Injecting Abstract Interpretations into Linear Cost Models

interpretation often considers Galois connections B −−−→ ←−−− α γ A where B is a powerset5 representing the concrete semantic domain, and A is a complete lattice representing the abstract Powersets are naturally equipped with a particular structure of complete lattice called boolean lattice [9]. 74 Injecting Abstract Interpretations into Linear Cost Models domain. In order to lift α into a linear mapping, we will focus on how to lift-order these particular structures. The easy case naturally is the one of boolean lattices. Lift-ordering boolean lattices. A boolean lattice B is generated by its set of atoms A (B), corresponding to the singletons in the case of a powerset. Indeed, for each b ∈ B, b = ∨{a ∈ A (B) | a ≤ b} [9]. Let us code atoms a as vectors a↑ in {⊥,e}|A (B)| as previously (we note a = a↑). Then, coding the other elements will follow from the use of ⊕. b =⊕{a | a ≤ b} We denote by B the complete moduloid constructed this way from B, where the ⊕ operator of B matches the ∪ operator of B by construction. Now that we have expressed boolean lattices as moduloids, we are able to easily lift-order the abstraction function of a Galois connection B1 −−−→ ←−−− α γ B2, where B1 and B2 are boolean lattices. By lift-ordering these lattices, we obtain two moduloids (B1,⊕1,⊗1) and (B2,⊕2,⊗2). Since ∪i and ⊕i coincide, and as α is a union morphism, its linear translation α is defined by its values on the basis vectors of B1, i.e. the vectors coding atoms of B1. α({b1} ∪1 {b2}) = α({b1}) ∪2 α({b2}) l l α(b1 ⊕1 b2) = α(b1) ⊕2 α(b2) Lift-ordering complete lattices. In most of the cases, A is not a powerset but a more general complete lattice for which the vectorial translation is not so straightforward. The representation theorem of finite distributive lattices [9] asserts that any such lattice A is isomorphic to a lattice of sets. Thus, A can be seen as a sublattice of a given powerset, which we will denote by B(A). The previous coding applies to B(A) and a fortiori to A. However, the set of vectors A constructed this way no more has a structure of complete moduloid, unlike B(A). This method provides a solution to the “state explosion” problem presented in Section 5. Nevertheless, our second problem remains unsolved. Indeed, there is still no match between the ⊕ operator and ∪, the join operator of the lattice. For instance, [−2]∪ [2] = [−2,2] and [−2]⊕ [2] = (e,⊥,e)T and [−2,2] = (e,e,e)T . This makes it impossible to express α as a linear mapping, since for instance α({−2}⊕{2}) = (e,e,e)T 6= α({−2})⊕α({2}) = (e,⊥,e)T . We thus have to weaken our requirement: in the following, we choose to lift-order Galois connections into non linear, but still residuable, mappings. 6.1.2 Lifting a Galois connection into a residuable mapping Since B(A) is a complete boolean lattice, we will decompose α into a linear part from B to B(A), and a projection from B(A) into its sublattice A we are interested in, representing the vector encodings of elements of A. Figure 1(b) illustrates this decomposition. The linear part of α , denoted by α1 is defined as in the case of a connection between two boolean lattices: α1 is defined on the set of atoms of B by α1(b) = α(b) where b is an atom of B, and then extended to B by linearity. As an example, Figure 1(c) shows the abstraction matrix for the abstraction by even intervals, for n = 2. Element {−1} of the concrete domain is mapped to interval [−2,0] of the abstract domain. Thus, α1 maps atom {−1} to [−2,0] which is the sum of atoms [−2] and [0]. D. Cachera & A. Jobin 75 / 0 [−2] [0] [2] [−2,0] [0,2] [−2,2]