The SSA Representation Framework: Semantics, Analyses and GCC Implementation

Interpretation. The semantics of a programming language can be more or less precise, depending on the degree of detail captured by the description of program execution. The framework of abstract interpretation [CC77, CC79] can be used to automatically define several semantics layers, corresponding to different levels of precision. Each semantics layer is represented by a partially ordered set, or domain. The relation between two semantics layers is defined using an approximation translation, which maps elements of the concrete domain to elements of the abstract domain. In order to guarantee the conservation of concrete semantics properties, the approximation and concretization functions have to satisfy the following properties, for some given elements from the concrete domain c1, c2 ∈ Σ1, and for some elements of the abstract domain a1, a2 ∈ Σ2: • α monotone: c1 ⊆ c2 ⇒ (α(c1) ⊆ α(c2)), • γ monotone: a1 ⊆ a2 ⇒ (γ(a1) ⊆ γ(a2)), • γ ◦ α extensive: c1 ⊆ γ(α(c1)), • α ◦ γ reductive: α(γ(a1)) ⊆ a1. A coarse approximation of a concrete domain leads to a less precise abstract domain on which the semantics can be practically computable. It is sometimes the case that the operations on the abstract domain are exact, i.e. computing on the abstract set produces exactly the same result, without loss of precision, as in the concrete set. In this case the order is trivially preserved, as the abstraction and concretization functions are bijections on some subsets of the abstract and concrete sets. 52 CHAPTER 3. FROM SSA TO ASSA IN PROLOG Theory of Abstract Domains. As we have seen, approximation functions map elements from a concrete semantics domain to an abstract domain. Once an abstract domain is formalized, it is possible to use the abstract interpretation framework for automatically defining the static analyzer corresponding to the translation of the concrete semantics domain to the abstract domain. The importance of abstract domains in the framework of abstract interpretation is illustrated by the number of abstract domains that have been formalized: