Algorithms for computing the single static assignment form

The Static Single Assignment (SSA) form is a program representation used in many optimizing compilers. The key step in converting a program to SSA form is called φ-placement. Many algorithms for φ-placement have been proposed in the literature, but the relationships between these algorithms are not well understood.In this article, we propose a framework within which we systematically derive (i) properties of the SSA form and (ii) φ-placement algorithms. This framework is based on a new relation called merge which captures succinctly the structure of a program's control flow graph that is relevant to its SSA form. The φ-placement algorithms we derive include most of the ones described in the literature, as well as several new ones. We also evaluate experimentally the performance of some of these algorithms on the SPEC92 benchmarks.Some of the algorithms described here are optimal for a single variable. However, their repeated application is not necessarily optimal for multiple variables. We conclude the article by describing such an optimal algorithm, based on the transitive reduction of the merge relation, for multi-variable φ-placement in structured programs. The problem for general programs remains open.