The Duality in Computing SSA Programs and Control Dependency

Abstract

Control dependency (CD) and Static Single Assignment (SSA) form are the basis of many program analyses, transformation, and optimization techniques, and these are implemented and used by modern compilers such as GCC and LLVM. Most state-of-the-art algorithms approximate these computations by using postdominator relations and dominance frontiers (DF) respectively for efficiency reasons which have been used for over three decades. Dominator-based SSA transformation and control dependencies exhibit a non-dual relationship. Recently, it has been shown that DF-based SSA computation is grossly imprecise, and Weak and Strong Control Closure (WCC and SCC) have wider applicability in capturing control dependencies than postdominator-based CD computation. Our main contribution in this article is the proof of duality between the generation of <inline-formula><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> functions and the computation of weakly deciding (WD) vertices which are the most computationally expensive part of SSA program construction and WCC/SCC computation respectively. We have provided a duality theorem and its constructive proof by means of an algorithm that can compute both the <inline-formula><tex-math notation="LaTeX">$\phi$</tex-math></inline-formula> functions and the WD vertices seamlessly. We have used this algorithm to compute SSA programs and WCC, and performed experiments on real-world industrial benchmarks. The practical efficiency of our algorithm is (i) almost equal to the best state-of-the-art algorithm in computing WCC, and (ii) closer to (but not as efficient as) the DF-based algorithms in computing SSA programs. Moreover, our algorithm achieves the ultimate precision in computing WCC and SSA programs with respect to the inputs of these algorithms and obtains wider applicability in the WCC computation (handling nonterminating programs)