Compositionality in SLD-derivations and their Abstractions

Interpretation is a theory developed to reason about the abstraction relation between two different semantics. The theory requires the two semantics to be defined on domains which are complete lattices. (C, ) (concrete domain) is the domain of the concrete semantics, while (A,≤) (abstract domain) is the domain of the abstract semantics. The partial order relations reflect an approximation relation. The two domains are related by a pair of functions α (abstraction) and γ (concretization), which form a Galois Insertion. Definition 4.1 (Galois Insertion) Let (C, ) be the concrete domain and (A,≤) be the abstract domain. A Galois insertion 〈α, γ〉 : C A is a pair of maps α : C → A and γ : A→ C such that α and γ are monotonic and ∀x ∈ C, x (γ ◦ α)(x) and ∀y ∈ A, (α ◦ γ)(y) = y. Given a Galois Insertion between the concrete and the abstract domain and a concrete semantics, we want to define an abstract semantics. The concrete semantics is the least fixpoint of a semantic function F : C → C. The abstract semantic function F̃ : A→ A is correct if ∀x ∈ C, F (x) ≤ γ(F̃ (α(x))). The same property holds for the least fixpoints of F and F̃ . F is in turn defined as composition of “primitive” operators. Let f : Cn → C be one such an operator and assume that f̃ is its abstract counterpart. Then f̃ is correct w.r.t. f if ∀x1, . . . , xn ∈ C, f(x1, . . . , xn) γ(f̃(α(x1), . . . , α(xn))). The correctness of all the primitive operators implies the correctness of F̃ . Hence, we can define an abstract semantics by defining correct abstract primitive semantic functions. An abstract computation is then related to the concrete computation, simply by replacing the concrete operators by the corresponding abstract operators. According to the theory, for each operator f , there exists an optimal (most precise) correct abstract operator f̃ defined as f̃(y1, . . . , yn) = α(f(γ(y1), . . . , γ(yn))). However the composition of optimal operators is not necessarily optimal [3]. The abstract interpretation theory allows us to model the approximation process which is typically involved in program analysis. In this paper we are only concerned with different semantics and therefore we look for precise abstractions. The abstract operator f̃ is precise if ∀x1, . . . , xn ∈ C,α(f(x1, . . . , xn)) = f̃(α(x1), . . . , α(xn)). Hence the optimal abstract operator f̃ is precise if and only if α(f(x1, . . . , xn)) = α(f((γ ◦ α)x1, . . . , (γ ◦ α)xn)). Hence the precision of the optimal abstract operators can be reformulated in terms of properties of α, γ and the corresponding concrete operators. The above definitions are naturally extended to “primitive” semantic operators from P(C) to C.