Bilateral Algorithms for Symbolic Abstraction

Given a concrete domain C, a concrete operation tau: C -> C, and an abstract domain A, a fundamental problem in abstract interpretation is to find the best abstract transformer tau#: A -> A that over-approximates tau. This problem, as well as several other operations needed by an abstract interpreter, can be reduced to the problem of symbolic bstraction: the symbolic abstraction of a formula phi in logic L, denoted by alphaHat(phi), is the best value in A that over-approximates the meaning of phi. When the concrete semantics of tau is defined in L using a formula psi that specifies the relation between input and output states, the best abstract transformer tau# can be computed as alphaHat(psi). 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. The advantage of having a non-trivial over-approximation is that it makes the technique resilient to timeouts