complex variable analysis

complex variable analysis

complex variable analysis, branch of mathematics that deals with the calculus of functions of a complex variable, i.e., a variable of the form z=x+iy, where x and y are real and i=-1 (see number). A function w=f(z) of a complex variable z is separable into two parts, w = g1(x,y) + ig2(x,y), where g1 and g2 are real-valued functions of the real variables x and y. The theory of functions of a complex variable is concerned mainly with functions that have a derivative at every point of a given domain of values for z; such functions are called analytic, regular, or holomorphic. If a function is analytic in a given domain, then it also has continuous derivatives of higher order and can be expanded in an infinite series in terms of these derivatives (i.e., a Taylor's series). The function can also be expressed in the infinite serieswhere z0 is a point in the domain. Also of interest in complex variable analysis are the points in a domain, called singular points, where a function fails to have a derivative. The theory of functions of a complex variable was developed during the 19th cent. by A. L. Cauchy, C. F. Gauss, B. Riemann, K. T. Weierstrass, and others.
In computer science, live variable analysis (or simply liveness analysis) is a classic data flow analysis performed by compilers 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.

L1: b := 3;
L2: c := 5;
L3: a := b + c;
goto L1;
The set of live variables at line L2 is {b, c}, but the set of live variables at line L1 is only {b} since variable c is updated in line 2. The value of variable a is never used, so the variable is never live.

The dataflow equations used for a given basic block s and exiting block f in live variable analysis are:

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

{rm LIVE}_{rm out}[final] = {emptyset}
{rm LIVE}_{rm out}[s] = bigcup_{p in succ[S]} {rm LIVE}_{rm in}[p]

{rm GEN}[d : y leftarrow f(x_1,cdots,x_n)] = {x_1,...,x_n}
{rm KILL}[d : y leftarrow f(x_1,cdots,x_n)] = {y}

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 in-state is the union of the out-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.

// out: {}
b1: a = 3;
    b = 5;
    d = 4;
    if a > b then
// in: {a,b,d} // out: {a,b}
b2:    c = a + b;
       d = 2;
// in: {b,d} // out: {b,d}
b3: endif
    c = 4;
    return b * d + c;
// in:{}

The out-state of b3 only contains b and d, since c has been written. The in-state of b1 is the union of the out-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 {b,d} {} {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 that 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 as of 2006, 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.


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