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| In [[compiler theory]], '''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 may be potentially read before their next write, that is, the variables that are ''live'' at the exit from each program point.
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| Stated simply: a variable is '''live''' if it holds a value that may be needed in the future.
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| It is a "backwards way" analysis. The analysis is done in a backwards order, and the dataflow [[confluence operator]] is [[set union]].
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| <table>
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| <tr>
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| <td>
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| 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
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| 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>.
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| </td>
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| <td> | |
| <code> | |
| L1: b := 3;
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| L2: c := 5;
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| L3: a := f(b + c);
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| goto L1;
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| </code>
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| </td>
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| </tr>
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| </table>
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| The dataflow equations used for a given basic block ''s'' and exiting block ''f'' in live variable analysis are the following:
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| :<math>
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| {\mbox{GEN}}[s] </math>: The set of variables that are used in s before any assignment.
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| :<math>
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| {\mbox{KILL}}[s] </math>: The set of variables that are assigned a value in s (in many books, KILL (s) is also defined as the set of variables assigned a value in s ''before any use'', but this doesn't change the solution of the dataflow equation):
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| :<math> | |
| {\mbox{LIVE}}_{in}[s] = {\mbox{GEN}}[s] \cup ({\mbox{LIVE}}_{out}[s] - {\mbox{KILL}}[s])
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| </math>
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| :<math>
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| {\mbox{LIVE}}_{out}[final] = {\emptyset}
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| </math>
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| :<math>
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| {\mbox{LIVE}}_{out}[s] = \bigcup_{p \in succ[S]} {\mbox{LIVE}}_{in}[p]
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| </math>
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| :<math>
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| {\mbox{GEN}}[d : y \leftarrow f(x_1,\cdots,x_n)] = \{x_1,...,x_n\}
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| </math>
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| :<math>
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| {\mbox{KILL}}[d : y \leftarrow f(x_1,\cdots,x_n)] = \{y\}
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| </math>
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| 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.
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| <table>
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| <tr><td>
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| // in: {}
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| b1: a = 3;
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| b = 5;
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| d = 4;
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| x = 100; //x is never being used later thus not in the out set {a,b,d}
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| if a > b then
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| // out: {a,b,d} //union of all (in) successors of b1 => b2: {a,b}, and b3:{b,d}
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|
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| // in: {a,b}
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| b2: c = a + b;
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| d = 2;
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| // out: {b,d}
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|
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| // in: {b,d}
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| b3: endif
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| c = 4;
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| return b * d + c;
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| // out:{}
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| </td>
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| </tr>
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| </table>
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| The in-state of b3 only contains ''b'' and ''d'', since ''c'' has been written. The out-state of b1 is the union of the in-states of b2 and b3. The definition of ''c'' in b2 can be removed, since ''c'' is not live immediately after the statement.
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| Solving the data flow equations starts with initializing all in-states and out-states to the empty set. The work list is initialized by inserting the exit point (b3) in the work list (typical for backward flow). Its computed out-state differs from the previous one, so its predecessors b1 and b2 are inserted and the process continues. The progress is summarized in the table below.
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| {| class="wikitable"
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| |-
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| ! processing
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| ! out-state
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| ! old in-state
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| ! new in-state
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| ! work list
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| |-
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| | b3
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| | {}
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| | {}
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| | {b,d}
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| | (b1,b2)
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| |-
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| | b1
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| | {b,d}
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| | {}
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| | {}
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| | (b2)
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| |-
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| | b2
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| | {b,d}
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| | {}
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| | {a,b}
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| | (b1)
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| |-
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| | b1
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| | {a,b,d}
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| | {}
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| | {}
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| | ()
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| |}
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| Note that b1 was entered in the list before b2, which forced processing b1 twice (b1 was re-entered as predecessor of b2). Inserting b2 before b1 would have allowed earlier completion.
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| Initializing with the empty set is an optimistic initialization: all variables start out as dead. Note that the out-states cannot shrink from one iteration to the next, although the out-state can be smaller than the in-state. This can be seen from the fact that after the first iteration the out-state can only change by a change of the in-state. Since the in-state starts as the empty set, it can only grow in further iterations.
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| <!-- Is this paragraph relevant? -->
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| 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 | display-authors=1}}</ref>
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| ==References==
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| <references/>
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| [[Category:Data-flow analysis]]
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