For each region **R** from innermost loop to outermost loop, and
from bottom to top in the call graph, we compute its transfer function
. A basic block's transfer function is derived directly
(using the function).
The transfer function of a procedure call takes the procedure body's
transfer function and maps it to the caller space, renaming variables
in its representation (using the operation).

The transfer function for a loop applies the Iteration operation to the
transfer function of its loop body to
obtain a transfer function
representing the effect of **i** iterations of the body, where
**i** is the loop's normalized index variable.
The final transfer function
showing the total effect of the loop is obtained by using the Closure
operation to eliminate the iteration counter **i**.

For loop bodies and procedure bodies, deriving
the transfer function involves the transfer functions of its
immediate subregions.
In a forward data-flow problem,
for each subregion , we compute
, the transfer function from
the entry of **R** to the entry of .
This calculation results from a meet over the predecessors
of .
If the immediate subregions graph
is cyclic, then an iterative solution may be required to
find the transfer function. Otherwise,
the subregions are simply visited in the appropriate
(reverse postorder) order within the region.
The final transfer function for the loop body or procedure body
is derived by composing the transfer function
for the subregion that represents the exit from region **R**,
with the transfer function of that region.

Data-flow problems that require a backward
propagation within the intervals are analogous.
For an acyclic subregion graph,
a postorder traversal over the subregions derives transfer functions
to describe
the effects from the exit of **R** up to the exit of .

Mon Oct 2 11:00:22 PDT 1995