Live variable analysis
Encyclopedia
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 variables
that may be potentially read before their next write, that is, the variables that are live at the exit from each program point.
Stated simply: a variable is live if it holds a value that may be needed in the future.
It is a "backwards may" analysis. The analysis is done in a backwards order, and the dataflow confluence operator is set union.
The dataflow equations used for a given basic block s and exiting block f in live variable analysis are GEN(s) means the set of variables used in s before any assignment; KILL (s) means the set of variables 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):
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.
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.
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.
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.
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.
Recently , 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.
Compiler
A compiler is a computer program that transforms source code written in a programming language into another computer language...
s to calculate for each program point the variables
Variable (programming)
In computer programming, a variable is a symbolic name given to some known or unknown quantity or information, for the purpose of allowing the name to be used independently of the information it represents...
that may be potentially read before their next write, that is, the variables that are live at the exit from each program point.
Stated simply: a variable is live if it holds a value that may be needed in the future.
It is a "backwards may" analysis. The analysis is done in a backwards order, and the dataflow confluence operator is set union.
| The set of live variables at line L3 is { b, c} because both are used in the addition, and thereby the call to f and assignment to a. But the set of live variables at line L1 isonly { b} since variable c is updated in line 2. The value of variable a is never used,so the variable is never live. Note that f may be stateful, so the never-live assignment to a can be eliminated, but there is insufficient information to rule on the entirety of L3. |
The dataflow equations used for a given basic block s and exiting block f in live variable analysis are GEN(s) means the set of variables used in s before any assignment; KILL (s) means the set of variables 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):
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.
// in: {} b1: a = 3; b = 5; d = 4; x = 100; //x is never being used later thus not in the out set {a,b,d} if a > b then // out: {a,b,d} //union of all (in) successors of b1 => b2: {a,b}, and b3:{b,d} // in: {a,b} b2: c = a + b; d = 2; // out: {b,d} // in: {b,d} b3: endif c = 4; return b * d + c; // out:{} |
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.
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.
| processing | in-state | old out-state | new out-state | work list |
|---|---|---|---|---|
| b3 | {} | {} | {b,d} | (b1,b2) |
| b1 | {b,d} | {} | {} | (b2) |
| b2 | {a,b} | {} | {a,b} | (b1) |
| b1 | {a,b,d} | {} | {} |
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.
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.
Recently , various program analyses such as live variable analysis have been solved using Datalog
Datalog
Datalog is a query and rule language for deductive databases that syntactically is a subset of Prolog. Its origins date back to the beginning of logic programming, but it became prominent as a separate area around 1977 when Hervé Gallaire and Jack Minker organized a workshop on logic and databases...
. 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.