Abstract Interpretation as Automated Deduction

Abstract

Automata theory, algorithmic deduction and abstract interpretation provide the foundation behind three approaches to implementing program verifiers. This article is a first step towards a mathematical translation between these approaches. By extending Buchi's theorem, we show that reachability in a control flow graph can be encoded as satisfiability in an extension of the weak, monadic, second-order logic of one successor. Abstract interpreters are, in a precise sense, sound but incomplete solvers for such formulae. The three components of an abstract interpreter: the lattice, transformers and iteration algorithm, respectively represent a fragment of a first-order theory, deduction in that theory, and second-order constraint propagation. By inverting the Lindenbaum---Tarski construction, we show that lattices used in practice are subclassical first-order theories.