Introduction

Our starting point is research and experiences in the context of the state-of-the art SMT solver Z3 [13], developed by the authors at Microsoft Research. We first cover a selection of the main challenges and techniques for making SMT solving practical, integrating algorithms for tractable subproblems, and pragmatics and heuristics used in practice. We then take a proof-theoretical perspective on the power and scope of the engines used by SMT solvers. Most modern SMT solvers are built around a tight integration with efficient SAT solving. The framework is commonly referred to as DPLL(T), where T refers to a theory or a combination of theories. The theoretical result we present compares DPLL(T) with unrestricted resolution. A straightforward adaption of DPLL(T) provides a weaker proof system than unrestricted resolution, and we investigate an extension we call Conflict Directed Theory Resolution as a candidate method for bridging this gap. Our results apply to the case where T is the theory of equality.