Lemmas on Demand for the Extensional Theory of Arrays

The quantifier-free extensional theory of arrays TA plays an important role in hardware and software verification. In this article we present a novel decision procedure that refines formula abstractions with lemmas on demand. We consider the case where TA is combined with a decidable quantifier-free first-order theory TB. Unlike traditional lazy SMT approaches, where lemmas are added on the boolean abstraction layer, our decision procedure adds lemmas in TB. We discuss our decision procedure in detail. In particular, we prove soundness and completeness, and discuss complexity. We present our decision procedure in a generic context and provide implementation details and optimizations, in particular for bit-vectors. Finally, we report on experiments and discuss related work.