Strategies for Combining Decision Procedures

Implementing efficient algorithms for combining decision procedures has been a challenge and their correctness precarious. In this paper we describe an inference system that has the classical Nelson-Oppen procedure at its core and includes several optimizations: variable abstraction with sharing, canonization of terms at the theory level, and Shostak’s streamlined generation of new equalities for theories with solvers. The transitions of our system are fine-grained enough to model most of the mechanisms currently used in designing combination procedures. In particular, with a simple language of regular expressions we are able to describe several combination algorithms as strategies for our inference system, from the basic Nelson-Oppen to the very highly optimized one recently given by Shankar and Rueß. Presenting the basic system at a high level of generality and nondeterminism allows transparent correctness proofs that can be extended in a modular fashion whenever a new feature is introduced in the system. Similarly, the correctness proof of any new strategy requires only minimal additional proof effort.