Bilateral algorithms for symbolic abstraction

  • Authors:

  • Aditya Thakur;Matt Elder;Thomas Reps

  • Affiliations:

  • University of Wisconsin, Madison, WI;University of Wisconsin, Madison, WI;University of Wisconsin, Madison, WI, USA,GrammaTech, Inc., Ithaca, NY

  • Venue:
  • SAS'12 Proceedings of the 19th international conference on Static Analysis
  • Year:
  • 2012

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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 ϕτ 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.