Transfer Functions.

The side-effect transfer function of a region R on a particular array is represented as the 4-tuple

where the elements are the sets informally defined as described above. The following operations are defined on tuples:

• : result of composing read and write accesses in block b.
  Read access: . Write access: .
  where a is a system describing the access indices, rewritten in loop-relative terms (from symbolic analysis), with relevant inequality constraints added.

• Composition: .

• Meet: .
• Identity element: .

• Iteration: . The given loop index variable i is used to perform dependence and privatization tests, as in the following section.

• Closure: , where contains the loop index i and other loop-modified variables.

• , as discussed in Section 5.2.