Complementation in abstract interpretation

Abstract

Abstract. The reduced product of abstract domains is a rather well known operation in abstract interpretation. In this paper we study the inverse operation, which we call complementation. Such an operation allows to systematically decompose domains; it provides a systematic way to design new abstract domains; it allows to simplify domain verification problems, like correctness proofs; and it yields space saving representations for domains. We show that the complement exists in most cases, and we apply complementation to two well known abstract domains, notably to the Cousot and Cousot's comportment domain for analysis of functional languages and to the complex domain Sharing for aliasing analysis of logic languages.

Making abstract interpretations complete

“…Let {A i } iʦI ʕ uco(C): (i) iʦI A i is the most concrete among the domains in ᑦ C which are abstractions of all the A i 's, that is, iʦI A i is the least (with respect to ) common abstraction of all the A i 's; (ii) iʦI A i is (isomorphic to) the well-known reduced product (basically cartesian product plus reduction) of all the A i 's, or, equivalently, it is the most abstract among the domains in ᑦ C which are more concrete than every A i . Let us remark that the reduced product can be also characterized as Moore-closure of set-union, that is, Cortesi et al [1997] present systematic methodologies for decomposing abstract domains.…”

Static analysis techniques for robotics software verification

“…Abstract interpretation can be seen as the most general setting to express and compute static analysis [4,9,10,11,12,19,25].…”

Decomposing bytecode verification by abstract interpretation

“…The lemma at the end of this section will describe such conditions. We first define the term decomposition [Cortesi et al 1997]. …”

“…An attractive goal of abstract interpretation theory is the systematic design of abstract domains [Cortesi et al 1997;Cousot and Cousot 1979]. The idea is to define operators that combine or refine abstract domains, in order to obtain new domains with a desired property, such as greater precision or expressiveness.…”

“…In this paper, simplification was motivated by the need to perform the analysis on a memory constrained device, but there are clearly other scenarios where a very complex analysis could benefit from being decomposed into a set of simpler analyses. Examples of domain operations that decompose or simplify abstract domains are complementation [Cortesi et al 1997] and least disjunctive basis [Giacobazzi and Ranzato 1998]. Parallel decomposition is not a new domain operator, but it may be regarded as a guideline on how to combine the use of relative and absolute complete shells [Bernardeschi et al 2006a] with reduced product [Cousot and Cousot 1979].…”