

// in: {b,d}; predecessor blocks: b1 and b2

b3: endif

c = 4;

Line 117 ⟶ 110:

==References==

{{reflist|30em}}

{{cite book|last1=Aho|first1=Alfred|last2=Lam|first2=Monica|last3=Sethi|first3=Ravi|last4=Ullman|first4=Jeffrey|year=2007|edition=2|title=Compilers: Principles, Techniques, and Tools|page=608}}

{{Compiler optimizations}}

[[Category:Compiler optimizations]]

[[Category:Data-flow analysis]]

[[Category:Static program analysis]]