Making abstract interpretations complete

Abstract

Completeness is an ideal, although uncommon, feature of abstract interpretations, formalizing the intuition that, relatively to the properties encoded by the underlying abstract domains, there is no loss of information accumulated in abstract computations. Thus, complete abstract interpretations can be rightly understood as optimal. We deal with both pointwise completeness, involving generic semantic operations, and (least) fixpoint completeness. Completeness and fixpoint completeness are shown to be properties that depend on the underlying abstract domains only. Our primary goal is then to solve the problem of making abstract interpretations complete by minimally extending or restricting the underlying abstract domains. Under the weak and reasonable hypothesis of dealing with continuous semantic operations, we provide constructive characterizations; for the least complete extensions and the greatest complete restrictions of abstract domains. As far as fixpoint completeness is concerned, for merely monotone semantic operators, the greatest restrictions of abstract domains are constructively characterized, while it is shown that the existence of least extensions of abstract domains cannot be, in general, guaranteed, even under strong hypotheses. These methodologies, which in finite settings give rise to effective algorithms, provide advanced formal tools for manipulating and comparing abstract interpretations, useful both in static program analysis and in semantics design. A number of examples illustrating these techniques are given

Typestate Verification: Abstraction Techniques and Complexity Results

“…More importantly, our verification algorithms utilize the function-space lattice Q → {f alse, maybe} for abstraction, with a worst-case complexity that is linear in the number of predicates. Our selection of predicates ensures that there is no loss of precision in using the smaller function-space lattice, i.e., we ensure that our abstraction is complete (e.g., see [13]). Our approach yields polynomial time verification algorithms for various problems, while none of the predicate abstraction based techniques mentioned above can guarantee a polynomial time worst-case complexity for the same problems (even though they may perform verification with a smaller number of predicates than our algorithm).…”

An Abstract Interpretation Perspective on Linear vs. Branching Time

“…Thus, the comparison between ∀LTL and ACTL boils down to the comparison between ∀LTL and LTL ∀ . As a consequence of the incomparability of ∀LTL and LTL ∀ we obtain that the abstraction map ∀ is incomplete in the abstract interpretation sense [3,9]. In fact, if ∀ would be complete for the operators of LTL then we would also have that ∀LTL = LTL ∀ whereas this is not the case.…”

“…Completeness in abstract interpretation [3,9] corresponds to require the following strengthening of soundness: α • f = f • α. Hence, completeness corresponds to require that, in addition to soundness, no loss of precision is introduced by the abstract function f on the approximation α(c) of a concrete object c ∈ C with respect to approximating by α the concrete computation f (c).…”

Abstract Interpretation of Combinational Asynchronous Circuits

“…The hierarchy of domains that results is shown in Fig. 2 -the relationship to 2-value Boolean logic B and 3-value ternary logic B 3 is shown 8 . Note that since B lacks an upper bound that corresponds with , it is not possible to define α : B 3 → B (though γ : B → B 3 can be trivially defined), so a Galois connection does not exist in that particular case.…”