Bilateral Algorithms for Symbolic Abstraction

Aditya V. Thakur, Matt Elder, Thomas W. Reps

Abstract: Given a concrete domain $\mathcal{C}$ , a concrete operation $\tau: \mathcal{C} \to \mathcal{C}$ , and an abstract domain $\mathcal{A}$ , a fundamental problem in abstract interpretation is to find the best abstract transformer $\tau^{\#}: \mathcal{A} \to \mathcal{A}$ that over-approximates τ. This problem, as well as several other operations needed by an abstract interpreter, can be reduced to the problem of symbolic abstraction: the symbolic abstraction of a formula ϕ in logic $\mathcal{L}$ , denoted by $\widehat{\alpha}(\varphi)$ , is the best value in $\mathcal{A}$ that over-approximates the meaning of ϕ. When the concrete semantics of τ is defined in $\mathcal{L}$ using a formula ϕ τ that specifies the relation between input and output states, the best abstract transformer τ # can be computed as $\widehat{\alpha}(\varphi_\tau)$ . In this paper, we present a new framework for performing symbolic abstraction, discuss its properties, and present several instantiations for various logics and abstract domains. The key innovation is to use a bilateral successive-approximation algorithm, which maintains both an over-approximation and an under-approximation of the desired answer.