Static Analysis, Abstract Interpretation and Verification in (Constraint Logic) Programming

values in A. This justifies the choice of the backward and forward terminology above. We denote by, respectively, F(C, f) and B(C, f) the set of Fand Bcomplete abstractions of C for f . It is worth noting that in general F(C, f) 6⊆ B(C, f) and F(C, f) 6⊆ B(C, f), namely Band F-completeness are incomparable notions. Example 1. Let Sign be the simple abstraction of 〈℘(Z),⊆〉 for analysing integer variables depicted in Fig. 1. Consider the pointwise square operation sq : ℘(Z)→ ℘(Z) defined as follows: sq(X) , {x | x ∈ X }. Let ρ ∈ uco(℘(Z)) be the closure operator associated with Sign, i.e. ρ = γSign+ ◦ αSign+ , where the abstraction and concretization maps are the obvious ones. The best correct approximation of sq in Sign is sq : Sign → Sign defined as sq(X) , ρ(sq(X)), with X ∈ Sign. It is easy to note that the closure operators ρa , {Z, [0,+∞], [0, 10]} and ρb , {Z, [0, 2], [0]}, defined by their images — the images of ρa and ρb are depicted as bullets in Fig. 1 — are such that: – ρa ∈ F(Sign, sq) but ρa 6∈ B(Sign, sq): for example, ρa(sq(ρa([0]))) = [0,+∞] while ρa(sq([0])) = [0, 10]; – ρb ∈ B(Sign, sq) but ρb 6∈ F(Sign, sq): for example, ρb(sq(ρb([0, 2]))) = Z while sq(ρb([0, 2])) = [0, 10]. u t One key result in [44] provides a constructive characterization of the structure of abstract domains that are B-complete for continuous functions. Given a function f : C → C and S ⊆ C, f−1(S) denotes the inverse image of f in S, i.e., {x ∈ C | f(x) ∈ S }. Then, [44] shows that ρ ∈ uco(C) is B-complete for f ⇔ ⋃ y∈ρ(C) max(f−1(↓y)) ⊆ ρ(C) (∗) Let us consider Example 1. It is easy to see that ρa is not B-complete because ρa does not include the maximal inverse image of sq of the subset ↓ [0, 10], namely max(sq](↓ [0, 10])) = {[0, 2]}. An analogous (and trivial to prove) result can be stated for F-completeness. In this case, F-complete domains can be characterized for merely monotone operations as